Finite Group Databasebeta

← #46845584

Subgroups — #46845584

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×#22576182C211yesmaximal, characteristic, complemented
3C2×#22576180C211yesmaximal, characteristic, complemented
4#31774409C211yesmaximal, characteristic, complemented
5#31774062C211yesmaximal, characteristic, complemented
6#31774060C211yesmaximal, characteristic, complemented
7#31774054C211yesmaximal, characteristic, complemented
8#31774049C211yesmaximal, characteristic, complemented
9#31774045C211yesmaximal, characteristic, complemented
10#31774043C211yesmaximal, characteristic, complemented
11#31774037C211yesmaximal, characteristic, complemented
12#31774032C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14C2×(C2×D271)⋊C8C211yesmaximal, characteristic, complemented
15#416×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic, complemented
20C2×#16864653—21no—
21C2×#16864653C2211yescharacteristic
22#41×D271—24no—
23#41×D271—24no—
24C2×Q8×D271C2211yescharacteristic
25C2×D4×D271—21no—
26C2×D4×D271C2211yescharacteristic
27C22×C4×D271—21no—
28C22×C4×D271C2211yescharacteristic
29#22576182—24no—
30#22576180—24no—
31C2×Dic1084C2211yescharacteristic, complemented
32C2×#16864644C2211yescharacteristic, complemented
33C2×#16864643C2211yescharacteristic, complemented
34C2×#16864643C2211yescharacteristic, complemented
35C2×#16864642C2211yescharacteristic, complemented
36C2×#16864641C2211yescharacteristic, complemented
37C2×D2168C2211yescharacteristic, complemented
38C2×#16864639C2211yescharacteristic, complemented
39C22×C271⋊C8C2211yescharacteristic, complemented
40#22576144—21no—
41#22576140C2211yescharacteristic
42#22576140C2211yescharacteristic, complemented
43#22576135C2211yescharacteristic
44#22576135C2211yescharacteristic, complemented
45#22576130—21no—
46C4×D1084—21no—
47C4×D1084—21no—
48#22576083C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53(C2×D271)⋊C8C2211yescharacteristic
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C2×#41×C271C2211yescharacteristic
57C22×C8×C271C2211yescharacteristic, complemented
58#117×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62#102×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271D411yescharacteristic
65#16864655—22no—
66#16864655—22no—
67#16864654—28no—
68#16864654—22no—
69#16864654—22no—
70#16864654—22no—
71#16864654D414yescomplemented
72#16864653—28no—
73#16864653—22no—
74#16864653—22no—
75C2×Dic542—21no—
76C2×Dic542C2311yescharacteristic
77C2×Dic542C2311yescharacteristic
78Q8×D271—22no—
79Q8×D271—22no—
80C2×D1084—21no—
81C2×D1084—21no—
82C2×D1084—21no—
83C2×D1084C2311yescharacteristic
84C2×D1084C2311yescharacteristic
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87C2×C22⋊D271—21no—
88C2×C22⋊D271—21no—
89D4×D271—28no—
90D4×D271—22no—
91D4×D271—22no—
92C22×Dic271—21no—
93C22×Dic271D411yescharacteristic
94C2×C4×D271—28no—
95C2×C4×D271—21no—
96C2×C4×D271—21no—
97C2×C4×D271—21no—
98C2×C4×D271—22no—
99C2×C4×D271—22no—
100C2×C4×D271—21no—
101C2×C4×D271D411yescharacteristic
102C2×C4×D271D411yescharacteristic
103C2×C4×D271C2311yescharacteristic
104C2×C4×D271C2311yescharacteristic
105Dic1084—22no—
106#16864644—22no—
107#16864643—22no—
108#16864643—22no—
109#16864642—22no—
110#16864641—22no—
111D2168—22no—
112#16864639—22no—
113C2×C271⋊C8—22no—
114C2×C271⋊C8C2311yescharacteristic
115C2×C271⋊C8C2311yescharacteristic
116C4⋊Dic271—21no—
117C4⋊Dic271—21no—
118(C2×D271)⋊C4—21no—
119(C2×D271)⋊C4—21no—
120(C2×D271)⋊C4—21no—
121(C2×D271)⋊C4—21no—
122#41×C271—22no—
123#41×C271D414yescomplemented
124C2×Q8×C271C2311yescharacteristic
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C22×C4×C271—21no—
128C22×C4×C271C2311yescharacteristic
129C2×C8×C271—22no—
130C2×C8×C271C2311yescharacteristic
131C2×C8×C271C2311yescharacteristic
132C4⋊C4×C271C2311yescharacteristic
133C4⋊C4×C271C2311yescharacteristic
134C22⋊C4×C271—21no—
135C42×C271—21no—
136C22×D271—24no—
137C22×D271—22no—
138C22×D271—21no—
139C22×D271—22no—
140C22×D271—21no—
141C22×D271—41no—
142C22×D271—42no—
143C22×D271—42no—
144C22×D271C2×D411yescharacteristic
145C22×D271C2×D411yescharacteristic
146Dic542—24no—
147Dic542—21no—
148Dic542—21no—
149Dic542—21no—
150Dic542C2×D412yes—
151D1084—24no—
152D1084—21no—
153D1084—21no—
154D1084—21no—
155D1084—42no—
156D1084—42no—
157D1084C2×D412yes—
158C22⋊D271—24no—
159C22⋊D271—24no—
160C22⋊D271—24no—
161C22⋊D271—42no—
162C2×Dic271—24no—
163C2×Dic271—22no—
164C2×Dic271—21no—
165C2×Dic271—21no—
166C2×Dic271C2×D411yescharacteristic
167C2×Dic271C2×D411yescharacteristic
168C4×D271—24no—
169C4×D271—24no—
170C4×D271—22no—
171C4×D271—24no—
172C4×D271—22no—
173C4×D271—21no—
174C4×D271—21no—
175C4×D271—21no—
176C4×D271—21no—
177C4×D271—42no—
178C271⋊C8—21no—
179C271⋊C8—21no—
180C23×C271—21no—
181C23×C271C2×D411yescharacteristic
182Q8×C271—21no—
183Q8×C271C2×D412yes—
184D4×C271—24no—
185D4×C271—21no—
186D4×C271C2×D412yes—
187C2×C4×C271—21no—
188C2×C4×C271—24no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
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#12012yes—
203C22×C271C22×D411yescharacteristic
204C2×D271—21no—
205C2×D271—21no—
206C2×D271—21no—
207C2×D271—22no—
208C2×D271—21no—
209C2×D271—21no—
210C2×D271—44no—
211C2×D271—48no—
212C2×D271—41no—
213C2×D271—41no—
214C2×D271—41no—
215C2×D271—41no—
216C2×D271—41no—
217C2×D271#13512yescomplemented
218C4×C271—22no—
219C4×C271—41no—
220C4×C271—41no—
221C4×C271#12012yes—
222C4×C271C22×D411yescharacteristic
223C4×C271C22×D411yescharacteristic
224Dic271—21no—
225Dic271—22no—
226Dic271—21no—
227Dic271—41no—
228C2×C271—22no—
229C2×C271—42no—
230C2×C271C2×#12011yescharacteristic
231C2×C271C2×#13511yescharacteristic
232C2×C271#57511yescharacteristic
233D271—22no—
234D271—41no—
235D271—42no—
236D271—81no—
237C271#300211yescharacteristic, complemented, Sylow
238#3002—2711nomaximal, Sylow
239C22×#41—2711no—
240C2×#135—2711no—
241C2×#135—2711no—
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#416D27111yescharacteristic, complemented
253C2×C22⋊C8—2711no—
254C2×#41—2711no—
255C2×#41—2711no—
256C2×#41—2711no—
257C2×#41—2711no—
258C2×#41—5421no—
259C2×#41—5421no—
260C2×#41—5424no—
261C2×#41—5424no—
262C2×#41C2×D27111yescharacteristic
263C22×Q8—2711no—
264C22×D4—2711no—
265C22×D4—5421no—
266C23×C4—2711no—
267C23×C4—5421no—
268#135—5424no—
269#135—5424no—
270C2×Q16—2711no—
271C2×Q16—2711no—
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16—2711no—
275C2×SD16—2711no—
276C2×D8—2711no—
277C2×D8—2711no—
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#117C2×D27111yescharacteristic, complemented
289#103—2711no—
290#103—2711no—
291#103C2×D27111yescharacteristic, complemented
292#103C2×D27111yescharacteristic, complemented
293#102—2711no—
294#102—2711no—
295#102C2×D27111yescharacteristic, complemented
296#102C2×D27111yescharacteristic, complemented
297C22⋊C8—2711no—
298C22⋊C8—2711no—
299C22⋊C8—2711no—
300C22⋊C8—2711no—
301C24—2711no—
302C24—5421no—
303#41—22no—
304#41—2714no—
305#41—5428no—
306#41—5422no—
307#41—5428no—
308#41—5422no—
309#41—5422no—
310#41—5422no—
311#41—5422no—
312#41—5422no—
313#41—5422no—
314#41D108414yescomplemented
315C2×Q8—2711no—
316C2×Q8—2711no—
317C2×Q8—5421no—
318C2×Q8—5422no—
319C2×Q8—5422no—
320C2×Q8C22×D27111yescharacteristic
321C2×D4—21no—
322C2×D4—2711no—
323C2×D4—2711no—
324C2×D4—5421no—
325C2×D4—5421no—
326C2×D4—5421no—
327C2×D4—5421no—
328C2×D4—5422no—
329C2×D4—5428no—
330C2×D4—5421no—
331C2×D4—5421no—
332C2×D4—5422no—
333C2×D4—5421no—
334C2×D4C22×D27111yescharacteristic
335C22×C4—21no—
336C22×C4—2711no—
337C22×C4—2711no—
338C22×C4—2711no—
339C22×C4—2711no—
340C22×C4—2711no—
341C22×C4—5421no—
342C22×C4—5421no—
343C22×C4—5428no—
344C22×C4—5421no—
345C22×C4—5422no—
346C22×C4—5421no—
347C22×C4—5422no—
348C22×C4—5421no—
349C22×C4C22×D27111yescharacteristic
350Q16—5422no—
351Q16—5422no—
352SD16—5422no—
353SD16—5422no—
354SD16—5422no—
355SD16—5422no—
356D8—5422no—
357D8—5422no—
358C2×C8—22no—
359C2×C8—2711no—
360C2×C8—2711no—
361C2×C8—5422no—
362C2×C8C22×D27111yescharacteristic
363C2×C8C22×D27111yescharacteristic
364C4⋊C4—5421no—
365C4⋊C4—5421no—
366C4⋊C4C22×D27111yescharacteristic
367C4⋊C4C22×D27111yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5421no—
370C22⋊C4—5421no—
371C22⋊C4—5421no—
372C22⋊C4—5421no—
373C42—21no—
374C23—21no—
375C23—2711no—
376C23—2711no—
377C23—5422no—
378C23—5424no—
379C23—5422no—
380C23—5421no—
381C23—5421no—
382C23—10841no—
383C23—10842no—
384C23—10842no—
385C23D4×D27111yescharacteristic
386Q8—21no—
387Q8—2712no—
388Q8—5424no—
389Q8—5421no—
390Q8—5421no—
391Q8—5421no—
392Q8C2×D108412yes—
393D4—24no—
394D4—21no—
395D4—2712no—
396D4—5424no—
397D4—5424no—
398D4—5424no—
399D4—5424no—
400D4—5421no—
401D4—5421no—
402D4—5421no—
403D4—10842no—
404D4—10842no—
405D4—10842no—
406D4C2×D108412yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
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—10842no—
427C2×C4D4×D27112yes—
428C2×C4D4×D27111yescharacteristic
429C2×C4C2×D108412yes—
430C2×C4C23×D27111yescharacteristic
431C8—21no—
432C8—21no—
433C8—5421no—
434C8—5421no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—2712no—
441C22—5421no—
442C22—5421no—
443C22—5422no—
444C22—5421no—
445C22—5421no—
446C22—5421no—
447C22—10844no—
448C22—10848no—
449C22—10841no—
450C22—10841no—
451C22—10841no—
452C22—10841no—
453C22—10841no—
454C22#2257613012yes—
455C22C2×D4×D27111yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—5421no—
460C4—5421no—
461C4—5422no—
462C4—10841no—
463C4#2257613012yes—
464C4C2×D4×D27111yescharacteristic
465C4C22×D108411yescharacteristic
466C2—22no—
467C2—42no—
468C2—5422no—
469C2—10841no—
470C2—10842no—
471C2—21681no—
472C2C2×#2257613011yescharacteristic
473C2#135×D27111yescharacteristic
474C2#3177469411yescharacteristic
subgroupG/H
centerC2×C4D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#416×C271C2
socleC22×C271C22×D4
hypercenter#416D271
hypocenterC271#3002
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D271C2semidirect
2. C2×#22576182C2semidirect
3. C2×#22576180C2semidirect
4. #31774409C2semidirect
5. #31774062C2semidirect
6. #31774060C2semidirect
7. #31774054C2semidirect
8. #31774049C2semidirect
9. #31774045C2semidirect
10. #31774043C2semidirect
11. #31774037C2semidirect
12. #31774032C2semidirect
13. #31773891C2semidirect
14. C2×(C2×D271)⋊C8C2semidirect
15. #416×C271C2semidirect
19. C2×#16864654C22semidirect
31. C2×Dic1084C22semidirect
32. C2×#16864644C22semidirect
33. C2×#16864643C22semidirect
34. C2×#16864643C22semidirect
35. C2×#16864642C22semidirect
36. C2×#16864641C22semidirect
37. C2×D2168C22semidirect
38. C2×#16864639C22semidirect
39. C22×C271⋊C8C22semidirect
42. #22576140C22semidirect
44. #22576135C22semidirect
48. #22576083C22semidirect
49. #22576083C22semidirect
50. #22576080C22semidirect
51. #22576080C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
57. C22×C8×C271C22semidirect
58. #117×C271C22semidirect
59. #103×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. #102×C271C22semidirect
71. #16864654D4semidirect
123. #41×C271D4semidirect
217. C2×D271#135semidirect
237. C271#3002semidirect
252. #416D271semidirect
279. C22×C8C2×D271semidirect
288. #117C2×D271semidirect
291. #103C2×D271semidirect
292. #103C2×D271semidirect
295. #102C2×D271semidirect
296. #102C2×D271semidirect
314. #41D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2712
C2×#225761822
C2×#225761802
#317744092
#317740622
#317740602
#317740542
#317740492
#317740452
#317740432
#317740372
#317740322
#317738912
C2×(C2×D271)⋊C82
#416×C2712
#3002271