Finite Group Databasebeta

← #32015813

Subgroups — #32015813

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986099C211yesmaximal, characteristic
2C2×#17986099C211yesmaximal, characteristic
3C2×C4×#11765853C211yesmaximal, characteristic
4C4⋊C4×Dic142C211yesmaximal, characteristic, complemented
5#22589075C211yesmaximal, characteristic, complemented
6#22589070C211yesmaximal, characteristic, complemented
7#22588607C211yesmaximal, characteristic, complemented
8#22588601C211yesmaximal, characteristic
9#22588593C211yesmaximal, characteristic, complemented
10#22588587C211yesmaximal, characteristic
11#22588509C211yesmaximal, characteristic, complemented
12#22588352C211yesmaximal, characteristic, complemented
13#22588340C211yesmaximal, characteristic
14#22588215C211yesmaximal, characteristic, complemented
15C71×#2349C211yesmaximal, characteristic
16C22×#11765853C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18C2×C4×Dic142C2211yescharacteristic
19C2×C4×Dic142C2211yescharacteristic
20#17986099—24no—
21#17986099C418yescomplemented
22C4×#11765853—24no—
23C2×#14556061C2211yescharacteristic
24C2×#14556061C2211yescharacteristic
25C2×#14556061C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28C2×C4⋊(C71⋊C8)C2211yescharacteristic
29C2×#14556051C2211yescharacteristic
30C2×#14556051C2211yescharacteristic
31C2×#14556051C2211yescharacteristic
32C2×#14556049C2211yescharacteristic
33C2×#14556049C2211yescharacteristic
34C2×#14556049C2211yescharacteristic
35C2×#14556039C2211yescharacteristic
36C2×#14556039C2211yescharacteristic
37C2×C4×C71⋊C8C2211yescharacteristic
38#17985897C2211yescharacteristic
39#17985897C2211yescharacteristic
40#17985896—21no—
41#17985896—21no—
42#17985869C2211yescharacteristic
43#17985868—21no—
44C4⋊C4×Dic71—21no—
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985754C2211yescharacteristic
50#17985754C2211yescharacteristic
51C2×C4×D4×C71C2211yescharacteristic
52C2×C4×D4×C71C2211yescharacteristic
53C2×C4×D4×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#389×C71C2211yescharacteristic
56#385×C71—21no—
57#385×C71—21no—
58#382×C71C2211yescharacteristic
59#380×C71—21no—
60C4×C4⋊C4×C71C2211yescharacteristic
61C2×#95×C71—21no—
62C22×Dic142C2311yescharacteristic
63C2×#11765853C2×C414yes—
64C2×#11765853D414yes—
65C2×#11765853Q814yes—
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C4×Dic142—22no—
71C4×Dic142—22no—
72C4×Dic142—22no—
73C4×Dic142—41no—
74C4×Dic142—41no—
75C4×Dic142C2×C414yescomplemented
76C2×C4⋊Dic71—21no—
77C2×C4⋊Dic71—21no—
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×Dic71⋊C4—21no—
82C2×Dic71⋊C4—21no—
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71—21no—
89#14556061—22no—
90#14556061—41no—
91#14556061C2×C414yescomplemented
92#14556059—22no—
93#14556059—41no—
94C4⋊(C71⋊C8)—22no—
95#14556051—22no—
96#14556051—22no—
97#14556051C2×C414yescomplemented
98#14556049—22no—
99#14556049—22no—
100#14556049C2×C414yes—
101#14556039—22no—
102#14556039C2×C414yescomplemented
103C4×C71⋊C8—22no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C22×D4×C71C2311yescharacteristic
107C23×C4×C71—21no—
108C23×C4×C71—21no—
109C23×C4×C71—21no—
110C4×D4×C71—22no—
111C4×D4×C71—22no—
112C4×D4×C71—22no—
113C4×D4×C71—41no—
114C4×D4×C71—41no—
115C4×D4×C71C2×C414yes—
116C2×C4⋊C4×C71—21no—
117C2×C4⋊C4×C71—21no—
118C2×C4⋊C4×C71D411yescharacteristic
119C2×C4⋊C4×C71D411yescharacteristic
120C2×C4⋊C4×C71C2311yescharacteristic
121C2×C4⋊C4×C71C2311yescharacteristic
122C2×C4⋊C4×C71C2311yescharacteristic
123C2×C22⋊C4×C71—21no—
124C2×C22⋊C4×C71—21no—
125C2×C22⋊C4×C71—21no—
126C2×C42×C71C2311yescharacteristic
127C2×C42×C71C2311yescharacteristic
128C2×C42×C71C2311yescharacteristic
129#95×C71—21no—
130#95×C71—21no—
131#95×C71—21no—
132#95×C71—21no—
133#95×C71D411yescharacteristic
134#95×C71D411yescharacteristic
135C2×Dic142—21no—
136C2×Dic142—21no—
137C2×Dic142—21no—
138C2×Dic142C22×C412yes—
139C2×Dic142C2×D412yes—
140C2×Dic142C2×Q812yes—
141C22×Dic71—21no—
142C22×Dic71—21no—
143C22×Dic71—21no—
144C22×Dic71—21no—
145#11765853C4⋊C4116yescomplemented
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8C22×C412yes—
155C2×C71⋊C8C22×C412yes—
156C2×C71⋊C8C2×D412yes—
157C2×C71⋊C8C2×Q812yes—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163C4⋊Dic71—42no—
164C4⋊Dic71—81no—
165C4⋊Dic71C22×C412yes—
166Dic71⋊C4—24no—
167Dic71⋊C4—42no—
168Dic71⋊C4—42no—
169Dic71⋊C4—81no—
170Dic71⋊C4—81no—
171C4×Dic71—24no—
172C4×Dic71—42no—
173C4×Dic71—42no—
174C24×C71—21no—
175C2×D4×C71—21no—
176C2×D4×C71—21no—
177C2×D4×C71—21no—
178C2×D4×C71C22×C412yes—
179C2×D4×C71C2×D412yes—
180C2×D4×C71C2×Q812yes—
181C22×C4×C71—21no—
182C22×C4×C71—24no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—42no—
186C22×C4×C71—42no—
187C22×C4×C71—42no—
188C22×C4×C71—42no—
189C22×C4×C71—81no—
190C22×C4×C71—81no—
191C22×C4×C71—81no—
192C22×C4×C71—81no—
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2×D411yescharacteristic
195C22×C4×C71C2×D411yescharacteristic
196C22×C4×C71C2×D411yescharacteristic
197C22×C4×C71C2×D411yescharacteristic
198C22×C4×C71C2×D411yescharacteristic
199C22×C4×C71C2411yescharacteristic
200C4⋊C4×C71—22no—
201C4⋊C4×C71—21no—
202C4⋊C4×C71—21no—
203C4⋊C4×C71—21no—
204C4⋊C4×C71—21no—
205C4⋊C4×C71—21no—
206C4⋊C4×C71—41no—
207C4⋊C4×C71—81no—
208C4⋊C4×C71—81no—
209C4⋊C4×C71C22×C412yes—
210C22⋊C4×C71—24no—
211C22⋊C4×C71—42no—
212C22⋊C4×C71—42no—
213C42×C71—21no—
214C42×C71—21no—
215C42×C71—21no—
216C42×C71—21no—
217C42×C71C22×C412yes—
218C42×C71#4112yes—
219C23×C71—22no—
220C23×C71—22no—
221C23×C71—22no—
222C23×C71—41no—
223C23×C71—41no—
224C23×C71—41no—
225C23×C71—41no—
226C23×C71C22×D411yescharacteristic
227D4×C71—22no—
228D4×C71—22no—
229D4×C71—22no—
230D4×C71C2×C4⋊C414yes—
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—22no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—21no—
241C2×C4×C71—21no—
242C2×C4×C71—21no—
243C2×C4×C71—21no—
244C2×C4×C71—21no—
245C2×C4×C71—21no—
246C2×C4×C71—21no—
247C2×C4×C71—41no—
248C2×C4×C71—41no—
249C2×C4×C71—41no—
250C2×C4×C71—41no—
251C2×C4×C71—41no—
252C2×C4×C71—41no—
253C2×C4×C71—41no—
254C2×C4×C71—41no—
255C2×C4×C71—84no—
256C2×C4×C71—84no—
257C2×C4×C71—84no—
258C2×C4×C71C4×D412yes—
259C2×C4×C71C4×D412yes—
260C2×C4×C71#13712yes—
261C2×C4×C71C23×C411yescharacteristic
262C2×C4×C71C22×D411yescharacteristic
263C2×C4×C71C22×Q811yescharacteristic
264C2×C4×C71C2×#4111yescharacteristic
265C2×C4×C71H(5,𝔽2)11yescharacteristic
266C2×C4×C712−1+411yescharacteristic
267C2×Dic71—22no—
268C2×Dic71—22no—
269C2×Dic71—22no—
270C2×Dic71—22no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—41no—
277C2×Dic71—41no—
278C2×Dic71—41no—
279Dic142—22no—
280Dic142—22no—
281Dic142—22no—
282Dic142C2×C4⋊C414yes—
283C71⋊C8—22no—
284C71⋊C8—41no—
285C71⋊C8—41no—
286C71⋊C8C2×C4⋊C414yes—
287C22×C71—24no—
288C22×C71—42no—
289C22×C71—42no—
290C22×C71—42no—
291C22×C71—42no—
292C22×C71—42no—
293C22×C71—42no—
294C22×C71C2×C4×D411yescharacteristic
295C22×C71D4211yescharacteristic
296C22×C71D4×Q811yescharacteristic
297C22×C71C2×#13711yescharacteristic
298C22×C71#57511yescharacteristic
299C22×C71#57611yescharacteristic
300C22×C71#57711yescharacteristic
301C4×C71—22no—
302C4×C71—22no—
303C4×C71—22no—
304C4×C71—42no—
305C4×C71—41no—
306C4×C71—41no—
307C4×C71—41no—
308C4×C71—81no—
309C4×C71—81no—
310C4×C71C22×C4⋊C411yescharacteristic
311C4×C71#52011yescharacteristic
312C4×C71#54711yescharacteristic
313C4×C71#55411yescharacteristic
314Dic71—24no—
315Dic71—42no—
316Dic71—81no—
317Dic71—81no—
318#22678—711nomaximal, Sylow
319C2×C71—44no—
320C2×C71D4×C4⋊C411yescharacteristic
321C2×C71C4×#13711yescharacteristic
322C2×C71#294611yescharacteristic
323C2×C71#326611yescharacteristic
324C2×C71#326811yescharacteristic
325C2×C71#338511yescharacteristic
326C2×C71#338811yescharacteristic
327C2×#440—711no—
328C2×#440—711no—
329C2×C4×SD16—711no—
330Q8×C4⋊C4—711no—
331#2349D7111yescharacteristic
332#1943—711no—
333#1942—711no—
334#1865—711no—
335#1864—711no—
336#1863—711no—
337#1862—711no—
338#1847—711no—
339#1805—711no—
340#1802—711no—
341#1771—711no—
342C71#2267811yescharacteristic, complemented, Sylow
343C22×SD16—711no—
344C2×C4×Q8—711no—
345C2×C4×Q8—711no—
346C2×C4×Q8—711no—
347C2×C4×D4C2×D7111yescharacteristic
348C2×C4×D4C2×D7111yescharacteristic
349C2×C4×D4C2×D7111yescharacteristic
350#440—718no—
351#440—1424no—
352C4×SD16—1424no—
353C2×#107—711no—
354C2×#107—711no—
355C2×#107—711no—
356C2×#106—711no—
357C2×#106—711no—
358C2×C4⋊C8—711no—
359C2×#103—711no—
360C2×#103—711no—
361C2×#103—711no—
362C2×#102—711no—
363C2×#102—711no—
364C2×#102—711no—
365C2×#97—711no—
366C2×#97—711no—
367C2×C4×C8—711no—
368#389—711no—
369#389—711no—
370#389—1421no—
371#389—1421no—
372#389C2×D7111yescharacteristic
373#389C2×D7111yescharacteristic
374#385—21no—
375#385—21no—
376#384—711no—
377#384—1421no—
378#382C2×D7111yescharacteristic
379#380—21no—
380C4×C4⋊C4—1421no—
381C4×C4⋊C4C2×D7111yescharacteristic
382C2×#95—21no—
383#340—711no—
384#340—711no—
385#340—711no—
386#340—711no—
387#336—711no—
388#336—711no—
389C22×Q8—711no—
390C22×D4C22×D7111yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—21no—
394C2×SD16—714no—
395C2×SD16—714no—
396C2×SD16—714no—
397C22×C8—711no—
398C22×C8—711no—
399C22×C8—711no—
400C22×C8—711no—
401C4×Q8—2841no—
402C4×Q8—2841no—
403C4×Q8—714no—
404C4×Q8—1422no—
405C4×Q8—1422no—
406C4×Q8—1422no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—41no—
412C4×D4C4×D7114yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—711no—
416C2×C4⋊C4—711no—
417C2×C4⋊C4—711no—
418C2×C4⋊C4—1421no—
419C2×C4⋊C4—1421no—
420C2×C4⋊C4—1421no—
421C2×C4⋊C4—1421no—
422C2×C4⋊C4—1421no—
423C2×C4⋊C4—1421no—
424C2×C4⋊C4—1421no—
425C2×C4⋊C4C22⋊D7111yescharacteristic
426C2×C4⋊C4C22⋊D7111yescharacteristic
427C2×C4⋊C4C22×D7111yescharacteristic
428C2×C4⋊C4C22×D7111yescharacteristic
429C2×C4⋊C4C22×D7111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C42—1421no—
434C2×C42—1421no—
435C2×C42—1421no—
436C2×C42C22×D7111yescharacteristic
437C2×C42C22×D7111yescharacteristic
438C2×C42C22×D7111yescharacteristic
439#107—2841no—
440#107—714no—
441#107—1422no—
442#106—2841no—
443#106—1422no—
444C4⋊C8—1422no—
445#103—714no—
446#103—1422no—
447#103—1422no—
448#102—714no—
449#102—1422no—
450#102—1422no—
451#97—714no—
452#97—1422no—
453C4×C8—1422no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—1421no—
459#95—1421no—
460#95C22⋊D7111yescharacteristic
461#95C22⋊D7111yescharacteristic
462C24—21no—
463C2×Q8—712no—
464C2×Q8—712no—
465C2×Q8—712no—
466C2×Q8—1421no—
467C2×Q8—1421no—
468C2×Q8—1421no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D7112yes—
473C2×D4D4×D7112yes—
474C2×D4Q8×D7112yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—1421no—
488C22×C4—1421no—
489C22×C4—1421no—
490C22×C4—1421no—
491C22×C4C2×C22⋊D7111yescharacteristic
492C22×C4C2×C22⋊D7111yescharacteristic
493C22×C4C2×C22⋊D7111yescharacteristic
494C22×C4C2×C22⋊D7111yescharacteristic
495C22×C4C2×C22⋊D7111yescharacteristic
496C22×C4C2×C22⋊D7111yescharacteristic
497C22×C4C23×D7111yescharacteristic
498SD16—7116no—
499C2×C8—712no—
500C2×C8—712no—
501C2×C8—712no—
502C2×C8—712no—
503C2×C8—1421no—
504C2×C8—1421no—
505C2×C8—1421no—
506C2×C8—1421no—
507C2×C8—1421no—
508C2×C8—1421no—
509C2×C8—1421no—
510C2×C8—1421no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—41no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—2842no—
521C4⋊C4—2842no—
522C4⋊C4—2842no—
523C4⋊C4—5681no—
524C4⋊C4—5681no—
525C4⋊C4—5681no—
526C4⋊C4—712no—
527C4⋊C4—1421no—
528C4⋊C4—1421no—
529C4⋊C4—1421no—
530C4⋊C4—1421no—
531C4⋊C4—1421no—
532C4⋊C4—1424no—
533C4⋊C4C2×C4×D7112yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—2842no—
542C42—2842no—
543C42—1424no—
544C42C2×C4×D7112yes—
545C42#1176586512yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C22×C22⋊D7111yescharacteristic
554Q8—714no—
555Q8—1422no—
556Q8—1422no—
557Q8—1422no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C4⋊C4×D7114yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2841no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2841no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4—1422no—
600C2×C4—1422no—
601C2×C4C4×C22⋊D7112yes—
602C2×C4C4×C22⋊D7112yes—
603C2×C4#1455615812yes—
604C2×C4C22×C4×D7111yescharacteristic
605C2×C4C2×D4×D7111yescharacteristic
606C2×C4C2×Q8×D7111yescharacteristic
607C2×C4C2×#1176586511yescharacteristic
608C2×C4#1455617311yescharacteristic
609C2×C4#1455617611yescharacteristic
610C8—2841no—
611C8—2841no—
612C8—714no—
613C8—1422no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×C4×C22⋊D7111yescharacteristic
622C22D4×C22⋊D7111yescharacteristic
623C22Q8×C22⋊D7111yescharacteristic
624C22C2×#1455615811yescharacteristic
625C22#1798679711yescharacteristic
626C22#1798680211yescharacteristic
627C22#1798681011yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—81no—
637C4—2842no—
638C4—5681no—
639C4—5681no—
640C4—1424no—
641C4C2×C4⋊C4×D7111yescharacteristic
642C4#1798648011yescharacteristic
643C4#1798665011yescharacteristic
644C4#1798670411yescharacteristic
645C2—44no—
646C2C4⋊C4×C22⋊D7111yescharacteristic
647C2C4×#1455615811yescharacteristic
648C2#2259717211yescharacteristic
649C2#2259984611yescharacteristic
650C2#2259987611yescharacteristic
651C2#2260125011yescharacteristic
652C2#2260128511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2349C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2349D71
hypocenterC71#22678
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic142C2semidirect
5. #22589075C2semidirect
6. #22589070C2semidirect
7. #22588607C2semidirect
9. #22588593C2semidirect
11. #22588509C2semidirect
12. #22588352C2semidirect
14. #22588215C2semidirect
21. #17986099C4semidirect
75. C4×Dic142C2×C4semidirect
91. #14556061C2×C4semidirect
97. #14556051C2×C4semidirect
102. #14556039C2×C4semidirect
145. #11765853C4⋊C4semidirect
342. C71#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#179860992
C2×C4×#117658532
C4⋊C4×Dic1422
#225890752
#225890702
#225886072
#225886012
#225885932
#225885872
#225885092
#225883522
#225883402
#225882152
C71×#23492
#2267871