Finite Group Databasebeta

← #32027501

Subgroups — #32027501

Order 18176.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590738C211yesmaximal, characteristic
2#22589908C211yesmaximal, characteristic, complemented
3#22589888C211yesmaximal, characteristic, complemented
4#22589821C211yesmaximal, characteristic, complemented
5#22589776C211yesmaximal, characteristic
6#22589766C211yesmaximal, characteristic
7#22589758C211yesmaximal, characteristic, complemented
8#22589540C211yesmaximal, characteristic
9#22589533C211yesmaximal, characteristic
10C4×#14556061C211yesmaximal, characteristic, complemented
11C4×#14556059C211yesmaximal, characteristic, complemented
12C4×#14556052C211yesmaximal, characteristic, complemented
13C4×#14556052C211yesmaximal, characteristic, complemented
14C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
15C4×C22⋊Q8×C71C211yesmaximal, characteristic
16C2×#14556061C2211yescharacteristic
17C2×#14556061C2211yescharacteristic
18C2×#14556059C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×#14556052C2211yescharacteristic
21C2×#14556052C2211yescharacteristic
22C2×#14556052C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25C2×C22⋊(C71⋊C8)C2211yescharacteristic
26C2×C4×C71⋊C8C2211yescharacteristic
27C2×C4×C71⋊C8C2211yescharacteristic
28#17985867—21no—
29#17985865C2211yescharacteristic
30#17985865C2211yescharacteristic
31#17985862—21no—
32#17985859C2211yescharacteristic
33#17985859C2211yescharacteristic
34C4×C4⋊Dic71C2211yescharacteristic
35C4×C4⋊Dic71C2211yescharacteristic
36C4×C22⋊Dic71—21no—
37#17985845—21no—
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×C22⋊Q8×C71C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50C22×C42×C71C2211yescharacteristic
51#391×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#387×C71—21no—
54#385×C71—21no—
55#380×C71C2211yescharacteristic
56C4×C4⋊C4×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C22×C71⋊C8C2311yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C2×C4⋊Dic71C2311yescharacteristic
64C2×C4⋊Dic71C2311yescharacteristic
65C2×C4⋊Dic71C2311yescharacteristic
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C22⋊Dic71—21no—
68C2×C22⋊Dic71—21no—
69C2×C4×Dic71—21no—
70C2×C4×Dic71—21no—
71#14556061—22no—
72#14556061—22no—
73#14556059—22no—
74#14556059—22no—
75#14556052—22no—
76#14556052—22no—
77#14556052—22no—
78#14556052—22no—
79C22⋊(C71⋊C8)—22no—
80C22⋊(C71⋊C8)—22no—
81C4×C71⋊C8—22no—
82C4×C71⋊C8—22no—
83C71⋊#95—21no—
84C71⋊#95—21no—
85C71⋊#95—21no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C22×Q8×C71C2311yescharacteristic
92C23×C4×C71D411yescharacteristic
93C23×C4×C71D411yescharacteristic
94C23×C4×C71C2311yescharacteristic
95C22⋊Q8×C71—22no—
96C22⋊Q8×C71—22no—
97C4×Q8×C71—22no—
98C4×Q8×C71—22no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—21no—
107C2×C42×C71—22no—
108C2×C42×C71—22no—
109C2×C42×C71—21no—
110C2×C42×C71—21no—
111C2×C42×C71D411yescharacteristic
112C2×C42×C71D411yescharacteristic
113C2×C42×C71C2311yescharacteristic
114C2×C42×C71C2311yescharacteristic
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8#4112yes—
135C2×C71⋊C8#4112yes—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C22⋊Dic71—42no—
145C22⋊Dic71—42no—
146C4×Dic71—42no—
147C4×Dic71—42no—
148C24×C71C2×D411yescharacteristic
149C2×Q8×C71—21no—
150C2×Q8×C71—21no—
151C2×Q8×C71—21no—
152C2×Q8×C71—21no—
153C2×Q8×C71—21no—
154C2×Q8×C71#4112yes—
155C22×C4×C71—22no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
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—41no—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2411yescharacteristic
177C4⋊C4×C71—24no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—42no—
184C4⋊C4×C71—42no—
185C4⋊C4×C71#4112yes—
186C22⋊C4×C71—24no—
187C22⋊C4×C71—42no—
188C42×C71—22no—
189C42×C71—22no—
190C42×C71—21no—
191C42×C71—21no—
192C42×C71—42no—
193C42×C71—42no—
194C42×C71#4112yes—
195C42×C71#4112yes—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—21no—
202C23×C71#13512yes—
203C23×C71C22×D411yescharacteristic
204Q8×C71—22no—
205Q8×C71—22no—
206Q8×C71—41no—
207Q8×C71—41no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—44no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—44no—
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#12312yes—
238C2×C4×C71#12312yes—
239C2×C4×C71#13512yes—
240C2×C4×C71#13712yes—
241C2×C4×C71C2×#4111yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C71⋊C8—22no—
260C71⋊C8—22no—
261C71⋊C8—41no—
262C71⋊C8—41no—
263C22×C71—22no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71C2×#12311yescharacteristic
271C22×C71#52511yescharacteristic
272C22×C71#53211yescharacteristic
273C22×C71C2×#13511yescharacteristic
274C22×C71C2×#13511yescharacteristic
275C22×C71C2×#13711yescharacteristic
276C22×C71#57511yescharacteristic
277C4×C71—22no—
278C4×C71—41no—
279C4×C71—41no—
280C4×C71—42no—
281C4×C71—41no—
282C4×C71—42no—
283C4×C71—81no—
284C4×C71—81no—
285C4×C71#48514yes—
286C4×C71#52911yescharacteristic
287C4×C71#52911yescharacteristic
288C4×C71#52911yescharacteristic
289C4×C71#52911yescharacteristic
290Dic71—81no—
291Dic71—81no—
292Dic71—81no—
293Dic71—81no—
294#23759—711nomaximal, Sylow
295C2×C71—42no—
296C2×C71#239511yescharacteristic
297C2×C71C2×#48511yescharacteristic
298C2×C71#309111yescharacteristic
299C2×C71#310211yescharacteristic
300C2×C71#310811yescharacteristic
301C2×C71#316811yescharacteristic
302C2×C71#317211yescharacteristic
303C4×C22⋊Q8D7111yescharacteristic
304#2283—711no—
305#2101—711no—
306#2097—711no—
307#2085—711no—
308#2077—711no—
309#2075—711no—
310#2074—711no—
311#2036—711no—
312#2035—711no—
313C4×#107—711no—
314C4×#106—711no—
315C4×#103—711no—
316C4×#103—711no—
317C4×C22⋊C8—711no—
318C71#2375911yescharacteristic, complemented, Sylow
319C2×C22⋊Q8C2×D7111yescharacteristic
320C2×C4×Q8C2×D7111yescharacteristic
321C22×C42C2×D7111yescharacteristic
322C2×#107—711no—
323C2×#107—711no—
324C2×#106—711no—
325C2×#106—711no—
326C2×#103—711no—
327C2×#103—711no—
328C2×#103—711no—
329C2×#103—711no—
330C2×C22⋊C8—711no—
331C2×C22⋊C8—711no—
332C2×C4×C8—711no—
333C2×C4×C8—711no—
334#391C2×D7111yescharacteristic
335#389C2×D7111yescharacteristic
336#387—21no—
337#385—21no—
338#383—1421no—
339#382—711no—
340#382—711no—
341#381—1421no—
342#380—711no—
343#380—711no—
344#380C2×D7111yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4—711no—
347C4×C4⋊C4—711no—
348C4×C4⋊C4C2×D7111yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4—1421no—
351#376—1421no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#336—711no—
361#336—711no—
362C22×Q8C22×D7111yescharacteristic
363C23×C4C22⋊D7111yescharacteristic
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22×D7111yescharacteristic
366C22×C8—711no—
367C22×C8—711no—
368C22×C8—711no—
369C22×C8—711no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—711no—
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1421no—
393C2×C42—1421no—
394C2×C42C22⋊D7111yescharacteristic
395C2×C42C22⋊D7111yescharacteristic
396C2×C42C22×D7111yescharacteristic
397C2×C42C22×D7111yescharacteristic
398#107—1422no—
399#107—1422no—
400#106—1422no—
401#106—1422no—
402#103—1422no—
403#103—1422no—
404#103—1422no—
405#103—1422no—
406C22⋊C8—1422no—
407C22⋊C8—1422no—
408C4×C8—1422no—
409C4×C8—1422no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423C24C2×C22⋊D7111yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1176586512yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1421no—
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4C2×C22⋊D7111yescharacteristic
452C22×C4C2×C22⋊D7111yescharacteristic
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C23×D7111yescharacteristic
456C2×C8—712no—
457C2×C8—712no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—1421no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4#1176586512yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—2842no—
488C22⋊C4—2842no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—2842no—
496C42—2842no—
497C42#1176586412yes—
498C42#1176586412yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1455614812yes—
506C23C22×C22⋊D7111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—2841no—
541C2×C4—2841no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4#1455610912yes—
553C2×C4#1455610912yes—
554C2×C4#1455614812yes—
555C2×C4#1455615912yes—
556C2×C4#41×D7111yescharacteristic
557C2×C4#41×D7111yescharacteristic
558C2×C4C2×#1176586411yescharacteristic
559C2×C4C2×#1176586411yescharacteristic
560C2×C4C2×#1176586511yescharacteristic
561C2×C4C2×#1176586511yescharacteristic
562C8—2841no—
563C8—2841no—
564C8—1422no—
565C8—1422no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1455610911yescharacteristic
574C22#1798651011yescharacteristic
575C22#1798654511yescharacteristic
576C22C2×#1455614811yescharacteristic
577C22C2×#1455614811yescharacteristic
578C22C2×#1455615911yescharacteristic
579C22#1798679711yescharacteristic
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—5681no—
589C4—5681no—
590C4—5681no—
591C4—5681no—
592C4#1798633714yes—
593C4#1798652911yescharacteristic
594C4#1798652911yescharacteristic
595C4#1798653111yescharacteristic
596C4#1798653111yescharacteristic
597C2—42no—
598C2#2259159911yescharacteristic
599C2C2×#1798633711yescharacteristic
600C2#2259824911yescharacteristic
601C2#2259835811yescharacteristic
602C2#2259842011yescharacteristic
603C2#2259890411yescharacteristic
604C2#2259894011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenterC4×C22⋊Q8D71
hypocenterC71#23759
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22589908C2semidirect
3. #22589888C2semidirect
4. #22589821C2semidirect
7. #22589758C2semidirect
10. C4×#14556061C2semidirect
11. C4×#14556059C2semidirect
12. C4×#14556052C2semidirect
13. C4×#14556052C2semidirect
318. C71#23759semidirect

Maximal subgroups

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

subgroupindex
#225907382
#225899082
#225898882
#225898212
#225897762
#225897662
#225897582
#225895402
#225895332
C4×#145560612
C4×#145560592
C4×#145560522
C4×C22⋊(C71⋊C8)2
C4×C22⋊Q8×C712
#2375971