Finite Group Databasebeta

← #32005231

Subgroups — #32005231

Order 18176.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986276C211yesmaximal, characteristic
2#22590702C211yesmaximal, characteristic
3#22589289C211yesmaximal, characteristic, complemented
4#22589106C211yesmaximal, characteristic, complemented
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588600C211yesmaximal, characteristic
8#22588586C211yesmaximal, characteristic
9#22588347C211yesmaximal, characteristic
10#22588238C211yesmaximal, characteristic, complemented
11C4×#14556061C211yesmaximal, characteristic, complemented
12#22588197C211yesmaximal, characteristic, complemented
13#22588168C211yesmaximal, characteristic
14C4×#14556048C211yesmaximal, characteristic
15C71×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D284C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19#17986276C418yescomplemented
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C4⋊(C71⋊C8)C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556048C2211yescharacteristic
29C2×#14556048C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C4×C4⋊Dic71C2211yescharacteristic
34C4×C4⋊Dic71C2211yescharacteristic
35#17985846—21no—
36#17985846—21no—
37#17985846—21no—
38#95×D71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4⋊Q8×C71C2211yescharacteristic
50#391×C71—21no—
51#391×C71—21no—
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#387×C71—21no—
55#387×C71—21no—
56#382×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59#376×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×D284—22no—
69C4×D284—22no—
70C4×D284—22no—
71C4×D284—22no—
72C4×D284C2×C414yes—
73C4×D284Q814yescomplemented
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×(C2×D71)⋊C4—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83#14556061—22no—
84#14556061C2×C414yescomplemented
85#14556061C2×C414yescomplemented
86#14556059—22no—
87C4⋊(C71⋊C8)—22no—
88C4⋊(C71⋊C8)C2×C414yescomplemented
89#14556048—22no—
90#14556048—22no—
91#14556048C2×C414yes—
92#14556048C2×C414yes—
93#14556039—22no—
94C4×C71⋊C8—22no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C22×Q8×C71—21no—
99C4⋊Q8×C71—22no—
100C4⋊Q8×C71C2×C414yescomplemented
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71D411yescharacteristic
120#95×C71D411yescharacteristic
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123C23×D71—21no—
124C2×D284—21no—
125C2×D284—21no—
126C2×D284—21no—
127C2×D284C22×C412yes—
128C2×D284C2×Q812yes—
129C2×D284#4112yes—
130C22×Dic71—21no—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C2×C4×D71—24no—
134C2×C4×D71—24no—
135C2×C4×D71—24no—
136C2×C4×D71—81no—
137C2×C4×D71—81no—
138C2×C4×D71—81no—
139C2×C4×D71—81no—
140C2×C4×D71—81no—
141C2×C4×D71—81no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
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⋊C8C22×C412yes—
153C2×C71⋊C8C22×C412yes—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71C22×C412yes—
158C4⋊Dic71C2×Q812yes—
159C4⋊Dic71#4112yes—
160(C2×D71)⋊C4—24no—
161(C2×D71)⋊C4—24no—
162(C2×D71)⋊C4—24no—
163C4×Dic71—81no—
164C4×Dic71—81no—
165C4×Dic71—81no—
166C2×Q8×C71—24no—
167C2×Q8×C71—81no—
168C2×Q8×C71—81no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—24no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—81no—
188C4⋊C4×C71—81no—
189C4⋊C4×C71C22×C412yes—
190C4⋊C4×C71C22×C412yes—
191C42×C71—21no—
192C42×C71—21no—
193C42×C71—81no—
194C42×C71—81no—
195C42×C71C22×C412yes—
196C42×C71C2×Q812yes—
197C42×C71#4112yes—
198C42×C71#4112yes—
199C23×C71C22×D411yescharacteristic
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—41no—
204C22×D71—41no—
205C22×D71—41no—
206C22×D71—41no—
207Q8×C71—84no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
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—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71C4×D412yes—
235C2×C4×C71C4×D412yes—
236C2×C4×C71C22⋊Q812yes—
237C2×C4×C71C22⋊Q812yes—
238C2×C4×C71C23×C411yescharacteristic
239C2×C4×C71C22×Q811yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253D284—22no—
254D284—22no—
255D284—22no—
256D284C4×Q814yes—
257C4×D71—84no—
258C4×D71—84no—
259C4×D71—84no—
260C71⋊C8—22no—
261C71⋊C8—22no—
262C71⋊C8—41no—
263C71⋊C8—41no—
264C22×C71C2×C4×D411yescharacteristic
265C22×C71C2×C22⋊Q811yescharacteristic
266C22×C71#52511yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57611yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57811yescharacteristic
271C2×D71—24no—
272C2×D71—42no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C4×C71—22no—
279C4×C71—22no—
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—41no—
283C4×C71—41no—
284C4×C71—42no—
285C4×C71—42no—
286C4×C71—81no—
287C4×C71—81no—
288C4×C71C2×C4×Q811yescharacteristic
289C4×C71C4×#4111yescharacteristic
290C4×C71#52911yescharacteristic
291C4×C71#53311yescharacteristic
292Dic71—42no—
293Dic71—42no—
294Dic71—42no—
295#21624—711nomaximal, Sylow
296C2×C71C4×C22⋊Q811yescharacteristic
297C2×C71#294711yescharacteristic
298C2×C71#294811yescharacteristic
299C2×C71#308111yescharacteristic
300C2×C71#308311yescharacteristic
301C2×C71#311811yescharacteristic
302C2×C71#312411yescharacteristic
303D71—44no—
304C2×#474—711no—
305#2331D7111yescharacteristic, complemented
306#2275—711no—
307#1987—711no—
308#1949—711no—
309#1921—711no—
310#1918—711no—
311#1864—711no—
312#1862—711no—
313#1804—711no—
314#1776—711no—
315C4×#107—711no—
316#1768—711no—
317#1762—711no—
318C4×#102—711no—
319C71#2162411yescharacteristic, complemented, Sylow
320C2×C4⋊Q8C2×D7111yescharacteristic
321C2×C4×D4—711no—
322C2×C4×D4—711no—
323C2×C4×D4—711no—
324#474—718no—
325C2×#107—711no—
326C2×#107—711no—
327C2×#107—711no—
328C2×#106—711no—
329C2×C4⋊C8—711no—
330C2×C4⋊C8—711no—
331C2×#102—711no—
332C2×#102—711no—
333C2×#102—711no—
334C2×#102—711no—
335C2×#97—711no—
336C2×C4×C8—711no—
337#391—21no—
338#391—21no—
339#389C2×D7111yescharacteristic
340#389C2×D7111yescharacteristic
341#387—21no—
342#387—21no—
343#382C2×D7111yescharacteristic
344C4×C4⋊C4—711no—
345C4×C4⋊C4—711no—
346C4×C4⋊C4—711no—
347C4×C4⋊C4C2×D7111yescharacteristic
348C4×C4⋊C4C2×D7111yescharacteristic
349C4×C22⋊C4—1421no—
350C4×C22⋊C4—1421no—
351C4×C22⋊C4—1421no—
352#376C2×D7111yescharacteristic
353C2×#95—1421no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#336—711no—
363#336—711no—
364C22×Q8—21no—
365C22×D4—711no—
366C23×C4—1421no—
367C23×C4—1421no—
368C23×C4—1421no—
369C22×C8—711no—
370C22×C8—711no—
371C22×C8—711no—
372C22×C8—711no—
373C4⋊Q8—22no—
374C4⋊Q8C4×D7114yescomplemented
375C4×D4—714no—
376C4×D4—714no—
377C4×D4—1422no—
378C4×D4—1422no—
379C4×D4—1422no—
380C4×D4—1422no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C4⋊C4C22×D7111yescharacteristic
389C2×C4⋊C4C22×D7111yescharacteristic
390C2×C4⋊C4C22×D7111yescharacteristic
391C2×C22⋊C4—1421no—
392C2×C22⋊C4—1421no—
393C2×C22⋊C4—1421no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—1421no—
397C2×C42—1421no—
398C2×C42—1421no—
399C2×C42C22×D7111yescharacteristic
400C2×C42C22×D7111yescharacteristic
401C2×C42C22×D7111yescharacteristic
402#107—714no—
403#107—714no—
404#107—1422no—
405#106—1422no—
406C4⋊C8—714no—
407C4⋊C8—1422no—
408#102—714no—
409#102—714no—
410#102—1422no—
411#102—1422no—
412#97—1422no—
413C4×C8—1422no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423#95C22⋊D7111yescharacteristic
424#95C22⋊D7111yescharacteristic
425#95C22⋊D7111yescharacteristic
426#95C22⋊D7111yescharacteristic
427C24—1421no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—712no—
432C2×D4—712no—
433C2×D4—712no—
434C2×D4—1421no—
435C2×D4—1421no—
436C2×D4—1421no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—5681no—
442C22×C4—5681no—
443C22×C4—5681no—
444C22×C4—5681no—
445C22×C4—5681no—
446C22×C4—5681no—
447C22×C4—1424no—
448C22×C4—1421no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1424no—
452C22×C4—1424no—
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C23×D7111yescharacteristic
460C2×C8—712no—
461C2×C8—712no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—712no—
482C4⋊C4—712no—
483C4⋊C4—712no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4—1421no—
487C4⋊C4C2×C4×D7112yes—
488C4⋊C4C2×C4×D7112yes—
489C22⋊C4—1424no—
490C22⋊C4—1424no—
491C22⋊C4—1424no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—5681no—
497C42—5681no—
498C42—5681no—
499C42C2×C4×D7112yes—
500C42Q8×D7112yes—
501C42#1176586512yes—
502C42#1176586612yes—
503C23—2841no—
504C23—2841no—
505C23—2841no—
506C23—2841no—
507C23—1422no—
508C23—1422no—
509C23—1422no—
510C23C22×C22⋊D7111yescharacteristic
511Q8—84no—
512D4—714no—
513D4—1422no—
514D4—1422no—
515D4—1422no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
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—41no—
541C2×C4—41no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—5684no—
549C2×C4—5684no—
550C2×C4—5684no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4C4×C22⋊D7112yes—
555C2×C4C4×C22⋊D7112yes—
556C2×C4#1455610412yes—
557C2×C4#1455610412yes—
558C2×C4C22×C4×D7111yescharacteristic
559C2×C4C2×Q8×D7111yescharacteristic
560C2×C4#41×D7111yescharacteristic
561C2×C4#41×D7111yescharacteristic
562C2×C4C2×#1176586511yescharacteristic
563C2×C4C2×#1176586611yescharacteristic
564C8—2841no—
565C8—2841no—
566C8—1422no—
567C8—1422no—
568C22—2842no—
569C22—2842no—
570C22—2842no—
571C22—2842no—
572C22—2842no—
573C22—2842no—
574C22—1424no—
575C22C2×C4×C22⋊D7111yescharacteristic
576C22C2×#1455610411yescharacteristic
577C22#1798650611yescharacteristic
578C22#1798680211yescharacteristic
579C22#1798680211yescharacteristic
580C22#1798680711yescharacteristic
581C22#1798681411yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—2842no—
593C4—2842no—
594C4—2842no—
595C4C4×Q8×D7111yescharacteristic
596C4C4×#1176586611yescharacteristic
597C4#1798653611yescharacteristic
598C4#1798655311yescharacteristic
599C2—2844no—
600C2C4×#1455610411yescharacteristic
601C2#2259718311yescharacteristic
602C2#2259719011yescharacteristic
603C2#2259816511yescharacteristic
604C2#2259818711yescharacteristic
605C2#2259851811yescharacteristic
606C2#2259859611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2331C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2331D71
hypocenterC71#21624
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #22589289C2semidirect
4. #22589106C2semidirect
5. #22588929C2semidirect
6. #22588908C2semidirect
10. #22588238C2semidirect
11. C4×#14556061C2semidirect
12. #22588197C2semidirect
15. C71×#2331C2semidirect
19. #17986276C4semidirect
73. C4×D284Q8semidirect
84. #14556061C2×C4semidirect
85. #14556061C2×C4semidirect
88. C4⋊(C71⋊C8)C2×C4semidirect
100. C4⋊Q8×C71C2×C4semidirect
305. #2331D71semidirect
319. C71#21624semidirect
374. C4⋊Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862762
#225907022
#225892892
#225891062
#225889292
#225889082
#225886002
#225885862
#225883472
#225882382
C4×#145560612
#225881972
#225881682
C4×#145560482
C71×#23312
#2162471