Finite Group Databasebeta

← #32000920

Subgroups — #32000920

Order 18176.

1082 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986040C211yesmaximal, characteristic
2C2×#17986038C211yesmaximal, characteristic
3C2×#17986021C211yesmaximal, characteristic
4#22590746C211yesmaximal, characteristic
5C4×C4⋊C4×D71C211yesmaximal, characteristic
6#22589110C211yesmaximal, characteristic, complemented
7#22589108C211yesmaximal, characteristic, complemented
8#22589106C211yesmaximal, characteristic, complemented
9#22589102C211yesmaximal, characteristic, complemented
10#22589100C211yesmaximal, characteristic, complemented
11#22589097C211yesmaximal, characteristic, complemented
12#22589095C211yesmaximal, characteristic, complemented
13#22588058C211yesmaximal, characteristic
14#22588058C211yesmaximal, characteristic
15C71×#1949C211yesmaximal, characteristic, complemented
16C22×#11765848C2211yescharacteristic
17C22×#11765848C2211yescharacteristic
18C2×#14556086C2211yescharacteristic
19C2×#14556086C2211yescharacteristic
20C2×C4⋊C4×D71C2211yescharacteristic
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C42×D71—21no—
23C2×C42×D71C2211yescharacteristic
24#17986040—24no—
25#17986038—24no—
26C2×#14556062C2211yescharacteristic
27C2×#14556061C2211yescharacteristic
28C2×#14556061C2211yescharacteristic
29C2×#14556060C2211yescharacteristic
30C2×#14556059C2211yescharacteristic
31C2×#14556059C2211yescharacteristic
32#17986021C418yescomplemented
33C2×C4⋊(C71⋊C8)C2211yescharacteristic
34C2×Dic71⋊C8C2211yescharacteristic
35C2×Dic71⋊C8C2211yescharacteristic
36#17985868C2211yescharacteristic
37#17985868C2211yescharacteristic
38#17985865C2211yescharacteristic
39#17985858—21no—
40#17985858—21no—
41C4⋊C4×Dic71C2211yescharacteristic
42C4⋊C4×Dic71C2211yescharacteristic
43C4×Dic71⋊C4—21no—
44C4×C4⋊Dic71C2211yescharacteristic
45#17985850—21no—
46#17985850—21no—
47#17985850—21no—
48#95×D71—21no—
49#17985762C2211yescharacteristic
50#17985762C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53#17985761C2211yescharacteristic
54#17985761C2211yescharacteristic
55C2×#107×C71C2211yescharacteristic
56C2×#106×C71C2211yescharacteristic
57C2×C4⋊C8×C71C2211yescharacteristic
58#382×C71C2211yescharacteristic
59C4×C4⋊C4×C71C2211yescharacteristic
60#340×C71C2211yescharacteristic
61#340×C71C2211yescharacteristic
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C22×C4×D71D411yescharacteristic
65C22×C4×D71D411yescharacteristic
66C22×C4×D71C2311yescharacteristic
67C2×#11765848—22no—
68C2×#11765848—22no—
69C2×#11765848—22no—
70C2×#11765848—22no—
71C2×#11765848C2×C414yes—
72C2×#11765848C2×C414yes—
73C22×C71⋊C8C2311yescharacteristic
74C22×C71⋊C8C2311yescharacteristic
75#14556086—22no—
76#14556086—22no—
77#14556086—22no—
78#14556086—22no—
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C2×Dic71⋊C4—21no—
87C2×Dic71⋊C4—21no—
88C2×Dic71⋊C4—21no—
89C4⋊C4×D71—22no—
90C4⋊C4×D71—22no—
91C4⋊C4×D71D414yes—
92C4⋊C4×D71Q814yes—
93C2×(C2×D71)⋊C4—21no—
94C2×(C2×D71)⋊C4—21no—
95C2×C4×Dic71—21no—
96C2×C4×Dic71—21no—
97C2×C4×Dic71—21no—
98C2×C4×Dic71—21no—
99C2×C4×Dic71—21no—
100C2×C4×Dic71D411yescharacteristic
101C2×C4×Dic71D411yescharacteristic
102C2×C4×Dic71C2311yescharacteristic
103C42×D71—44no—
104C42×D71C2×C414yes—
105C42×D71Q814yes—
106#14556062—22no—
107#14556061—22no—
108#14556061—22no—
109#14556060—22no—
110#14556059—22no—
111#14556059—22no—
112C4⋊(C71⋊C8)C2×C414yescomplemented
113Dic71⋊C8C2×C414yescomplemented
114Dic71⋊C8C2×C414yescomplemented
115C71⋊#95—21no—
116C71⋊#95—21no—
117C71⋊#95—21no—
118C71⋊#95—21no—
119C71⋊#95—21no—
120C22×C8×C71C2311yescharacteristic
121C22×C8×C71C2311yescharacteristic
122C2×C4⋊C4×C71C2311yescharacteristic
123C2×C4⋊C4×C71C2311yescharacteristic
124C2×C4⋊C4×C71C2311yescharacteristic
125C2×C4⋊C4×C71C2311yescharacteristic
126C2×C42×C71—21no—
127C2×C42×C71C2311yescharacteristic
128#107×C71—22no—
129#106×C71—22no—
130C4⋊C8×C71C2×C414yescomplemented
131#95×C71—21no—
132#95×C71—21no—
133#95×C71—21no—
134C23×D71C2×D411yescharacteristic
135C22×Dic71—21no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C22×Dic71C2×D411yescharacteristic
140C22×Dic71C2×D411yescharacteristic
141C22×Dic71C2×D411yescharacteristic
142C2×C4×D71—22no—
143C2×C4×D71—22no—
144C2×C4×D71—24no—
145C2×C4×D71—24no—
146C2×C4×D71—41no—
147C2×C4×D71—41no—
148C2×C4×D71—42no—
149C2×C4×D71—42no—
150C2×C4×D71—42no—
151C2×C4×D71—42no—
152C2×C4×D71C4⋊C414yes—
153C2×C4×D71C4⋊C414yes—
154C2×C4×D71C22×C412yes—
155C2×C4×D71C2×D412yes—
156C2×C4×D71C2×Q812yes—
157C2×C4×D71C2×Q812yes—
158C2×C4×D71#4112yes—
159C2×C4×D71#4112yes—
160#11765848—28no—
161#11765848—28no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8—21no—
165C2×C71⋊C8—21no—
166C2×C71⋊C8C22×C412yes—
167C2×C71⋊C8C22×C412yes—
168C4⋊Dic71—21no—
169C4⋊Dic71—21no—
170C4⋊Dic71—21no—
171C4⋊Dic71—21no—
172C4⋊Dic71—21no—
173C4⋊Dic71—21no—
174C4⋊Dic71C2×D412yes—
175C4⋊Dic71C2×Q812yes—
176Dic71⋊C4—24no—
177Dic71⋊C4—24no—
178Dic71⋊C4—42no—
179Dic71⋊C4—42no—
180Dic71⋊C4—42no—
181Dic71⋊C4—42no—
182(C2×D71)⋊C4—42no—
183(C2×D71)⋊C4—42no—
184C4×Dic71—42no—
185C4×Dic71—42no—
186C4×Dic71—42no—
187C4×Dic71—42no—
188C4×Dic71—42no—
189C4×Dic71C4⋊C414yes—
190C4×Dic71C4⋊C414yes—
191C4×Dic71C22×C412yes—
192C4×Dic71C2×Q812yes—
193C22×C4×C71—21no—
194C22×C4×C71—21no—
195C22×C4×C71—21no—
196C22×C4×C71—21no—
197C22×C4×C71C2×D411yescharacteristic
198C22×C4×C71C2×D411yescharacteristic
199C22×C4×C71C2411yescharacteristic
200C2×C8×C71—21no—
201C2×C8×C71—21no—
202C2×C8×C71—21no—
203C2×C8×C71—21no—
204C2×C8×C71C22×C412yes—
205C2×C8×C71C22×C412yes—
206C4⋊C4×C71—21no—
207C4⋊C4×C71—21no—
208C4⋊C4×C71—21no—
209C4⋊C4×C71—21no—
210C4⋊C4×C71—21no—
211C4⋊C4×C71—21no—
212C4⋊C4×C71C2×D412yes—
213C4⋊C4×C71C2×Q812yes—
214C42×C71—42no—
215C42×C71C22×C412yes—
216C42×C71C2×Q812yes—
217C23×C71C22×D411yescharacteristic
218C22×D71—21no—
219C22×D71—21no—
220C22×D71—21no—
221C22×D71—21no—
222C22×D71C2×C4⋊C412yes—
223C22×D71#12112yes—
224C22×D71C22⋊Q812yes—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71C2×C4⋊C412yes—
242C2×C4×C71C2×C4⋊C412yes—
243C2×C4×C71C23×C411yescharacteristic
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C22×Q811yescharacteristic
246C2×C4×C71C22×Q811yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×Dic71—21no—
250C2×Dic71—21no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71C2×C4⋊C412yes—
264C2×Dic71C2×C4⋊C412yes—
265C2×Dic71C2×C4⋊C412yes—
266C2×Dic71#12112yes—
267C2×Dic71C22⋊Q812yes—
268C2×Dic71AGL(1,ℤ/8)12yes—
269C2×Dic71#13712yes—
270C4×D71—28no—
271C4×D71—44no—
272C4×D71—48no—
273C4×D71—48no—
274C4×D71C4×D414yes—
275C4×D71C4×Q814yes—
276C4×D71#12514yes—
277C4×D71C4⋊Q814yes—
278C8×C71—22no—
279C8×C71—22no—
280C71⋊C8—22no—
281C71⋊C8—22no—
282C22×C71C22×C4⋊C411yescharacteristic
283C22×C71C2×#12111yescharacteristic
284C22×C71C2×C22⋊Q811yescharacteristic
285C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C71C2×#13711yescharacteristic
287C22×C71#57511yescharacteristic
288C22×C71#57511yescharacteristic
289C2×D71—22no—
290C2×D71—22no—
291C2×D71—22no—
292C2×D71—22no—
293C2×D71—22no—
294C2×D71—22no—
295C2×D71#38914yes—
296C4×C71—22no—
297C4×C71—22no—
298C4×C71—42no—
299C4×C71—42no—
300C4×C71—81no—
301C4×C71—81no—
302C4×C71C2×C4×D411yescharacteristic
303C4×C71C2×C4×Q811yescharacteristic
304C4×C71C2×#12511yescharacteristic
305C4×C71C2×C4⋊Q811yescharacteristic
306Dic71—22no—
307Dic71—42no—
308Dic71—42no—
309Dic71—81no—
310Dic71—81no—
311Dic71#38914yes—
312Dic71#42814yescomplemented
313#21170—711nomaximal, Sylow
314C2×C71C2×#38911yescharacteristic
315C2×C71C2×#42811yescharacteristic
316C2×C71#291111yescharacteristic
317C2×C71#303611yescharacteristic
318C2×C71#303711yescharacteristic
319C2×C71#307711yescharacteristic
320C2×C71#307811yescharacteristic
321D71—24no—
322C2×#428—711no—
323C2×#428—711no—
324C2×#423—711no—
325#2285—711no—
326C2×C4×C4⋊C4—711no—
327#1949—711no—
328#1949—711no—
329#1949—711no—
330#1949D7111yescharacteristic, complemented
331#1948—711no—
332#1948—711no—
333#1947—711no—
334#1947—711no—
335#1736—711no—
336#1736—711no—
337C71#2117011yescharacteristic, complemented, Sylow
338C22×M16—711no—
339C22×M16—711no—
340C2×#117—711no—
341C2×#117—711no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—711no—
344C22×C42—711no—
345C22×C42—1421no—
346#428—1424no—
347#428—1424no—
348C2×#107—711no—
349C2×#107—711no—
350C2×#107—711no—
351C2×#107C2×D7111yescharacteristic
352C2×#106—711no—
353C2×#106—711no—
354C2×#106—711no—
355C2×#106C2×D7111yescharacteristic
356#423—718no—
357C2×C4⋊C8—711no—
358C2×C4⋊C8—711no—
359C2×C4⋊C8—711no—
360C2×C4⋊C8C2×D7111yescharacteristic
361#384—711no—
362#384—711no—
363#382—711no—
364#382C2×D7111yescharacteristic
365#380—1421no—
366#380—1421no—
367C4×C4⋊C4—711no—
368C4×C4⋊C4—711no—
369C4×C4⋊C4—711no—
370C4×C4⋊C4—1421no—
371C4×C4⋊C4—1421no—
372C4×C4⋊C4—1421no—
373C4×C4⋊C4—1421no—
374C4×C4⋊C4C2×D7111yescharacteristic
375C2×#95—1421no—
376#340—711no—
377#340—711no—
378#340—711no—
379#340—711no—
380#340—711no—
381#340—711no—
382#340C2×D7111yescharacteristic
383#340C2×D7111yescharacteristic
384C23×C4—711no—
385C23×C4—711no—
386C23×C4—711no—
387C23×C4—1421no—
388C23×C4—1421no—
389C2×M16—714no—
390C2×M16—714no—
391C2×M16—1422no—
392C2×M16—1422no—
393C2×M16—1422no—
394C2×M16—1422no—
395C22×C8—711no—
396C22×C8—711no—
397C22×C8C22×D7111yescharacteristic
398C22×C8C22×D7111yescharacteristic
399#117—1422no—
400#117—1422no—
401#117—1422no—
402#117—1422no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—711no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—714no—
407C2×C4⋊C4—711no—
408C2×C4⋊C4—711no—
409C2×C4⋊C4—1422no—
410C2×C4⋊C4—1422no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1421no—
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C4⋊C4C22×D7111yescharacteristic
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C22⋊C4—1421no—
422C2×C22⋊C4—1421no—
423C2×C42—21no—
424C2×C42—2844no—
425C2×C42—711no—
426C2×C42—711no—
427C2×C42—711no—
428C2×C42—714no—
429C2×C42—714no—
430C2×C42—1421no—
431C2×C42—1421no—
432C2×C42—1421no—
433C2×C42—1421no—
434C2×C42—1421no—
435C2×C42C22×D7111yescharacteristic
436#107—22no—
437#107—1422no—
438#107—1422no—
439#107—1422no—
440#106—22no—
441#106—1422no—
442#106—1422no—
443#106—1422no—
444C4⋊C8—714no—
445C4⋊C8—714no—
446C4⋊C8—714no—
447C4⋊C8C4×D7114yescomplemented
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1421no—
452#95—1421no—
453#95—1421no—
454#95—1421no—
455#95—1421no—
456C24—711no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—2841no—
462C22×C4—2841no—
463C22×C4—2842no—
464C22×C4—2842no—
465C22×C4—2842no—
466C22×C4—2842no—
467C22×C4—714no—
468C22×C4—711no—
469C22×C4—711no—
470C22×C4—712no—
471C22×C4—712no—
472C22×C4—712no—
473C22×C4—712no—
474C22×C4—712no—
475C22×C4—712no—
476C22×C4—711no—
477C22×C4—714no—
478C22×C4—1424no—
479C22×C4—1424no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1421no—
483C22×C4—1421no—
484C22×C4—1421no—
485C22×C4—1421no—
486C22×C4D4×D7111yescharacteristic
487C22×C4D4×D7111yescharacteristic
488C22×C4C23×D7111yescharacteristic
489M16—1428no—
490M16—1428no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—712no—
496C2×C8—712no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8—1421no—
500C2×C8—1421no—
501C2×C8C2×C4×D7112yes—
502C2×C8C2×C4×D7112yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—2842no—
510C4⋊C4—2842no—
511C4⋊C4—2842no—
512C4⋊C4—2842no—
513C4⋊C4—712no—
514C4⋊C4—712no—
515C4⋊C4—1424no—
516C4⋊C4—1424no—
517C4⋊C4—1421no—
518C4⋊C4—1421no—
519C4⋊C4—1421no—
520C4⋊C4—1421no—
521C4⋊C4—1421no—
522C4⋊C4—1421no—
523C4⋊C4D4×D7112yes—
524C4⋊C4Q8×D7112yes—
525C22⋊C4—2842no—
526C22⋊C4—2842no—
527C42—42no—
528C42—2842no—
529C42—2842no—
530C42—2842no—
531C42—2842no—
532C42—2842no—
533C42—712no—
534C42—712no—
535C42—714no—
536C42—714no—
537C42C2×C4×D7112yes—
538C42Q8×D7112yes—
539C23—712no—
540C23—712no—
541C23—712no—
542C23—1421no—
543C23—1421no—
544C23—1421no—
545C23—1421no—
546C23C2×D4×D7111yescharacteristic
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—2844no—
564C2×C4—2848no—
565C2×C4—2848no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—2841no—
570C2×C4—2841no—
571C2×C4—2841no—
572C2×C4—2841no—
573C2×C4—2841no—
574C2×C4—2841no—
575C2×C4—2841no—
576C2×C4—712no—
577C2×C4—712no—
578C2×C4—712no—
579C2×C4—712no—
580C2×C4—712no—
581C2×C4—712no—
582C2×C4—714no—
583C2×C4—714no—
584C2×C4—714no—
585C2×C4—714no—
586C2×C4—712no—
587C2×C4—1428no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4—1421no—
591C2×C4—1421no—
592C2×C4C4⋊C4×D7112yes—
593C2×C4C4⋊C4×D7112yes—
594C2×C4C22×C4×D7111yescharacteristic
595C2×C4C2×D4×D7111yescharacteristic
596C2×C4C2×Q8×D7111yescharacteristic
597C2×C4C2×Q8×D7111yescharacteristic
598C2×C4#41×D7111yescharacteristic
599C2×C4#41×D7111yescharacteristic
600C8—22no—
601C8—22no—
602C8—1422no—
603C8—1422no—
604C22—714no—
605C22—1422no—
606C22—1422no—
607C22—1422no—
608C22—1422no—
609C22—1422no—
610C22—1422no—
611C22C2×C4⋊C4×D7111yescharacteristic
612C22#121×D7111yescharacteristic
613C22C22⋊Q8×D7111yescharacteristic
614C22AGL(1,ℤ/8)×D7111yescharacteristic
615C22#137×D7111yescharacteristic
616C22#1798679611yescharacteristic
617C22#1798679611yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—81no—
623C4—81no—
624C4—2842no—
625C4—2842no—
626C4—5681no—
627C4—5681no—
628C4—714no—
629C4—714no—
630C4—1422no—
631C4C4×D4×D7111yescharacteristic
632C4C4×Q8×D7111yescharacteristic
633C4#125×D7111yescharacteristic
634C4C4⋊Q8×D7111yescharacteristic
635C2—1424no—
636C2#389×D7111yescharacteristic
637C2#428×D7111yescharacteristic
638C2#2259692811yescharacteristic
639C2#2259780211yescharacteristic
640C2#2259781011yescharacteristic
641C2#2259813511yescharacteristic
642C2#2259814311yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1949C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1949D71
hypocenterC71#21170
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #22589110C2semidirect
7. #22589108C2semidirect
8. #22589106C2semidirect
9. #22589102C2semidirect
10. #22589100C2semidirect
11. #22589097C2semidirect
12. #22589095C2semidirect
15. C71×#1949C2semidirect
32. #17986021C4semidirect
112. C4⋊(C71⋊C8)C2×C4semidirect
113. Dic71⋊C8C2×C4semidirect
114. Dic71⋊C8C2×C4semidirect
130. C4⋊C8×C71C2×C4semidirect
312. Dic71#428semidirect
330. #1949D71semidirect
337. C71#21170semidirect
447. C4⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860402
C2×#179860382
C2×#179860212
#225907462
C4×C4⋊C4×D712
#225891102
#225891082
#225891062
#225891022
#225891002
#225890972
#225890952
#225880582
C71×#19492
#2117071