Finite Group Databasebeta

← #32015006

Subgroups — #32015006

Order 18176.

924 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986055C211yesmaximal, characteristic
2C2×#17986054C211yesmaximal, characteristic
3C4×C4⋊C4×D71C211yesmaximal, characteristic
4#22589057C211yesmaximal, characteristic, complemented
5#22589055C211yesmaximal, characteristic, complemented
6#22588906C211yesmaximal, characteristic, complemented
7#22588905C211yesmaximal, characteristic, complemented
8#22588903C211yesmaximal, characteristic, complemented
9#22588407C211yesmaximal, characteristic
10#22588407C211yesmaximal, characteristic
11#22588319C211yesmaximal, characteristic
12#22588208C211yesmaximal, characteristic, complemented
13#22588206C211yesmaximal, characteristic, complemented
14#22588097C211yesmaximal, characteristic
15C71×#1917C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D71C411yescharacteristic
17C2×C4⋊C4×D71C411yescharacteristic
18C2×C42×D71C2211yescharacteristic
19C2×C42×D71C2211yescharacteristic
20C2×C42×D71C2211yescharacteristic
21#17986055—24no—
22#17986054—24no—
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×Dic71⋊C8C2211yescharacteristic
25C2×Dic71⋊C8C2211yescharacteristic
26C2×Dic71⋊C8C2211yescharacteristic
27C2×Dic71⋊C8C2211yescharacteristic
28C2×(C2×D71)⋊C8C2211yescharacteristic
29C2×(C2×D71)⋊C8C2211yescharacteristic
30C2×(C2×D71)⋊C8C2211yescharacteristic
31C2×(C2×D71)⋊C8C2211yescharacteristic
32C2×#14556039C2211yescharacteristic
33C2×#14556038C2211yescharacteristic
34C2×#14556038C2211yescharacteristic
35C2×C8×Dic71C2211yescharacteristic
36C2×C8×Dic71C2211yescharacteristic
37C4⋊C4×Dic71C411yescharacteristic
38C4⋊C4×Dic71C411yescharacteristic
39C4×Dic71⋊C4C2211yescharacteristic
40C4×Dic71⋊C4C2211yescharacteristic
41C4×C4⋊Dic71C2211yescharacteristic
42#17985850C411yescharacteristic
43#17985850C411yescharacteristic
44#95×D71C411yescharacteristic
45#95×D71C411yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49#17985754C2211yescharacteristic
50#17985753C2211yescharacteristic
51#17985753C2211yescharacteristic
52#17985753C2211yescharacteristic
53#17985753C2211yescharacteristic
54C2×C4⋊C8×C71C2211yescharacteristic
55C2×#97×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57#336×C71C2211yescharacteristic
58#336×C71C2211yescharacteristic
59#336×C71C2211yescharacteristic
60#336×C71C2211yescharacteristic
61C22×C4×D71C2×C411yescharacteristic
62C22×C4×D71C2×C411yescharacteristic
63C22×C4×D71C2×C411yescharacteristic
64C22×C4×D71C2×C411yescharacteristic
65C22×C4×D71C2×C411yescharacteristic
66C22×C4×D71C2×C411yescharacteristic
67C22×C4×D71C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C22×C71⋊C8C2311yescharacteristic
71C22×C71⋊C8C2311yescharacteristic
72C2×C4⋊Dic71C2×C411yescharacteristic
73C2×C4⋊Dic71C2×C411yescharacteristic
74C2×Dic71⋊C4C2×C411yescharacteristic
75C2×Dic71⋊C4C2×C411yescharacteristic
76C2×Dic71⋊C4C2×C411yescharacteristic
77C2×Dic71⋊C4C2×C411yescharacteristic
78C4⋊C4×D71—24no—
79C4⋊C4×D71—24no—
80C2×C4×Dic71C2×C411yescharacteristic
81C2×C4×Dic71C2×C411yescharacteristic
82C2×C4×Dic71C2×C411yescharacteristic
83C2×C4×Dic71C2×C411yescharacteristic
84C2×C4×Dic71C2×C411yescharacteristic
85C2×C4×Dic71C2×C411yescharacteristic
86C2×C4×Dic71C2311yescharacteristic
87C2×C4×Dic71C2311yescharacteristic
88C2×C4×Dic71C2311yescharacteristic
89C42×D71—22no—
90C42×D71—22no—
91C42×D71—22no—
92C42×D71—22no—
93C42×D71—22no—
94C42×D71—22no—
95C4⋊(C71⋊C8)—22no—
96Dic71⋊C8—22no—
97Dic71⋊C8—22no—
98Dic71⋊C8—22no—
99Dic71⋊C8—22no—
100(C2×D71)⋊C8—22no—
101(C2×D71)⋊C8—22no—
102(C2×D71)⋊C8—22no—
103(C2×D71)⋊C8D414yes—
104#14556039—22no—
105#14556038—22no—
106#14556038—22no—
107C8×Dic71—22no—
108C8×Dic71—22no—
109C71⋊#95C2×C411yescharacteristic
110C71⋊#95C2×C411yescharacteristic
111C71⋊#95C2×C411yescharacteristic
112C71⋊#95C2×C411yescharacteristic
113C71⋊#95C2×C411yescharacteristic
114C71⋊#95C2×C411yescharacteristic
115C22×C8×C71C2311yescharacteristic
116C22×C8×C71C2311yescharacteristic
117C22×C8×C71C2311yescharacteristic
118C22×C8×C71C2311yescharacteristic
119C2×C4⋊C4×C71C2×C411yescharacteristic
120C2×C4⋊C4×C71C2×C411yescharacteristic
121C2×C42×C71C2311yescharacteristic
122C2×C42×C71C2311yescharacteristic
123C2×C42×C71C2311yescharacteristic
124C4⋊C8×C71—22no—
125#97×C71—22no—
126#95×C71C2×C411yescharacteristic
127#95×C71C2×C411yescharacteristic
128C23×D71C22×C411yescharacteristic
129C22×Dic71C22×C411yescharacteristic
130C22×Dic71C22×C411yescharacteristic
131C22×Dic71C22×C411yescharacteristic
132C22×Dic71C22×C411yescharacteristic
133C22×Dic71C22×C411yescharacteristic
134C22×Dic71C22×C411yescharacteristic
135C22×Dic71C22×C411yescharacteristic
136C2×C4×D71—21no—
137C2×C4×D71—21no—
138C2×C4×D71—21no—
139C2×C4×D71—21no—
140C2×C4×D71—22no—
141C2×C4×D71—22no—
142C2×C4×D71—22no—
143C2×C4×D71—22no—
144C2×C4×D71—22no—
145C2×C4×D71—22no—
146C2×C4×D71—22no—
147C2×C4×D71—22no—
148C2×C4×D71—22no—
149C2×C4×D71—22no—
150C2×C4×D71—22no—
151C2×C4×D71—22no—
152C2×C4×D71—22no—
153C2×C4×D71—22no—
154C2×C4×D71—41no—
155C2×C4×D71—41no—
156C2×C4×D71—41no—
157C2×C4×D71M1614yes—
158C2×C4×D71C2×D412yes—
159C2×C4×D71#4112yes—
160C2×C71⋊C8—21no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8—21no—
165C2×C71⋊C8—21no—
166C2×C71⋊C8—21no—
167C2×C71⋊C8—21no—
168C2×C71⋊C8—21no—
169C2×C71⋊C8—21no—
170C2×C71⋊C8—21no—
171C2×C71⋊C8C2×D412yes—
172C4⋊Dic71—22no—
173C4⋊Dic71—22no—
174Dic71⋊C4—22no—
175Dic71⋊C4—22no—
176Dic71⋊C4—22no—
177Dic71⋊C4M1614yes—
178C4×Dic71—21no—
179C4×Dic71—21no—
180C4×Dic71—21no—
181C4×Dic71—21no—
182C4×Dic71—22no—
183C4×Dic71—22no—
184C4×Dic71—22no—
185C4×Dic71—22no—
186C4×Dic71—22no—
187C4×Dic71—22no—
188C4×Dic71#4112yes—
189C4×Dic71#4112yes—
190C22×C4×C71C22×C411yescharacteristic
191C22×C4×C71C22×C411yescharacteristic
192C22×C4×C71C22×C411yescharacteristic
193C22×C4×C71C22×C411yescharacteristic
194C22×C4×C71C22×C411yescharacteristic
195C22×C4×C71C22×C411yescharacteristic
196C22×C4×C71C2411yescharacteristic
197C2×C8×C71—21no—
198C2×C8×C71—21no—
199C2×C8×C71—21no—
200C2×C8×C71—21no—
201C2×C8×C71—21no—
202C2×C8×C71—21no—
203C2×C8×C71—21no—
204C2×C8×C71—21no—
205C2×C8×C71—21no—
206C2×C8×C71—21no—
207C2×C8×C71—21no—
208C2×C8×C71C2×D412yes—
209C4⋊C4×C71—22no—
210C4⋊C4×C71—22no—
211C42×C71—21no—
212C42×C71—21no—
213C42×C71—21no—
214C42×C71—21no—
215C42×C71—21no—
216C42×C71—21no—
217C23×C71C23×C411yescharacteristic
218C22×D71—21no—
219C22×D71—21no—
220C22×D71—21no—
221C22×D71—21no—
222C22×D71C4×D412yes—
223C22×D71C2×M1612yes—
224C22×D71#13112yes—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—21no—
241C2×C4×C71C2×M1612yes—
242C2×C4×C71C2×M1612yes—
243C2×C4×C71C22×D411yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×C4×C712−1+411yescharacteristic
248C2×C4×C712−1+411yescharacteristic
249C2×Dic71—21no—
250C2×Dic71—21no—
251C2×Dic71—21no—
252C2×Dic71—21no—
253C2×Dic71—21no—
254C2×Dic71—21no—
255C2×Dic71—21no—
256C2×Dic71—21no—
257C2×Dic71—21no—
258C2×Dic71—21no—
259C2×Dic71—21no—
260C2×Dic71—21no—
261C2×Dic71—21no—
262C2×Dic71—21no—
263C2×Dic71—21no—
264C2×Dic71#11712yes—
265C2×Dic71#11712yes—
266C2×Dic71C4×D412yes—
267C2×Dic71C2×M1612yes—
268C2×Dic71C2×M1612yes—
269C2×Dic71C2×M1612yes—
270C4×D71—22no—
271C4×D71—22no—
272C4×D71—22no—
273C4×D71—22no—
274C4×D71—28no—
275C4×D71—44no—
276C4×D71—44no—
277C4×D71—44no—
278C4×D71—44no—
279C4×D71—44no—
280C8×C71—22no—
281C8×C71—41no—
282C8×C71—41no—
283C8×C71—41no—
284C71⋊C8—22no—
285C71⋊C8—41no—
286C71⋊C8—41no—
287C71⋊C8—41no—
288C22×C71C2×#11711yescharacteristic
289C22×C71C2×C4×D411yescharacteristic
290C22×C71#51911yescharacteristic
291C22×C71C22×M1611yescharacteristic
292C22×C71C22×M1611yescharacteristic
293C22×C71C2×#13111yescharacteristic
294C22×C71#56811yescharacteristic
295C2×D71—22no—
296C2×D71—22no—
297C2×D71—22no—
298C2×D71—22no—
299C2×D71—22no—
300C2×D71—22no—
301C2×D71#43614yes—
302C4×C71—22no—
303C4×C71—22no—
304C4×C71—41no—
305C4×C71—41no—
306C4×C71—41no—
307C4×C71—41no—
308C4×C71#54711yescharacteristic
309C4×C71#54711yescharacteristic
310C4×C71#55211yescharacteristic
311C4×C71#55211yescharacteristic
312Dic71—22no—
313Dic71—22no—
314Dic71—22no—
315Dic71—41no—
316Dic71—41no—
317Dic71—41no—
318Dic71#43214yescomplemented
319#22603—711nomaximal, Sylow
320C2×C71#234411yescharacteristic
321C2×C71C2×#43211yescharacteristic
322C2×C71#292411yescharacteristic
323C2×C71C2×#43611yescharacteristic
324C2×C71D4×M1611yescharacteristic
325C2×C71#298311yescharacteristic
326C2×C71#298411yescharacteristic
327D71—24no—
328C2×#432—711no—
329C2×#432—711no—
330C2×C4×C4⋊C4—711no—
331#1939—711no—
332#1939—711no—
333#1917—711no—
334#1917—711no—
335#1917—711no—
336#1917D7111yescharacteristic, complemented
337#1817—711no—
338#1817—711no—
339#1797—711no—
340#1770—711no—
341#1770—711no—
342C4×C22⋊C8—711no—
343C71#2260311yescharacteristic, complemented, Sylow
344C22×C4⋊C4—711no—
345C22×C4⋊C4—711no—
346C22×C42—711no—
347C22×C42—711no—
348C22×C42—711no—
349#432—1424no—
350#432—1424no—
351C2×C4⋊C8—711no—
352C2×C4⋊C8—711no—
353C2×C4⋊C8—711no—
354C2×C4⋊C8—711no—
355C2×C4⋊C8—711no—
356C2×C4⋊C8C2×D7111yescharacteristic
357C2×C22⋊C8—711no—
358C2×C22⋊C8—711no—
359C2×C22⋊C8—711no—
360C2×C22⋊C8—711no—
361C2×#97—711no—
362C2×#97—711no—
363C2×#97—711no—
364C2×#97C2×D7111yescharacteristic
365C2×C4×C8—711no—
366C2×C4×C8—711no—
367C4×C4⋊C4—711no—
368C4×C4⋊C4—711no—
369C4×C4⋊C4—711no—
370C4×C4⋊C4—711no—
371C4×C4⋊C4—711no—
372C4×C4⋊C4—711no—
373C4×C4⋊C4—711no—
374C4×C4⋊C4C2×D7111yescharacteristic
375C2×#95—711no—
376C2×#95—711no—
377#336—711no—
378#336—711no—
379#336—711no—
380#336—711no—
381#336—711no—
382#336—711no—
383#336—711no—
384#336—711no—
385#336C2×D7111yescharacteristic
386#336C2×D7111yescharacteristic
387#336C2×D7111yescharacteristic
388#336C2×D7111yescharacteristic
389C23×C4—711no—
390C23×C4—711no—
391C23×C4—711no—
392C23×C4—711no—
393C23×C4—711no—
394C23×C4—711no—
395C23×C4—711no—
396C22×C8—711no—
397C22×C8—711no—
398C22×C8—711no—
399C22×C8—711no—
400C22×C8C22×D7111yescharacteristic
401C22×C8C22×D7111yescharacteristic
402C22×C8C22×D7111yescharacteristic
403C22×C8C22×D7111yescharacteristic
404C2×C4⋊C4—711no—
405C2×C4⋊C4—711no—
406C2×C4⋊C4—711no—
407C2×C4⋊C4—711no—
408C2×C4⋊C4—711no—
409C2×C4⋊C4—711no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4C4×D7111yescharacteristic
413C2×C4⋊C4C4×D7111yescharacteristic
414C2×C42—711no—
415C2×C42—711no—
416C2×C42—711no—
417C2×C42—711no—
418C2×C42—711no—
419C2×C42—711no—
420C2×C42—711no—
421C2×C42—711no—
422C2×C42—711no—
423C2×C42—1422no—
424C2×C42—1422no—
425C2×C42—1422no—
426C2×C42—1422no—
427C2×C42—1422no—
428C2×C42—1422no—
429C2×C42C22×D7111yescharacteristic
430C2×C42C22×D7111yescharacteristic
431C2×C42C22×D7111yescharacteristic
432C4⋊C8—22no—
433C4⋊C8—1422no—
434C4⋊C8—1422no—
435C4⋊C8—1422no—
436C4⋊C8—1422no—
437C4⋊C8—1422no—
438C22⋊C8—714no—
439C22⋊C8—1422no—
440C22⋊C8—1422no—
441C22⋊C8—1422no—
442#97—22no—
443#97—1422no—
444#97—1422no—
445#97—1422no—
446C4×C8—1422no—
447C4×C8—1422no—
448#95—711no—
449#95—711no—
450#95—711no—
451#95—711no—
452#95—711no—
453#95—711no—
454#95C4×D7111yescharacteristic
455#95C4×D7111yescharacteristic
456C24—711no—
457C22×C4—2841no—
458C22×C4—2841no—
459C22×C4—2841no—
460C22×C4—711no—
461C22×C4—711no—
462C22×C4—711no—
463C22×C4—711no—
464C22×C4—712no—
465C22×C4—712no—
466C22×C4—714no—
467C22×C4—711no—
468C22×C4—711no—
469C22×C4—711no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1422no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1421no—
485C22×C4—1421no—
486C22×C4—1421no—
487C22×C4—1421no—
488C22×C4C2×C4×D7111yescharacteristic
489C22×C4C2×C4×D7111yescharacteristic
490C22×C4C2×C4×D7111yescharacteristic
491C22×C4C2×C4×D7111yescharacteristic
492C22×C4C2×C4×D7111yescharacteristic
493C22×C4C2×C4×D7111yescharacteristic
494C22×C4C23×D7111yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—712no—
507C2×C8—1421no—
508C2×C8—1421no—
509C2×C8—1421no—
510C2×C8—1421no—
511C2×C8—1421no—
512C2×C8—1421no—
513C2×C8—1421no—
514C2×C8—1421no—
515C2×C8—1421no—
516C2×C8—1421no—
517C2×C8—1421no—
518C2×C8D4×D7112yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—714no—
522C4⋊C4—1422no—
523C4⋊C4—1422no—
524C4⋊C4—1422no—
525C4⋊C4—1422no—
526C4⋊C4—1422no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—712no—
534C42—712no—
535C42—1422no—
536C42—1422no—
537C42—1422no—
538C42—1422no—
539C42—1422no—
540C42—1422no—
541C42—1421no—
542C42—1421no—
543C42—1421no—
544C42—1421no—
545C23—712no—
546C23—712no—
547C23—712no—
548C23—1421no—
549C23—1421no—
550C23—1421no—
551C23—1421no—
552C23C22×C4×D7111yescharacteristic
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—2844no—
570C2×C4—2844no—
571C2×C4—2844no—
572C2×C4—2844no—
573C2×C4—2844no—
574C2×C4—712no—
575C2×C4—712no—
576C2×C4—712no—
577C2×C4—712no—
578C2×C4—712no—
579C2×C4—712no—
580C2×C4—1428no—
581C2×C4—1421no—
582C2×C4—1421no—
583C2×C4—1421no—
584C2×C4—1421no—
585C2×C4—1421no—
586C2×C4—1421no—
587C2×C4—1421no—
588C2×C4—1421no—
589C2×C4—1421no—
590C2×C4—1421no—
591C2×C4—1421no—
592C2×C4—1421no—
593C2×C4—1421no—
594C2×C4—1421no—
595C2×C4—1421no—
596C2×C4—1422no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4—1422no—
600C2×C4M16×D7112yes—
601C2×C4M16×D7112yes—
602C2×C4C2×D4×D7111yescharacteristic
603C2×C4#41×D7111yescharacteristic
604C2×C4#41×D7111yescharacteristic
605C2×C4#41×D7111yescharacteristic
606C2×C4#1455617511yescharacteristic
607C2×C4#1455617611yescharacteristic
608C8—22no—
609C8—41no—
610C8—41no—
611C8—41no—
612C8—2841no—
613C8—2841no—
614C8—2841no—
615C8—1422no—
616C22—714no—
617C22—1422no—
618C22—1422no—
619C22—1422no—
620C22—1422no—
621C22—1422no—
622C22—1422no—
623C22#117×D7111yescharacteristic
624C22C4×D4×D7111yescharacteristic
625C22#1798647511yescharacteristic
626C22C2×M16×D7111yescharacteristic
627C22C2×M16×D7111yescharacteristic
628C22#131×D7111yescharacteristic
629C22#1798676411yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—2841no—
637C4—2841no—
638C4—2841no—
639C4—714no—
640C4—1422no—
641C4—1422no—
642C4—1422no—
643C4#1798664711yescharacteristic
644C4#1798665111yescharacteristic
645C4#1798668111yescharacteristic
646C4#1798668511yescharacteristic
647C2—1424no—
648C2#2259120611yescharacteristic
649C2#432×D7111yescharacteristic
650C2#2259699611yescharacteristic
651C2#436×D7111yescharacteristic
652C2#2259707311yescharacteristic
653C2#2259742211yescharacteristic
654C2#2259742611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1917C2
prime radicalC23×D71C22×C4
cosocleC22×C4×C71C24
hypercenter#1917D71
hypocenterC71#22603
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1917

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #22589057C2semidirect
5. #22589055C2semidirect
6. #22588906C2semidirect
7. #22588905C2semidirect
8. #22588903C2semidirect
12. #22588208C2semidirect
13. #22588206C2semidirect
15. C71×#1917C2semidirect
318. Dic71#432semidirect
336. #1917D71semidirect
343. C71#22603semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179860552
C2×#179860542
C4×C4⋊C4×D712
#225890572
#225890552
#225889062
#225889052
#225889032
#225884072
#225883192
#225882082
#225882062
#225880972
C71×#19172
#2260371