Finite Group Databasebeta

← #31968444

Subgroups — #31968444

Order 18176.

594 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598377C211yesmaximal, characteristic
2Q8×#11765848C211yesmaximal, characteristic
3#22597282C211yesmaximal, characteristic
4#22587250C211yesmaximal, characteristic, complemented
5#22587246C211yesmaximal, characteristic, complemented
6#22587212C211yesmaximal, characteristic, complemented
7#22587211C211yesmaximal, characteristic, complemented
8#22587211C211yesmaximal, characteristic, complemented
9#22587210C211yesmaximal, characteristic, complemented
10#22587156C211yesmaximal, characteristic, complemented
11#22587109C211yesmaximal, characteristic
12#22586793C211yesmaximal, characteristic
13Q8⋊C8×D71C211yesmaximal, characteristic
14#22586628C211yesmaximal, characteristic
15C71×#1579C211yesmaximal, characteristic, complemented
16#17986553C2211yescharacteristic
17C4×Q8×D71C2211yescharacteristic
18#17986336C411yescharacteristic
19#17986333C411yescharacteristic
20#17986293C411yescharacteristic
21#17986292C411yescharacteristic
22#17986208C411yescharacteristic
23#17986207C411yescharacteristic
24#17986156C411yescharacteristic
25#17986153C411yescharacteristic
26#17986130C2211yescharacteristic
27#17986129—21no—
28#17986129—21no—
29#17986129C2211yescharacteristic
30#17986129C2211yescharacteristic
31#17986128—21no—
32#17986126—21no—
33#17986126C2211yescharacteristic
34Q8×C71⋊C8C2211yescharacteristic
35#17986123C2211yescharacteristic
36C4×Dic284C2211yescharacteristic
37C4×#11765855C2211yescharacteristic
38C4×#11765855C2211yescharacteristic
39#17986050C2211yescharacteristic
40#17986024—21no—
41#17986021—21no—
42#17986021C2211yescharacteristic
43C4⋊C8×D71C2211yescharacteristic
44#17985953—21no—
45C4×#11765848—21no—
46C4×#11765848C2211yescharacteristic
47C4×C8×D71C2211yescharacteristic
48#17985750C2211yescharacteristic
49#17985749C2211yescharacteristic
50#17985749C2211yescharacteristic
51#17985731C2211yescharacteristic
52#17985731C2211yescharacteristic
53Q8⋊(C71⋊C8)C2211yescharacteristic
54Q8⋊(C71⋊C8)C2211yescharacteristic
55#17985729C2211yescharacteristic
56#17985729C2211yescharacteristic
57#17985721C2211yescharacteristic
58#17985721C2211yescharacteristic
59#17985721C2211yescharacteristic
60C8×Q8×C71C2211yescharacteristic
61C8×Q8×C71C2211yescharacteristic
62C4×Q16×C71C2211yescharacteristic
63#334×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65Q8⋊C8×C71C2211yescharacteristic
66#322×C71C2211yescharacteristic
67C2×Q8×D71C2×C411yescharacteristic
68C2×Dic284C2×C411yescharacteristic
69C2×#11765855C2×C411yescharacteristic
70C2×#11765855C2×C411yescharacteristic
71C2×#11765848—21no—
72C2×#11765848D411yescharacteristic
73C2×#11765848D411yescharacteristic
74C2×C8×D71—21no—
75C2×C8×D71—21no—
76#14556127C2×C411yescharacteristic
77#14556126C2×C411yescharacteristic
78#14556118—21no—
79#14556108C2×C411yescharacteristic
80#14556106—21no—
81#14556104C2×C411yescharacteristic
82Q8×Dic71C2×C411yescharacteristic
83C4×Dic142—21no—
84C4×Dic142—21no—
85C4×Dic142C2311yescharacteristic
86C4×Dic142C2311yescharacteristic
87#14556093—21no—
88#14556093C2×C411yescharacteristic
89#14556085—21no—
90C4⋊C4×D71—21no—
91C4⋊C4×D71C2×C411yescharacteristic
92C42×D71—21no—
93C42×D71C2311yescharacteristic
94#14556062C2×C411yescharacteristic
95#14556061C2×C411yescharacteristic
96#14556061C2×C411yescharacteristic
97C4⋊(C71⋊C8)—21no—
98C4⋊(C71⋊C8)—21no—
99C4⋊(C71⋊C8)C2311yescharacteristic
100C4⋊(C71⋊C8)C2311yescharacteristic
101Dic71⋊C8—21no—
102Dic71⋊C8—21no—
103Dic71⋊C8—21no—
104Dic71⋊C8—21no—
105Dic71⋊C8—21no—
106Dic71⋊C8—21no—
107Dic71⋊C8C2×C411yescharacteristic
108Dic71⋊C8C2×C411yescharacteristic
109#14556053C2×C411yescharacteristic
110#14556053C2×C411yescharacteristic
111#14556052C2×C411yescharacteristic
112#14556052C2×C411yescharacteristic
113#14556051C2×C411yescharacteristic
114#14556051C2×C411yescharacteristic
115(C2×D71)⋊C8C2×C411yescharacteristic
116(C2×D71)⋊C8C2×C411yescharacteristic
117#14556039—21no—
118#14556038—21no—
119#14556038—21no—
120#14556038—21no—
121#14556038D411yescharacteristic
122#14556038D411yescharacteristic
123C4×C71⋊C8—21no—
124C4×C71⋊C8C2311yescharacteristic
125C4×C71⋊C8C2311yescharacteristic
126C4×C71⋊C8C2311yescharacteristic
127C8×Dic71—21no—
128C8×Dic71—21no—
129C2×Q16×C71C2×C411yescharacteristic
130C4×Q8×C71C2311yescharacteristic
131C4×Q8×C71C2311yescharacteristic
132#107×C71C2×C411yescharacteristic
133C4⋊C8×C71—21no—
134C4⋊C8×C71—21no—
135C4⋊C8×C71C2311yescharacteristic
136C4⋊C8×C71C2311yescharacteristic
137#103×C71C2×C411yescharacteristic
138#103×C71C2×C411yescharacteristic
139C4×C8×C71—21no—
140C4×C8×C71—21no—
141C4×C8×C71C2311yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142—21no—
145C2×Dic142—21no—
146C2×Dic142C22×C411yescharacteristic
147C2×Dic142C22×C411yescharacteristic
148Q8×D71—22no—
149Q8×D71—41no—
150C2×C4×D71—21no—
151C2×C4×D71—21no—
152C2×C4×D71C22×C411yescharacteristic
153C2×C4×D71C2×D411yescharacteristic
154C2×C4×D71#4111yescharacteristic
155Dic284—41no—
156#11765855—22no—
157#11765855—41no—
158#11765848—22no—
159#11765848—22no—
160#11765848—42no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8C22×C411yescharacteristic
165C2×C71⋊C8C22×C411yescharacteristic
166C2×C71⋊C8C2×D411yescharacteristic
167C2×C71⋊C8C2×D411yescharacteristic
168C8×D71—42no—
169C8×D71—42no—
170C4⋊Dic71—21no—
171C4⋊Dic71—21no—
172C4⋊Dic71C22×C411yescharacteristic
173C4⋊Dic71C22×C411yescharacteristic
174Dic71⋊C4—21no—
175Dic71⋊C4—21no—
176Dic71⋊C4—21no—
177Dic71⋊C4—21no—
178Dic71⋊C4—21no—
179Dic71⋊C4—21no—
180Dic71⋊C4—21no—
181Dic71⋊C4—21no—
182(C2×D71)⋊C4—21no—
183(C2×D71)⋊C4—21no—
184C4×Dic71—21no—
185C4×Dic71—21no—
186C4×Dic71—21no—
187C4×Dic71—21no—
188C4×Dic71C22×C411yescharacteristic
189C4×Dic71C2×D411yescharacteristic
190C4×Dic71#4111yescharacteristic
191C2×Q8×C71C22×C411yescharacteristic
192C2×Q8×C71C22×C411yescharacteristic
193Q16×C71—22no—
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71C22×C411yescharacteristic
198C2×C8×C71C22×C411yescharacteristic
199C2×C8×C71C2×D411yescharacteristic
200C2×C8×C71C2×D411yescharacteristic
201C4⋊C4×C71—21no—
202C4⋊C4×C71—21no—
203C4⋊C4×C71C22×C411yescharacteristic
204C4⋊C4×C71C22×C411yescharacteristic
205C42×C71—21no—
206C42×C71—21no—
207C42×C71C2411yescharacteristic
208C22×D71C4×D411yescharacteristic
209Q8×C71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71#13112yes—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71C23×C411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C2×#4111yescharacteristic
220C2×Dic71—21no—
221C2×Dic71—21no—
222C2×Dic71—21no—
223C2×Dic71—21no—
224C2×Dic71C4×D411yescharacteristic
225C2×Dic71C4×D411yescharacteristic
226C2×Dic71C4×D411yescharacteristic
227C4×D71—21no—
228C4×D71—21no—
229C4×D71—41no—
230C4×D71—42no—
231C4×D71—81no—
232C4×D71#13712yes—
233C4×D71#13712yes—
234Dic142—21no—
235Dic142—21no—
236Dic142—21no—
237Dic142—21no—
238Dic142—42no—
239Dic142—81no—
240C8×C71—21no—
241C8×C71—21no—
242C8×C71—21no—
243C8×C71—21no—
244C8×C71—41no—
245C8×C71—41no—
246C8×C71—41no—
247C71⋊C8—22no—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71C2×C4×D411yescharacteristic
255C2×D71—21no—
256C2×D71—21no—
257C2×D71#44112yes—
258C4×C71—22no—
259C4×C71—21no—
260C4×C71—41no—
261C4×C71—41no—
262C4×C71—41no—
263C4×C71C2×#13111yescharacteristic
264C4×C71#56811yescharacteristic
265C4×C71C2×#13711yescharacteristic
266C4×C71C2×#13711yescharacteristic
267Dic71—21no—
268Dic71—21no—
269Dic71—41no—
270Dic71—41no—
271Dic71—41no—
272Dic71—41no—
273Dic71#44112yes—
274#18003—711nomaximal, Sylow
275C2×C71#293311yescharacteristic
276C2×C71C2×#44111yescharacteristic
277C2×C71#295611yescharacteristic
278D71—22no—
279#3104—711no—
280Q8×M16—711no—
281#2962—711no—
282#1593—711no—
283#1593—711no—
284#1587—711no—
285#1587—711no—
286#1587—711no—
287#1587—711no—
288#1579—711no—
289#1579D7111yescharacteristic, complemented
290#1570—711no—
291#1493—711no—
292C2×Q8⋊C8—711no—
293#1451—711no—
294C71#1800311yescharacteristic, complemented, Sylow
295#533—711no—
296C2×C4×Q8—711no—
297#484—711no—
298#484—711no—
299#477—711no—
300#477—711no—
301#462—711no—
302#462—711no—
303#451—711no—
304#451—711no—
305#446—711no—
306#446—711no—
307#446—711no—
308#446—711no—
309#446—1421no—
310#446—1421no—
311#446—1421no—
312#446—1421no—
313C8×Q8—711no—
314C8×Q8—711no—
315C8×Q8C2×D7111yescharacteristic
316C8×Q8C2×D7111yescharacteristic
317C4×Q16—711no—
318C4×Q16—711no—
319C4×Q16—711no—
320C4×Q16C2×D7111yescharacteristic
321#431—711no—
322#424—1421no—
323#423—711no—
324#423—1421no—
325C2×C4⋊C8—711no—
326#405—1421no—
327C4×M16—711no—
328C4×M16—1421no—
329C2×C4×C8—711no—
330#334—711no—
331#334—711no—
332#334—711no—
333#334C2×D7111yescharacteristic
334Q8⋊C8—711no—
335Q8⋊C8—711no—
336Q8⋊C8—711no—
337Q8⋊C8—711no—
338Q8⋊C8—711no—
339Q8⋊C8—711no—
340Q8⋊C8C2×D7111yescharacteristic
341Q8⋊C8C2×D7111yescharacteristic
342#322—711no—
343#322—711no—
344#322—711no—
345#322C2×D7111yescharacteristic
346C22×Q8—711no—
347C2×Q16—711no—
348C2×Q16—711no—
349C2×Q16—711no—
350C2×Q16C4×D7111yescharacteristic
351C2×M16—711no—
352C2×M16—711no—
353C2×M16—1421no—
354C22×C8—1421no—
355C22×C8—1421no—
356C4⋊Q8—711no—
357C4⋊Q8—711no—
358#125—1421no—
359C22⋊Q8—711no—
360C22⋊Q8—711no—
361C22⋊Q8—1421no—
362C4×Q8—711no—
363C4×Q8—711no—
364C4×Q8—711no—
365C4×Q8—711no—
366C4×Q8—1421no—
367C4×Q8—1421no—
368C4×Q8—1421no—
369C4×Q8C22×D7111yescharacteristic
370C4×Q8C22×D7111yescharacteristic
371#117—1421no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—1421no—
374C2×C42—711no—
375C2×C42—1421no—
376#107—711no—
377#107—711no—
378#107—711no—
379#107C4×D7111yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—711no—
383C4⋊C8—711no—
384C4⋊C8—711no—
385C4⋊C8—711no—
386C4⋊C8—1421no—
387C4⋊C8—1421no—
388C4⋊C8—1421no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
394C4⋊C8C22×D7111yescharacteristic
395C4⋊C8C22×D7111yescharacteristic
396#103—711no—
397#103—711no—
398#103—711no—
399#103—711no—
400#103—711no—
401#103—711no—
402#103C4×D7111yescharacteristic
403#103C4×D7111yescharacteristic
404C22⋊C8—711no—
405C22⋊C8—711no—
406#97—711no—
407#97—711no—
408#97—1421no—
409#97—1421no—
410#97—1421no—
411#97—1421no—
412C4×C8—21no—
413C4×C8—21no—
414C4×C8—711no—
415C4×C8—711no—
416C4×C8—711no—
417C4×C8—1421no—
418C4×C8—1421no—
419C4×C8—1421no—
420C4×C8C22×D7111yescharacteristic
421C4×C8C22×D7111yescharacteristic
422C4×C8C22×D7111yescharacteristic
423C2×Q8—2841no—
424C2×Q8—711no—
425C2×Q8—711no—
426C2×Q8—1421no—
427C2×Q8—1421no—
428C2×Q8—1422no—
429C2×Q8C2×C4×D7111yescharacteristic
430C2×Q8C2×C4×D7111yescharacteristic
431C22×C4—711no—
432C22×C4—711no—
433C22×C4—711no—
434C22×C4—1421no—
435C22×C4—1421no—
436Q16—22no—
437Q16—2841no—
438Q16—2841no—
439Q16—1422no—
440M16—2842no—
441M16—1422no—
442M16—1422no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—2842no—
447C2×C8—2842no—
448C2×C8—711no—
449C2×C8—711no—
450C2×C8—711no—
451C2×C8—711no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C4×D7111yescharacteristic
456C2×C8C2×C4×D7111yescharacteristic
457C2×C8D4×D7111yescharacteristic
458C2×C8D4×D7111yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—711no—
462C4⋊C4—711no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4C2×C4×D7111yescharacteristic
474C4⋊C4C2×C4×D7111yescharacteristic
475C22⋊C4—1421no—
476C22⋊C4—1421no—
477C42—21no—
478C42—21no—
479C42—711no—
480C42—711no—
481C42—711no—
482C42—1421no—
483C42—1421no—
484C42—1421no—
485C42—1421no—
486C42C23×D7111yescharacteristic
487C23—711no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—2842no—
492Q8—5681no—
493Q8—1421no—
494Q8—1421no—
495Q8—1421no—
496Q8—1421no—
497Q8#1455613412yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—2842no—
503C2×C4—2841no—
504C2×C4—5681no—
505C2×C4—711no—
506C2×C4—712no—
507C2×C4—712no—
508C2×C4—711no—
509C2×C4—711no—
510C2×C4—1421no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1421no—
514C2×C4—1421no—
515C2×C4—1421no—
516C2×C4C22×C4×D7111yescharacteristic
517C2×C4C2×D4×D7111yescharacteristic
518C2×C4#41×D7111yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—2841no—
527C8—2841no—
528C8—1422no—
529C8—1421no—
530C8—1421no—
531C8—1421no—
532C8—1421no—
533C22—712no—
534C22—1421no—
535C22—1421no—
536C22C4×D4×D7111yescharacteristic
537C4—21no—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—2841no—
543C4—2841no—
544C4—2841no—
545C4—2841no—
546C4—712no—
547C4—1421no—
548C4—1421no—
549C4C2×#1455613411yescharacteristic
550C4#1798676411yescharacteristic
551C4#137×D7111yescharacteristic
552C4#137×D7111yescharacteristic
553C2—1422no—
554C2#2259706011yescharacteristic
555C2#441×D7111yescharacteristic
556C2#2259724811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1579C2
socleC22×C71C2×C4×D4
prime radicalC22×D71C4×D4
cosocleC42×C71C24
hypercenter#1579D71
hypocenterC71#18003
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
4. #22587250C2semidirect
5. #22587246C2semidirect
6. #22587212C2semidirect
7. #22587211C2semidirect
8. #22587211C2semidirect
9. #22587210C2semidirect
10. #22587156C2semidirect
15. C71×#1579C2semidirect
289. #1579D71semidirect
294. C71#18003semidirect

Maximal subgroups

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

subgroupindex
#225983772
Q8×#117658482
#225972822
#225872502
#225872462
#225872122
#225872112
#225872102
#225871562
#225871092
#225867932
Q8⋊C8×D712
#225866282
C71×#15792
#1800371