Finite Group Databasebeta

← #32004688

Subgroups — #32004688

Order 18176.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986337C211yesmaximal, characteristic
2C4×#14556087C211yesmaximal, characteristic
3#22588978C211yesmaximal, characteristic, complemented
4#22588971C211yesmaximal, characteristic, complemented
5#22588929C211yesmaximal, characteristic, complemented
6#22588922C211yesmaximal, characteristic, complemented
7#22588653C211yesmaximal, characteristic
8#22588537C211yesmaximal, characteristic
9#22588432C211yesmaximal, characteristic
10#22588238C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12#22588170C211yesmaximal, characteristic, complemented
13#22588170C211yesmaximal, characteristic, complemented
14#22588102C211yesmaximal, characteristic
15C4×C22⋊Q8×C71C211yesmaximal, characteristic
16C2×#14556087C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18#17986337C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×#14556052C2211yescharacteristic
25C2×#14556052C2211yescharacteristic
26C2×#14556052C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×C22⋊(C71⋊C8)C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C4×C4⋊Dic71C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C4×C4⋊Dic71C2211yescharacteristic
34C4×C4⋊Dic71C2211yescharacteristic
35C4×C22⋊Dic71—21no—
36C4×C22⋊Dic71—21no—
37#17985845—21no—
38C42×Dic71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C22⋊Q8×C71C2211yescharacteristic
50C2×C4×Q8×C71C2211yescharacteristic
51C22×C42×C71C2211yescharacteristic
52#391×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#387×C71—21no—
55#385×C71—21no—
56#380×C71C2211yescharacteristic
57C4×C4⋊C4×C71—21no—
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C22⋊C4×C71—21no—
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556087—22no—
65#14556087—22no—
66#14556087—22no—
67#14556087C2×C414yes—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79C2×C4×Dic71—21no—
80#14556061—22no—
81#14556061C2×C414yescomplemented
82#14556059—22no—
83#14556059C2×C414yescomplemented
84#14556052—22no—
85#14556052—22no—
86#14556052C2×C414yescomplemented
87#14556052C2×C414yescomplemented
88C22⋊(C71⋊C8)—22no—
89C22⋊(C71⋊C8)C2×C414yes—
90#14556039—22no—
91#14556039—22no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C22×Q8×C71C2311yescharacteristic
97C23×C4×C71D411yescharacteristic
98C23×C4×C71D411yescharacteristic
99C23×C4×C71C2311yescharacteristic
100C22⋊Q8×C71—22no—
101C22⋊Q8×C71C2×C414yes—
102C4×Q8×C71—22no—
103C4×Q8×C71—22no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C22⋊C4×C71—21no—
111C2×C22⋊C4×C71—21no—
112C2×C42×C71—22no—
113C2×C42×C71—22no—
114C2×C42×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124#95×C71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8C22×C412yes—
140C2×C71⋊C8C22×C412yes—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71C22×C412yes—
146C4⋊Dic71C22×C412yes—
147C4⋊Dic71#4112yes—
148C4⋊Dic71#4112yes—
149C22⋊Dic71—24no—
150C22⋊Dic71—24no—
151C4×Dic71—24no—
152C4×Dic71—24no—
153C4×Dic71—42no—
154C4×Dic71—42no—
155C4×Dic71—42no—
156C4×Dic71—42no—
157C24×C71C2×D411yescharacteristic
158C2×Q8×C71—21no—
159C2×Q8×C71—21no—
160C2×Q8×C71—21no—
161C2×Q8×C71—21no—
162C2×Q8×C71—21no—
163C2×Q8×C71C22×C412yes—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—41no—
176C22×C4×C71—41no—
177C22×C4×C71—41no—
178C22×C4×C71C22×C412yes—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71#4112yes—
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—24no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—42no—
193C4⋊C4×C71—42no—
194C4⋊C4×C71C22×C412yes—
195C22⋊C4×C71—24no—
196C22⋊C4×C71—42no—
197C42×C71—22no—
198C42×C71—22no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—42no—
202C42×C71—42no—
203C42×C71#4112yes—
204C42×C71#4112yes—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71C4×D412yes—
212C23×C71C22×D411yescharacteristic
213Q8×C71—22no—
214Q8×C71—22no—
215Q8×C71—41no—
216Q8×C71—41no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—44no—
235C2×C4×C71—44no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71C4×D412yes—
247C2×C4×C71#12312yes—
248C2×C4×C71#12312yes—
249C2×C4×C71#13712yes—
250C2×C4×C71C23×C411yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×C4×C71C2×#4111yescharacteristic
253C2×C4×C71C2×#4111yescharacteristic
254C2×C4×C71C2×#4111yescharacteristic
255C2×C4×C71C2×#4111yescharacteristic
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C71⋊C8—22no—
269C71⋊C8—22no—
270C71⋊C8—41no—
271C71⋊C8—41no—
272C22×C71—22no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—41no—
276C22×C71—41no—
277C22×C71—41no—
278C22×C71—41no—
279C22×C71C2×C4×D411yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71#52511yescharacteristic
282C22×C71C2×#13711yescharacteristic
283C22×C71#57511yescharacteristic
284C22×C71#57511yescharacteristic
285C22×C71#57511yescharacteristic
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—41no—
291C4×C71—42no—
292C4×C71—42no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71C4×#4111yescharacteristic
296C4×C71C4×#4111yescharacteristic
297C4×C71#52911yescharacteristic
298C4×C71#52911yescharacteristic
299Dic71—42no—
300Dic71—42no—
301Dic71—42no—
302Dic71—42no—
303#21583—711nomaximal, Sylow
304C2×C71—42no—
305C2×C71C4×#12311yescharacteristic
306C2×C71#294311yescharacteristic
307C2×C71C4×#13711yescharacteristic
308C2×C71#309311yescharacteristic
309C2×C71#309511yescharacteristic
310C2×C71#311111yescharacteristic
311C2×C71#311611yescharacteristic
312C2×#485—711no—
313C4×C22⋊Q8D7111yescharacteristic
314C4×#117—711no—
315#1928—711no—
316#1927—711no—
317#1921—711no—
318#1920—711no—
319#1873—711no—
320#1854—711no—
321#1825—711no—
322#1776—711no—
323#1776—711no—
324#1763—711no—
325#1763—711no—
326#1749—711no—
327C71#2158311yescharacteristic, complemented, Sylow
328C2×C22⋊Q8C2×D7111yescharacteristic
329C2×C4×Q8C2×D7111yescharacteristic
330C2×#117—711no—
331C2×#117—711no—
332C22×C42C2×D7111yescharacteristic
333#485—718no—
334C2×#107—711no—
335C2×#107—711no—
336C2×#106—711no—
337C2×#106—711no—
338C2×#103—711no—
339C2×#103—711no—
340C2×#103—711no—
341C2×#103—711no—
342C2×C22⋊C8—711no—
343C2×C22⋊C8—711no—
344C2×#97—711no—
345C2×#97—711no—
346#391C2×D7111yescharacteristic
347#389C2×D7111yescharacteristic
348#387—21no—
349#385—21no—
350#380C2×D7111yescharacteristic
351C4×C4⋊C4—21no—
352C4×C4⋊C4—711no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4—711no—
356C4×C4⋊C4C2×D7111yescharacteristic
357C4×C22⋊C4—21no—
358C4×C22⋊C4—1421no—
359C4×C22⋊C4—1421no—
360#376—1421no—
361C43—1421no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#336—711no—
371#336—711no—
372C22×Q8C22×D7111yescharacteristic
373C23×C4C22⋊D7111yescharacteristic
374C23×C4C22⋊D7111yescharacteristic
375C23×C4C22×D7111yescharacteristic
376C22×C8—711no—
377C22×C8—711no—
378C22×C8—711no—
379C22×C8—711no—
380C22⋊Q8—22no—
381C22⋊Q8C4×D7114yes—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—714no—
385#117—1422no—
386#117—1422no—
387#117—1422no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—711no—
393C2×C4⋊C4—711no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1421no—
407C2×C42—1421no—
408C2×C42—1421no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42C22⋊D7111yescharacteristic
413C2×C42C22⋊D7111yescharacteristic
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416#107—714no—
417#107—1422no—
418#106—714no—
419#106—1422no—
420#103—714no—
421#103—714no—
422#103—1422no—
423#103—1422no—
424C22⋊C8—714no—
425C22⋊C8—1422no—
426#97—1422no—
427#97—1422no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95—1421no—
437C24C2×C22⋊D7111yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D7112yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4C2×C4×D7112yes—
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4#1176586512yes—
469C22×C4C23×D7111yescharacteristic
470C2×C8—712no—
471C2×C8—712no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—712no—
491C4⋊C4—712no—
492C4⋊C4—712no—
493C4⋊C4—712no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4C2×C4×D7112yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1424no—
502C22⋊C4—1424no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—2842no—
510C42—2842no—
511C42—2842no—
512C42—2842no—
513C42—1424no—
514C42—1424no—
515C42#1176586412yes—
516C42#1176586412yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D7112yes—
524C23C22×C22⋊D7111yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4C4×C22⋊D7112yes—
571C2×C4#1455610912yes—
572C2×C4#1455610912yes—
573C2×C4#1455615912yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4#41×D7111yescharacteristic
577C2×C4C2×#1176586411yescharacteristic
578C2×C4C2×#1176586411yescharacteristic
579C2×C4C2×#1176586511yescharacteristic
580C8—2841no—
581C8—2841no—
582C8—1422no—
583C8—1422no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D7111yescharacteristic
592C22C2×#1455610911yescharacteristic
593C22#1798650611yescharacteristic
594C22C2×#1455615911yescharacteristic
595C22#1798679711yescharacteristic
596C22#1798679711yescharacteristic
597C22#1798679711yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—2842no—
608C4—2842no—
609C4—2842no—
610C4—2842no—
611C4C4×#1176586411yescharacteristic
612C4C4×#1176586411yescharacteristic
613C4#1798652911yescharacteristic
614C4#1798652911yescharacteristic
615C2—42no—
616C2C4×#1455610911yescharacteristic
617C2#2259713811yescharacteristic
618C2C4×#1455615911yescharacteristic
619C2#2259826911yescharacteristic
620C2#2259828411yescharacteristic
621C2#2259844311yescharacteristic
622C2#2259849611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC4×C22⋊Q8D71
hypocenterC71#21583
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22588978C2semidirect
4. #22588971C2semidirect
5. #22588929C2semidirect
6. #22588922C2semidirect
10. #22588238C2semidirect
11. #22588238C2semidirect
12. #22588170C2semidirect
13. #22588170C2semidirect
18. #17986337C4semidirect
81. #14556061C2×C4semidirect
83. #14556059C2×C4semidirect
86. #14556052C2×C4semidirect
87. #14556052C2×C4semidirect
327. C71#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#179863372
C4×#145560872
#225889782
#225889712
#225889292
#225889222
#225886532
#225885372
#225884322
#225882382
#225881702
#225881022
C4×C22⋊Q8×C712
#2158371