Finite Group Databasebeta

← #46845578

Subgroups — #46845578

Order 34688.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D271C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3#31774404C211yesmaximal, characteristic, complemented
4#31774058C211yesmaximal, characteristic, complemented
5#31774056C211yesmaximal, characteristic, complemented
6#31774051C211yesmaximal, characteristic, complemented
7#31774046C211yesmaximal, characteristic, complemented
8#31774041C211yesmaximal, characteristic, complemented
9#31774039C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11#31774029C211yesmaximal, characteristic, complemented
12#31773894C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14C2×(C2×D271)⋊C8C211yesmaximal, characteristic, complemented
15C2×#135×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864654C2211yescharacteristic
19C2×#16864653—21no—
20C2×#16864653C2211yescharacteristic
21#41×D271—24no—
22#41×D271—24no—
23C2×Q8×D271C2211yescharacteristic
24C2×D4×D271—21no—
25C2×D4×D271C2211yescharacteristic
26C22×C4×D271—21no—
27C22×C4×D271C2211yescharacteristic
28#22576180—24no—
29C2×#16864644C2211yescharacteristic, complemented
30C2×#16864642C2211yescharacteristic, complemented
31C2×#16864641C2211yescharacteristic, complemented
32C2×#16864639C2211yescharacteristic, complemented
33C22×C271⋊C8C2211yescharacteristic, complemented
34#22576141—21no—
35#22576136C2211yescharacteristic
36#22576136C2211yescharacteristic, complemented
37#22576131C2211yescharacteristic
38#22576131C2211yescharacteristic, complemented
39#22576128—21no—
40C4×C22⋊D271—21no—
41C4×C22⋊D271—21no—
42#22576119C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576084C2211yescharacteristic, complemented
47#22576082C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53(C2×D271)⋊C8C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55C2×#41×C271C2211yescharacteristic
56C2×#41×C271C2211yescharacteristic, complemented
57#135×C271—24no—
58C2×Q16×C271C2211yescharacteristic, complemented
59C2×SD16×C271C2211yescharacteristic, complemented
60C2×SD16×C271C2211yescharacteristic, complemented
61C2×D8×C271C2211yescharacteristic, complemented
62C22×C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271D411yescharacteristic
65#16864655—22no—
66#16864655—22no—
67#16864654—28no—
68#16864654—22no—
69#16864654D414yescomplemented
70#16864653—28no—
71#16864653—22no—
72#16864653—22no—
73C2×Dic542—21no—
74C2×Dic542C2311yescharacteristic
75Q8×D271—22no—
76Q8×D271—22no—
77C2×D1084—21no—
78C2×D1084C2311yescharacteristic
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81C2×C22⋊D271—21no—
82C2×C22⋊D271—21no—
83C2×C22⋊D271—21no—
84D4×D271—28no—
85D4×D271—22no—
86D4×D271—22no—
87C22×Dic271—21no—
88C22×Dic271D411yescharacteristic
89C2×C4×D271—28no—
90C2×C4×D271—21no—
91C2×C4×D271—21no—
92C2×C4×D271—21no—
93C2×C4×D271—22no—
94C2×C4×D271—22no—
95C2×C4×D271D411yescharacteristic
96C2×C4×D271D411yescharacteristic
97C2×C4×D271C2311yescharacteristic
98C2×C4×D271C2311yescharacteristic
99#16864644—22no—
100#16864642—22no—
101#16864641—22no—
102#16864639—22no—
103C2×C271⋊C8—22no—
104C2×C271⋊C8C2311yescharacteristic
105C2×C271⋊C8C2311yescharacteristic
106C4⋊Dic271C2311yescharacteristic
107C4⋊Dic271C2311yescharacteristic
108Dic271⋊C4—21no—
109Dic271⋊C4—21no—
110C22⋊Dic271—21no—
111C22⋊Dic271—21no—
112C22⋊Dic271—21no—
113(C2×D271)⋊C4—21no—
114(C2×D271)⋊C4—21no—
115C4×Dic271—21no—
116#41×C271—22no—
117#41×C271—22no—
118#41×C271—22no—
119#41×C271D414yescomplemented
120C2×Q8×C271C2311yescharacteristic
121C2×Q8×C271C2311yescharacteristic
122C2×D4×C271—21no—
123C2×D4×C271—21no—
124C2×D4×C271C2311yescharacteristic
125C2×D4×C271C2311yescharacteristic
126C22×C4×C271—21no—
127C22×C4×C271—21no—
128C22×C4×C271C2311yescharacteristic
129Q16×C271—22no—
130SD16×C271—22no—
131SD16×C271—22no—
132D8×C271—22no—
133C2×C8×C271—22no—
134C2×C8×C271C2311yescharacteristic
135C2×C8×C271C2311yescharacteristic
136C22×D271—24no—
137C22×D271—22no—
138C22×D271—21no—
139C22×D271—22no—
140C22×D271—41no—
141C22×D271—42no—
142C22×D271—42no—
143C22×D271C2×D411yescharacteristic
144C22×D271C2×D411yescharacteristic
145Dic542—24no—
146Dic542—21no—
147Dic542C2×D412yes—
148D1084—24no—
149D1084—21no—
150D1084C2×D412yes—
151C22⋊D271—24no—
152C22⋊D271—24no—
153C22⋊D271—24no—
154C22⋊D271—42no—
155C22⋊D271—42no—
156C2×Dic271—24no—
157C2×Dic271—22no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271C2×D411yescharacteristic
162C2×Dic271C2×D411yescharacteristic
163C4×D271—24no—
164C4×D271—24no—
165C4×D271—22no—
166C4×D271—24no—
167C4×D271—22no—
168C4×D271—21no—
169C4×D271—21no—
170C4×D271—21no—
171C4×D271—21no—
172C271⋊C8—21no—
173C271⋊C8—21no—
174C23×C271—21no—
175C23×C271—21no—
176C23×C271C2×D411yescharacteristic
177Q8×C271—21no—
178Q8×C271—21no—
179Q8×C271—21no—
180Q8×C271C2×D412yes—
181D4×C271—24no—
182D4×C271—21no—
183D4×C271—21no—
184D4×C271—21no—
185D4×C271—42no—
186D4×C271C2×D412yes—
187C2×C4×C271—21no—
188C2×C4×C271—24no—
189C2×C4×C271—21no—
190C2×C4×C271—42no—
191C2×C4×C271C2×D412yes—
192C2×C4×C271C2×D411yescharacteristic
193C2×C4×C271C2×D412yes—
194C2×C4×C271C2411yescharacteristic
195C8×C271—21no—
196C8×C271—21no—
197C22×C271—21no—
198C22×C271—21no—
199C22×C271—22no—
200C22×C271—41no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271#12012yes—
206C22×C271C22×D411yescharacteristic
207C2×D271—21no—
208C2×D271—21no—
209C2×D271—21no—
210C2×D271—22no—
211C2×D271—21no—
212C2×D271—21no—
213C2×D271—44no—
214C2×D271—48no—
215C2×D271—41no—
216C2×D271—41no—
217C2×D271#13512yescomplemented
218C4×C271—22no—
219C4×C271—41no—
220C4×C271#12012yes—
221C4×C271C22×D411yescharacteristic
222C4×C271C22×D411yescharacteristic
223Dic271—21no—
224Dic271—22no—
225Dic271—21no—
226Dic271—41no—
227Dic271—41no—
228C2×C271—22no—
229C2×C271—42no—
230C2×C271—81no—
231C2×C271C2×#12011yescharacteristic
232C2×C271C2×#13511yescharacteristic
233C2×C271#57511yescharacteristic
234D271—22no—
235D271—41no—
236D271—42no—
237C271#300211yescharacteristic, complemented, Sylow
238#3002—2711nomaximal, Sylow
239C22×#41—2711no—
240C2×#135—2711no—
241C2×#135D27111yescharacteristic, complemented
242#525—2711no—
243#452—2711no—
244#452—2711no—
245#451—2711no—
246#450—2711no—
247#449—2711no—
248#449—2711no—
249#448—2711no—
250#447—2711no—
251#416—2711no—
252#416—2711no—
253C2×C22⋊C8—2711no—
254C2×#41—2711no—
255C2×#41—2711no—
256C2×#41—2711no—
257C2×#41—5421no—
258C2×#41—5421no—
259C2×#41—5424no—
260C2×#41—5424no—
261C2×#41C2×D27111yescharacteristic
262C2×#41C2×D27111yescharacteristic, complemented
263C22×Q8—2711no—
264C22×D4—2711no—
265C22×D4—5421no—
266C23×C4—2711no—
267C23×C4—5421no—
268#135—24no—
269#135—5424no—
270C2×Q16—2711no—
271C2×Q16C2×D27111yescharacteristic, complemented
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16C2×D27111yescharacteristic, complemented
275C2×SD16C2×D27111yescharacteristic, complemented
276C2×D8—2711no—
277C2×D8C2×D27111yescharacteristic, complemented
278C22×C8—2711no—
279C22×C8C2×D27111yescharacteristic, complemented
280#123—5421no—
281C22⋊Q8—2711no—
282C22⋊Q8—2711no—
283#121—2711no—
284#121—2711no—
285#120—5421no—
286C4×D4—5421no—
287C4×D4—5421no—
288#117—2711no—
289#103—2711no—
290#103—2711no—
291#103—2711no—
292#103—2711no—
293#102—2711no—
294#102—2711no—
295#102—2711no—
296#102—2711no—
297C22⋊C8—2711no—
298C22⋊C8—2711no—
299C22⋊C8—2711no—
300C22⋊C8—2711no—
301C24—2711no—
302C24—5421no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—2714no—
307#41—5428no—
308#41—5422no—
309#41—5428no—
310#41—5422no—
311#41—5422no—
312#41—5422no—
313#41—5422no—
314#41C22⋊D27114yescomplemented
315C2×Q8—2711no—
316C2×Q8—5421no—
317C2×Q8—5422no—
318C2×Q8—5422no—
319C2×Q8C22×D27111yescharacteristic
320C2×Q8C22×D27111yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—2711no—
324C2×D4—5421no—
325C2×D4—5421no—
326C2×D4—5422no—
327C2×D4—5428no—
328C2×D4—5421no—
329C2×D4—5421no—
330C2×D4—5422no—
331C2×D4—5421no—
332C2×D4—5421no—
333C2×D4C22×D27111yescharacteristic
334C2×D4C22×D27111yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—2711no—
338C22×C4—2711no—
339C22×C4—2711no—
340C22×C4—2711no—
341C22×C4—2711no—
342C22×C4—5421no—
343C22×C4—5421no—
344C22×C4—5428no—
345C22×C4—5421no—
346C22×C4—5422no—
347C22×C4—5421no—
348C22×C4—5422no—
349C22×C4C22×D27111yescharacteristic
350Q16—22no—
351Q16—5422no—
352SD16—22no—
353SD16—22no—
354SD16—5422no—
355SD16—5422no—
356D8—22no—
357D8—5422no—
358C2×C8—22no—
359C2×C8—2711no—
360C2×C8—2711no—
361C2×C8—5422no—
362C2×C8C22×D27111yescharacteristic
363C2×C8C22×D27111yescharacteristic
364C4⋊C4—2711no—
365C4⋊C4—2711no—
366C4⋊C4—5421no—
367C4⋊C4—5421no—
368C22⋊C4—5421no—
369C22⋊C4—5421no—
370C22⋊C4—5421no—
371C22⋊C4—5421no—
372C22⋊C4—5421no—
373C42—5421no—
374C23—21no—
375C23—21no—
376C23—2711no—
377C23—2711no—
378C23—5422no—
379C23—5424no—
380C23—5422no—
381C23—5421no—
382C23—10841no—
383C23—10842no—
384C23—10842no—
385C23D4×D27111yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—2712no—
390Q8—5424no—
391Q8—5421no—
392Q8C2×C22⋊D27112yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—2712no—
399D4—5424no—
400D4—5424no—
401D4—5424no—
402D4—5424no—
403D4—5421no—
404D4—10842no—
405D4—10842no—
406D4C2×C22⋊D27112yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—2711no—
412C2×C4—2711no—
413C2×C4—5424no—
414C2×C4—5424no—
415C2×C4—5424no—
416C2×C4—5422no—
417C2×C4—5421no—
418C2×C4—5422no—
419C2×C4—5422no—
420C2×C4—5424no—
421C2×C4—5421no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4D4×D27112yes—
428C2×C4D4×D27111yescharacteristic
429C2×C4C2×C22⋊D27112yes—
430C2×C4C23×D27111yescharacteristic
431C8—21no—
432C8—21no—
433C8—5421no—
434C8—5421no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—2712no—
444C22—5421no—
445C22—5421no—
446C22—5422no—
447C22—5421no—
448C22—5421no—
449C22—5421no—
450C22—10844no—
451C22—10848no—
452C22—10841no—
453C22—10841no—
454C22#2257612812yes—
455C22C2×D4×D27111yescharacteristic
456C4—22no—
457C4—41no—
458C4—5421no—
459C4—5421no—
460C4—5422no—
461C4—10841no—
462C4—10841no—
463C4#2257612812yes—
464C4C2×D4×D27111yescharacteristic
465C4C22×C22⋊D27111yescharacteristic
466C2—22no—
467C2—42no—
468C2—81no—
469C2—5422no—
470C2—10841no—
471C2—10842no—
472C2C2×#2257612811yescharacteristic
473C2#135×D27111yescharacteristic
474C2#3177469411yescharacteristic
subgroupG/H
centerC2×C4D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#135×C271C2
socleC22×C271C22×D4
hypercenterC2×#135D271
hypocenterC271#3002
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#41×D271C2semidirect
2. C2×#22576180C2semidirect
3. #31774404C2semidirect
4. #31774058C2semidirect
5. #31774056C2semidirect
6. #31774051C2semidirect
7. #31774046C2semidirect
8. #31774041C2semidirect
9. #31774039C2semidirect
10. #31774034C2semidirect
11. #31774029C2semidirect
12. #31773894C2semidirect
13. #31773893C2semidirect
14. C2×(C2×D271)⋊C8C2semidirect
15. C2×#135×C271C2semidirect
29. C2×#16864644C22semidirect
30. C2×#16864642C22semidirect
31. C2×#16864641C22semidirect
32. C2×#16864639C22semidirect
33. C22×C271⋊C8C22semidirect
36. #22576136C22semidirect
38. #22576131C22semidirect
42. #22576119C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
45. #22576084C22semidirect
46. #22576084C22semidirect
47. #22576082C22semidirect
48. #22576082C22semidirect
49. #22576081C22semidirect
50. #22576081C22semidirect
53. (C2×D271)⋊C8C22semidirect
54. (C2×D271)⋊C8C22semidirect
56. C2×#41×C271C22semidirect
58. C2×Q16×C271C22semidirect
59. C2×SD16×C271C22semidirect
60. C2×SD16×C271C22semidirect
61. C2×D8×C271C22semidirect
62. C22×C8×C271C22semidirect
69. #16864654D4semidirect
119. #41×C271D4semidirect
217. C2×D271#135semidirect
237. C271#3002semidirect
241. C2×#135D271semidirect
262. C2×#41C2×D271semidirect
271. C2×Q16C2×D271semidirect
274. C2×SD16C2×D271semidirect
275. C2×SD16C2×D271semidirect
277. C2×D8C2×D271semidirect
279. C22×C8C2×D271semidirect
314. #41C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2712
C2×#225761802
#317744042
#317740582
#317740562
#317740512
#317740462
#317740412
#317740392
#317740342
#317740292
#317738942
#317738932
C2×(C2×D271)⋊C82
C2×#135×C2712
#3002271