Finite Group Databasebeta

← #32861856

Subgroups — #32861856

Order 18176.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22601743C211yesmaximal, characteristic
2#22597285C218yesmaximal, complemented
3#22597282C218yesmaximal, complemented
4#22597281C211yesmaximal, characteristic, complemented
5#22597278C212yesmaximal, complemented
6#22597277C211yesmaximal, characteristic, complemented
7#22597274C211yesmaximal, characteristic, complemented
8#22597273C212yesmaximal, complemented
9#22596902C211yesmaximal, characteristic
10#424×D71C212yesmaximal
11#22596756C212yesmaximal
12C2×#17985953C211yesmaximal, characteristic
13C71×#2960C211yesmaximal, characteristic, complemented
14C22×#11765848C2211yescharacteristic
15C2×M16×D71C2212yes—
16C22×C8×D71C2211yescharacteristic
17#17986629C414yes—
18#17986628C414yes—
19#17986626C414yes—
20#17986624C414yes—
21#17986556C2211yescharacteristic
22#17986553C2218yes—
23#17986552C2212yes—
24C2×#14556085C2211yescharacteristic
25#117×D71C2212yes—
26#17986130C2218yescomplemented
27#17986129C2214yescomplemented
28#17986129C2218yescomplemented
29#17986128C2218yescomplemented
30#17986128C2218yescomplemented
31#17986127C2218yescomplemented
32C8×Dic142C2214yescomplemented
33#17986123C2218yescomplemented
34#17986027C2212yes—
35#17986026C2212yes—
36#17986025C2212yes—
37#17986024C2218yes—
38#17986024C2218yes—
39#17986023C2211yescharacteristic
40#17986022C2212yes—
41C4⋊C8×D71C2218yes—
42#17985957C2212yes—
43#17985956C2214yes—
44#17985955C2212yes—
45#17985954C2212yes—
46#17985953C2214yes—
47#17985953C2214yes—
48#17985953C2214yes—
49#17985951C2214yes—
50C4×#11765848C2214yes—
51C2×#14556039C2211yescharacteristic
52C2×#14556038C2211yescharacteristic
53C2×C8×Dic71C2211yescharacteristic
54#533×C71C2211yescharacteristic
55#446×C71C2214yescomplemented
56C8×Q8×C71C2214yescomplemented
57#424×C71C2212yes—
58#423×C71C2211yescharacteristic
59#405×C71C2212yes—
60C2×C4×C8×C71C2211yescharacteristic
61C22×C4×D71C2311yescharacteristic
62C2×#11765849C2312yes—
63C2×#11765848Q818yescomplemented
64C2×#11765848C2314yes—
65C2×#11765848C2312yes—
66C2×#11765848C2312yes—
67M16×D71—28no—
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C2×C8×D71—24no—
71C2×C8×D71C2314yes—
72C2×C8×D71C2314yes—
73#14556128C2×C412yes—
74#14556127C2×C414yes—
75#14556127C2×C418yes—
76#14556126C2×C418yes—
77#14556120C2×C412yes—
78#14556119C2×C414yes—
79#14556118C2×C418yes—
80#14556118C2×C418yes—
81#14556108C2×C418yes—
82#14556107C2×C414yes—
83#14556106C2×C418yes—
84#14556106C2×C418yes—
85#14556105C2×C418yes—
86#14556104C2×C418yes—
87C4×Dic142C2318yes—
88C4×Dic142C2314yes—
89#14556087C2312yes—
90#14556086C2×C414yes—
91#14556085C2×C412yes—
92#14556085C2×C412yes—
93#14556085C2314yes—
94#14556085C2314yes—
95#14556084C2×C414yes—
96C2×Dic71⋊C4C2×C412yes—
97C4⋊C4×D71C2×C414yes—
98C2×(C2×D71)⋊C4C2×C412yes—
99C22⋊C4×D71C2×C414yes—
100C2×C4×Dic71C2311yescharacteristic
101C42×D71C2314yes—
102C4⋊(C71⋊C8)C2314yes—
103C4⋊(C71⋊C8)C2318yes—
104Dic71⋊C8C2318yes—
105Dic71⋊C8C2318yes—
106Dic71⋊C8C2318yes—
107#14556039C2314yes—
108#14556039C2314yes—
109#14556038C2312yes—
110#14556038C2314yes—
111#14556038C2312yes—
112#14556038C2314yes—
113C4×C71⋊C8C2314yes—
114C8×Dic71C2314yes—
115C8×Dic71C2314yes—
116C8×Dic71C2314yes—
117C2×M16×C71C2312yes—
118C22×C8×C71C2311yescharacteristic
119C22×C8×C71C2311yescharacteristic
120C4⋊Q8×C71C2×C412yes—
121#125×C71C2×C412yes—
122C22⋊Q8×C71C2×C414yes—
123C4×Q8×C71C2314yes—
124#117×C71C2312yes—
125C2×C42×C71C2311yescharacteristic
126C4⋊C8×C71C2314yes—
127C4⋊C8×C71C2318yes—
128#97×C71C2314yes—
129C4×C8×C71C2314yes—
130C4×C8×C71C2312yes—
131C4×C8×C71C2312yes—
132C23×D71C22×C411yescharacteristic
133C2×Dic142C22×C418yes—
134C2×Dic142C22×C414yes—
135C22×Dic71C22×C412yes—
136C22×Dic71C22×C411yescharacteristic
137C2×C4×D71C22×C418yes—
138C2×C4×D71C22×C412yes—
139C2×C4×D71C2×Q814yes—
140C2×C4×D71#4114yes—
141C2×C4×D71C2412yes—
142#11765849—24no—
143#11765848—28no—
144#11765848C2×Q818yes—
145#11765848C2×Q818yes—
146C2×C71⋊C8—22no—
147C2×C71⋊C8C2×Q814yes—
148C2×C71⋊C8C2414yes—
149C2×C71⋊C8C2412yes—
150C2×C71⋊C8C2412yes—
151C8×D71—28no—
152C8×D71—28no—
153C4⋊Dic71C22×C414yes—
154C4⋊Dic71C22×C418yes—
155C4⋊Dic71C22×C414yes—
156Dic71⋊C4C22×C418yes—
157Dic71⋊C4C22×C414yes—
158Dic71⋊C4C22×C414yes—
159Dic71⋊C4C22×C418yes—
160Dic71⋊C4C22×C418yes—
161Dic71⋊C4C22×C418yes—
162C22⋊Dic71C22×C414yes—
163(C2×D71)⋊C4C22×C418yes—
164(C2×D71)⋊C4C22×C414yes—
165(C2×D71)⋊C4C22×C414yes—
166C4×Dic71C22×C418yes—
167C4×Dic71C22×C412yes—
168C4×Dic71C2414yes—
169C4×Dic71C2412yes—
170C2×Q8×C71C22×C414yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2411yescharacteristic
173M16×C71—24no—
174C2×C8×C71—22no—
175C2×C8×C71C2×Q814yes—
176C2×C8×C71C2412yes—
177C2×C8×C71C2414yes—
178C2×C8×C71C2411yescharacteristic
179C2×C8×C71C2411yescharacteristic
180C4⋊C4×C71C22×C414yes—
181C4⋊C4×C71C22×C418yes—
182C4⋊C4×C71C22×C414yes—
183C22⋊C4×C71C22×C414yes—
184C42×C71C22×C411yescharacteristic
185C42×C71C22×C411yescharacteristic
186C42×C71C2412yes—
187C42×C71C2414yes—
188C23×C71C23×C411yescharacteristic
189C22×D71—42no—
190C22×D71C4×Q814yes—
191C22×D71C23×C412yes—
192Q8×C71—44no—
193C2×C4×C71—42no—
194C2×C4×C71C23×C412yes—
195C2×C4×C71C23×C412yes—
196C2×C4×C71C23×C418yes—
197C2×C4×C71C23×C411yescharacteristic
198C2×C4×C71C22×Q812yes—
199C2×C4×C71C2×#4112yes—
200C2×C4×C71C2511yescharacteristic
201C2×Dic71—42no—
202C2×Dic71C4×Q814yes—
203C2×Dic71C23×C414yes—
204C2×Dic71C23×C418yes—
205C2×Dic71C23×C412yes—
206C4×D71—48no—
207C4×D71C4×Q818yes—
208C4×D71C22×Q814yes—
209C4×D71C2×#4114yes—
210Dic142—44no—
211Dic142—48no—
212C8×C71—22no—
213C8×C71—24no—
214C8×C71C22×Q812yes—
215C8×C71C22×Q812yes—
216C71⋊C8—22no—
217C71⋊C8—24no—
218C71⋊C8C22×Q814yes—
219C22×C71—22no—
220C22×C71C2×C4×Q812yes—
221C22×C71C24×C411yescharacteristic
222C2×D71—24no—
223C2×D71—44no—
224C2×D71C2×C4×Q814yes—
225C4×C71—22no—
226C4×C71—28no—
227C4×C71—22no—
228C4×C71C2×C4×Q812yes—
229C4×C71C23×Q811yescharacteristic
230C4×C71C22×#4111yescharacteristic
231Dic71—28no—
232Dic71—24no—
233Dic71C2×C4×Q814yes—
234#60193—711nomaximal, Sylow
235C2×C71—22no—
236C2×C71C22×C4×Q811yescharacteristic
237C2×C71#357312yes—
238D71—24no—
239#3458—711no—
240#2962—718no—
241#2962—718no—
242#2961—711no—
243#2961—712no—
244#2961—711no—
245#2960—712no—
246#2960—711no—
247#2960D7111yescharacteristic, complemented
248#2905—711no—
249C2×#424—712no—
250#2873—712no—
251C2×#405—711no—
252C71#6019311yescharacteristic, complemented, Sylow
253C22×M16—712no—
254C22×M16—711no—
255C23×C8—711no—
256#544—714no—
257#544—714no—
258#544—714no—
259#543—714no—
260#533—718no—
261#533—712no—
262#533—711no—
263#533C2×D7111yescharacteristic
264C2×#117—712no—
265C2×#117—711no—
266#446—718no—
267#446—718no—
268#446—718no—
269#446—718no—
270#446—714no—
271#446—718no—
272#446C2×D7114yescomplemented
273C8×Q8—718no—
274C8×Q8—714no—
275C8×Q8C2×D7114yescomplemented
276#424—712no—
277#424—712no—
278#424—712no—
279#424—718no—
280#424—718no—
281#424C2×D7112yes—
282#423—712no—
283#423—711no—
284#423C2×D7111yescharacteristic
285C2×C4⋊C8—718no—
286#405—712no—
287#405—712no—
288#405—712no—
289#405—714no—
290#405—714no—
291#405—714no—
292#405—714no—
293#405C2×D7112yes—
294C4×M16—714no—
295C4×M16—714no—
296C2×#97—711no—
297C2×#97—711no—
298C2×C4×C8—711no—
299C2×C4×C8C2×D7111yescharacteristic
300C23×C4—711no—
301C2×M16—712no—
302C2×M16—718no—
303C2×M16—712no—
304C2×M16—712no—
305C2×M16—714no—
306C2×M16—1428no—
307C2×M16C22×D7112yes—
308C22×C8—711no—
309C22×C8—711no—
310C22×C8—714no—
311C22×C8—714no—
312C22×C8—1424no—
313C22×C8C22×D7111yescharacteristic
314C22×C8C22×D7111yescharacteristic
315C4⋊Q8—718no—
316C4⋊Q8—718no—
317C4⋊Q8—714no—
318C4⋊Q8—712no—
319C4⋊Q8C4×D7112yes—
320#125—718no—
321#125—718no—
322#125—714no—
323#125—712no—
324#125C4×D7112yes—
325C22⋊Q8—718no—
326C22⋊Q8—714no—
327C22⋊Q8—718no—
328C22⋊Q8—718no—
329C22⋊Q8—718no—
330C22⋊Q8—718no—
331C22⋊Q8C4×D7114yes—
332C4×Q8—718no—
333C4×Q8—714no—
334C4×Q8C22×D7114yes—
335#117—712no—
336#117—714no—
337#117—714no—
338#117—712no—
339#117—712no—
340#117—714no—
341#117—714no—
342#117C22×D7112yes—
343C2×C4⋊C4—712no—
344C2×C4⋊C4—714no—
345C2×C22⋊C4—712no—
346C2×C22⋊C4—714no—
347C2×C42—711no—
348C2×C42—714no—
349C2×C42C22×D7111yescharacteristic
350C4⋊C8—718no—
351C4⋊C8—718no—
352C4⋊C8—718no—
353C4⋊C8—718no—
354C4⋊C8—714no—
355C4⋊C8C22×D7118yes—
356C4⋊C8C22×D7114yes—
357#97—712no—
358#97—712no—
359#97—714no—
360#97—714no—
361#97—714no—
362#97—714no—
363#97C22×D7114yes—
364C4×C8—714no—
365C4×C8—714no—
366C4×C8—714no—
367C4×C8—714no—
368C4×C8C22×D7112yes—
369C4×C8C22×D7112yes—
370C4×C8C22×D7114yes—
371C24—711no—
372C2×Q8—718no—
373C2×Q8—714no—
374C2×Q8C2×C4×D7114yes—
375C22×C4—712no—
376C22×C4—718no—
377C22×C4—711no—
378C22×C4—712no—
379C22×C4—714no—
380C22×C4—714no—
381C22×C4—712no—
382C22×C4C2×C4×D7112yes—
383C22×C4C23×D7111yescharacteristic
384M16—24no—
385M16—718no—
386M16—718no—
387M16—1428no—
388M16—1424no—
389C2×C8—22no—
390C2×C8—714no—
391C2×C8—712no—
392C2×C8—714no—
393C2×C8—712no—
394C2×C8—1428no—
395C2×C8—1428no—
396C2×C8—1422no—
397C2×C8Q8×D7114yes—
398C2×C8C23×D7112yes—
399C2×C8C23×D7114yes—
400C2×C8C23×D7111yescharacteristic
401C2×C8C23×D7111yescharacteristic
402C4⋊C4—718no—
403C4⋊C4—714no—
404C4⋊C4—714no—
405C4⋊C4—718no—
406C4⋊C4—718no—
407C4⋊C4—718no—
408C4⋊C4—714no—
409C4⋊C4—718no—
410C4⋊C4—714no—
411C4⋊C4C2×C4×D7114yes—
412C4⋊C4C2×C4×D7114yes—
413C4⋊C4C2×C4×D7118yes—
414C22⋊C4—714no—
415C22⋊C4—718no—
416C22⋊C4—714no—
417C22⋊C4—714no—
418C22⋊C4C2×C4×D7114yes—
419C42—718no—
420C42—712no—
421C42—714no—
422C42—712no—
423C42C2×C4×D7111yescharacteristic
424C42C2×C4×D7111yescharacteristic
425C42C23×D7114yes—
426C42C23×D7112yes—
427C23—2842no—
428C23—714no—
429C23—712no—
430C23C22×C4×D7111yescharacteristic
431Q8—44no—
432Q8—2844no—
433Q8—2848no—
434C2×C4—42no—
435C2×C4—2848no—
436C2×C4—2842no—
437C2×C4—712no—
438C2×C4—714no—
439C2×C4—714no—
440C2×C4—714no—
441C2×C4—718no—
442C2×C4—718no—
443C2×C4—714no—
444C2×C4C22×C4×D7118yes—
445C2×C4C22×C4×D7112yes—
446C2×C4C22×C4×D7112yes—
447C2×C4C22×C4×D7111yescharacteristic
448C2×C4C2×Q8×D7112yes—
449C2×C4#41×D7112yes—
450C2×C4C24×D7111yescharacteristic
451C8—22no—
452C8—24no—
453C8—714no—
454C8—1422no—
455C8—1424no—
456C8C2×Q8×D7112yes—
457C8C2×Q8×D7112yes—
458C22—22no—
459C22—2844no—
460C22—714no—
461C22—1424no—
462C22C4×Q8×D7112yes—
463C22C23×C4×D7111yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—714no—
468C4—1428no—
469C4—1424no—
470C4C4×Q8×D7112yes—
471C4C22×Q8×D7111yescharacteristic
472C4C2×#41×D7111yescharacteristic
473C2—22no—
474C2—1424no—
475C2C2×C4×Q8×D7111yescharacteristic
476C2#2260305112yes—
subgroupG/H
center, FrattiniC2×C4C24×D71
commutator, socleC22×C71C24×C4
FittingC71×#2960C2
prime radicalC23×D71C22×C4
cosocleC2×C4×C71C25
hypercenter#2960D71
hypocenterC71#60193
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #2960

Complements

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

NH ≅ G/Ntype
2. #22597285C2semidirect
3. #22597282C2semidirect
4. #22597281C2semidirect
5. #22597278C2semidirect
6. #22597277C2semidirect
7. #22597274C2semidirect
8. #22597273C2semidirect
13. C71×#2960C2semidirect
26. #17986130C22semidirect
27. #17986129C22semidirect
28. #17986129C22semidirect
29. #17986128C22semidirect
30. #17986128C22semidirect
31. #17986127C22semidirect
32. C8×Dic142C22semidirect
33. #17986123C22semidirect
55. #446×C71C22semidirect
56. C8×Q8×C71C22semidirect
63. C2×#11765848Q8semidirect
247. #2960D71semidirect
252. C71#60193semidirect
272. #446C2×D71semidirect
275. C8×Q8C2×D71semidirect

Maximal subgroups

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

subgroupindex
#226017432
#225972852
#225972822
#225972812
#225972782
#225972772
#225972742
#225972732
#225969022
#424×D712
#225967562
C2×#179859532
C71×#29602
#6019371