Finite Group Databasebeta

← #16828706

Subgroups — #16828706

Order 3712.

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×D29C211yesmaximal, characteristic, complemented
2C2×#11745312C211yesmaximal, characteristic, complemented
3#14271670C211yesmaximal, characteristic, complemented
4#14271324C211yesmaximal, characteristic, complemented
5#14271322C211yesmaximal, characteristic, complemented
6#14271317C211yesmaximal, characteristic, complemented
7#14271312C211yesmaximal, characteristic, complemented
8#14271307C211yesmaximal, characteristic, complemented
9#14271305C211yesmaximal, characteristic, complemented
10#14271300C211yesmaximal, characteristic, complemented
11#14271295C211yesmaximal, characteristic, complemented
12#14271160C211yesmaximal, characteristic, complemented
13#14271159C211yesmaximal, characteristic, complemented
14C2×#11745168C211yesmaximal, characteristic, complemented
15C2×C29×#135C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C2×#41×C29C2211yescharacteristic, complemented
18#41×D29—24no—
19#41×D29—24no—
20C2×#90809C2211yescharacteristic
21C2×Q8×D29C2211yescharacteristic
22C2×#90807—21no—
23C2×#90807C2211yescharacteristic
24C2×D4×D29—21no—
25C2×D4×D29C2211yescharacteristic
26C2×#90805—21no—
27C2×#90805C2211yescharacteristic
28C22×C4×D29—21no—
29C22×C4×D29C2211yescharacteristic
30C29×#135—24no—
31C2×Q16×C29C2211yescharacteristic, complemented
32C2×SD16×C29C2211yescharacteristic, complemented
33C2×SD16×C29C2211yescharacteristic, complemented
34C2×D8×C29C2211yescharacteristic, complemented
35C22×C8×C29C2211yescharacteristic, complemented
36#11745312—24no—
37#11745308C2211yescharacteristic
38#11745308C2211yescharacteristic, complemented
39C2×#90785C2211yescharacteristic, complemented
40C2×#90784C2211yescharacteristic, complemented
41#11745300C2211yescharacteristic
42#11745300C2211yescharacteristic, complemented
43#11745299—21no—
44#11745297—21no—
45C2×#90783C2211yescharacteristic, complemented
46C2×#90782C2211yescharacteristic, complemented
47C4×C22⋊D29—21no—
48C4×C22⋊D29—21no—
49#11745288C2211yescharacteristic, complemented
50C22×#6597C2211yescharacteristic, complemented
51#11745183C2211yescharacteristic, complemented
52#11745183C2211yescharacteristic, complemented
53#11745180C2211yescharacteristic, complemented
54#11745180C2211yescharacteristic, complemented
55#11745169C2211yescharacteristic, complemented
56#11745169C2211yescharacteristic, complemented
57#11745168C2211yescharacteristic
58#11745168C2211yescharacteristic
59#11745168C2211yescharacteristic, complemented
60#11745168C2211yescharacteristic, complemented
61#11745164C2211yescharacteristic, complemented
62#11745164C2211yescharacteristic, complemented
63C23×D29—21no—
64C23×D29D411yescharacteristic
65#41×C29—22no—
66#41×C29—22no—
67#41×C29—22no—
68#41×C29D414yescomplemented
69C2×Q8×C29C2311yescharacteristic
70C2×Q8×C29C2311yescharacteristic
71C2×D4×C29—21no—
72C2×D4×C29—21no—
73C2×D4×C29C2311yescharacteristic
74C2×D4×C29C2311yescharacteristic
75C22×C4×C29—21no—
76C22×C4×C29—21no—
77C22×C4×C29C2311yescharacteristic
78C2×C22⋊D29—21no—
79C2×C22⋊D29—21no—
80C2×C22⋊D29—21no—
81C2×C22⋊D29—21no—
82C2×C22⋊D29—21no—
83C22×Dic29—21no—
84C22×Dic29D411yescharacteristic
85#90809—22no—
86#90809—22no—
87Q8×D29—22no—
88Q8×D29—22no—
89#90807—28no—
90#90807—22no—
91#90807—22no—
92D4×D29—28no—
93D4×D29—22no—
94D4×D29—22no—
95#90805—28no—
96#90805—22no—
97#90805D414yescomplemented
98C2×D116—21no—
99C2×D116C2311yescharacteristic
100C2×C4×D29—28no—
101C2×C4×D29—21no—
102C2×C4×D29—21no—
103C2×C4×D29—22no—
104C2×C4×D29—22no—
105C2×C4×D29—21no—
106C2×C4×D29D411yescharacteristic
107C2×C4×D29D411yescharacteristic
108C2×C4×D29C2311yescharacteristic
109C2×C4×D29C2311yescharacteristic
110C2×Dic58—21no—
111C2×Dic58C2311yescharacteristic
112Q16×C29—22no—
113SD16×C29—22no—
114SD16×C29—22no—
115D8×C29—22no—
116C2×C8×C29—22no—
117C2×C8×C29C2311yescharacteristic
118C2×C8×C29C2311yescharacteristic
119C22⋊Dic29—21no—
120C22⋊Dic29—21no—
121C22⋊Dic29—21no—
122#90785—22no—
123#90784—22no—
124#90783—22no—
125#90782—22no—
126#90781—21no—
127#90781—21no—
128C4⋊Dic29C2311yescharacteristic
129C4⋊Dic29C2311yescharacteristic
130#90779—21no—
131#90779—21no—
132C4×Dic29—21no—
133C2×#6597—22no—
134C2×#6597C2311yescharacteristic
135C2×#6597C2311yescharacteristic
136C23×C29—21no—
137C23×C29—21no—
138C23×C29C2×D411yescharacteristic
139C22×D29—24no—
140C22×D29—22no—
141C22×D29—21no—
142C22×D29—22no—
143C22×D29—41no—
144C22×D29—42no—
145C22×D29—42no—
146C22×D29C2×D411yescharacteristic
147C22×D29C2×D411yescharacteristic
148Q8×C29—21no—
149Q8×C29—21no—
150Q8×C29—21no—
151Q8×C29C2×D412yes—
152D4×C29—24no—
153D4×C29—21no—
154D4×C29—21no—
155D4×C29—21no—
156D4×C29—42no—
157D4×C29C2×D412yes—
158C2×C4×C29—21no—
159C2×C4×C29—24no—
160C2×C4×C29—21no—
161C2×C4×C29—42no—
162C2×C4×C29C2×D412yes—
163C2×C4×C29C2×D411yescharacteristic
164C2×C4×C29C2×D412yes—
165C2×C4×C29C2411yescharacteristic
166C22⋊D29—24no—
167C22⋊D29—24no—
168C22⋊D29—24no—
169C22⋊D29—42no—
170C22⋊D29—42no—
171C2×Dic29—24no—
172C2×Dic29—22no—
173C2×Dic29—21no—
174C2×Dic29—21no—
175C2×Dic29—21no—
176C2×Dic29C2×D411yescharacteristic
177C2×Dic29C2×D411yescharacteristic
178D116—24no—
179D116—21no—
180D116C2×D412yes—
181C4×D29—24no—
182C4×D29—24no—
183C4×D29—22no—
184C4×D29—24no—
185C4×D29—22no—
186C4×D29—21no—
187C4×D29—21no—
188C4×D29—21no—
189C4×D29—21no—
190Dic58—24no—
191Dic58—21no—
192Dic58C2×D412yes—
193C8×C29—21no—
194C8×C29—21no—
195#6597—21no—
196#6597—21no—
197#3002—291nomaximal, Sylow
198C22×C29—21no—
199C22×C29—21no—
200C22×C29—22no—
201C22×C29—41no—
202C22×C29—41no—
203C22×C29—41no—
204C22×C29—41no—
205C22×C29—41no—
206C22×C29#12012yes—
207C22×C29C22×D411yescharacteristic
208C2×D29—21no—
209C2×D29—21no—
210C2×D29—21no—
211C2×D29—22no—
212C2×D29—21no—
213C2×D29—21no—
214C2×D29—44no—
215C2×D29—48no—
216C2×D29—41no—
217C2×D29—41no—
218C2×D29#13512yescomplemented
219C4×C29—22no—
220C4×C29—41no—
221C4×C29#12012yes—
222C4×C29C22×D411yescharacteristic
223C4×C29C22×D411yescharacteristic
224Dic29—21no—
225Dic29—22no—
226Dic29—21no—
227Dic29—41no—
228Dic29—41no—
229C22×#41—291no—
230C2×#135—291no—
231C2×#135D2911yescharacteristic, complemented
232#525—291no—
233#452—291no—
234#452—291no—
235#451—291no—
236#450—291no—
237#449—291no—
238#449—291no—
239#448—291no—
240#447—291no—
241#416—291no—
242#416—291no—
243C2×C22⋊C8—291no—
244C2×C29—22no—
245C2×C29—42no—
246C2×C29—81no—
247C2×C29C2×#12011yescharacteristic
248C2×C29C2×#13511yescharacteristic
249C2×C29#57511yescharacteristic
250D29—22no—
251D29—41no—
252D29—42no—
253C2×#41—291no—
254C2×#41—291no—
255C2×#41—291no—
256C2×#41—581no—
257C2×#41—581no—
258C2×#41—584no—
259C2×#41—584no—
260C2×#41C2×D2911yescharacteristic
261C2×#41C2×D2911yescharacteristic, complemented
262C22×Q8—291no—
263C22×D4—291no—
264C22×D4—581no—
265C23×C4—291no—
266C23×C4—581no—
267#135—24no—
268#135—584no—
269C2×Q16—291no—
270C2×Q16C2×D2911yescharacteristic, complemented
271C2×SD16—291no—
272C2×SD16—291no—
273C2×SD16C2×D2911yescharacteristic, complemented
274C2×SD16C2×D2911yescharacteristic, complemented
275C2×D8—291no—
276C2×D8C2×D2911yescharacteristic, complemented
277C22×C8—291no—
278C22×C8C2×D2911yescharacteristic, complemented
279#123—581no—
280C22⋊Q8—291no—
281C22⋊Q8—291no—
282#121—291no—
283#121—291no—
284#120—581no—
285C4×D4—581no—
286C4×D4—581no—
287#117—291no—
288#103—291no—
289#103—291no—
290#103—291no—
291#103—291no—
292#102—291no—
293#102—291no—
294#102—291no—
295#102—291no—
296C22⋊C8—291no—
297C22⋊C8—291no—
298C22⋊C8—291no—
299C22⋊C8—291no—
300C29#300211yescharacteristic, complemented, Sylow
301C24—291no—
302C24—581no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—294no—
307#41—588no—
308#41—582no—
309#41—588no—
310#41—582no—
311#41—582no—
312#41—582no—
313#41—582no—
314#41C22⋊D2914yescomplemented
315C2×Q8—291no—
316C2×Q8—581no—
317C2×Q8—582no—
318C2×Q8—582no—
319C2×Q8C22×D2911yescharacteristic
320C2×Q8C22×D2911yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—291no—
324C2×D4—581no—
325C2×D4—581no—
326C2×D4—582no—
327C2×D4—588no—
328C2×D4—581no—
329C2×D4—581no—
330C2×D4—582no—
331C2×D4—581no—
332C2×D4—581no—
333C2×D4C22×D2911yescharacteristic
334C2×D4C22×D2911yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—291no—
338C22×C4—291no—
339C22×C4—291no—
340C22×C4—291no—
341C22×C4—291no—
342C22×C4—581no—
343C22×C4—581no—
344C22×C4—588no—
345C22×C4—581no—
346C22×C4—582no—
347C22×C4—581no—
348C22×C4—582no—
349C22×C4C22×D2911yescharacteristic
350Q16—22no—
351Q16—582no—
352SD16—22no—
353SD16—22no—
354SD16—582no—
355SD16—582no—
356D8—22no—
357D8—582no—
358C2×C8—22no—
359C2×C8—291no—
360C2×C8—291no—
361C2×C8—582no—
362C2×C8C22×D2911yescharacteristic
363C2×C8C22×D2911yescharacteristic
364C4⋊C4—291no—
365C4⋊C4—291no—
366C4⋊C4—581no—
367C4⋊C4—581no—
368C22⋊C4—581no—
369C22⋊C4—581no—
370C22⋊C4—581no—
371C22⋊C4—581no—
372C22⋊C4—581no—
373C42—581no—
374C23—21no—
375C23—21no—
376C23—291no—
377C23—291no—
378C23—582no—
379C23—584no—
380C23—582no—
381C23—581no—
382C23—1161no—
383C23—1162no—
384C23—1162no—
385C23D4×D2911yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—292no—
390Q8—584no—
391Q8—581no—
392Q8C2×C22⋊D2912yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—292no—
399D4—584no—
400D4—584no—
401D4—584no—
402D4—584no—
403D4—581no—
404D4—1162no—
405D4—1162no—
406D4C2×C22⋊D2912yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—291no—
412C2×C4—291no—
413C2×C4—584no—
414C2×C4—584no—
415C2×C4—584no—
416C2×C4—582no—
417C2×C4—581no—
418C2×C4—582no—
419C2×C4—582no—
420C2×C4—584no—
421C2×C4—581no—
422C2×C4—581no—
423C2×C4—581no—
424C2×C4—581no—
425C2×C4—581no—
426C2×C4—581no—
427C2×C4D4×D2912yes—
428C2×C4D4×D2911yescharacteristic
429C2×C4C2×C22⋊D2912yes—
430C2×C4C23×D2911yescharacteristic
431C8—21no—
432C8—21no—
433C8—581no—
434C8—581no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—292no—
444C22—581no—
445C22—581no—
446C22—582no—
447C22—581no—
448C22—581no—
449C22—581no—
450C22—1164no—
451C22—1168no—
452C22—1161no—
453C22—1161no—
454C22#1174529912yes—
455C22C2×D4×D2911yescharacteristic
456C4—22no—
457C4—41no—
458C4—581no—
459C4—581no—
460C4—582no—
461C4—1161no—
462C4—1161no—
463C4#1174529912yes—
464C4C2×D4×D2911yescharacteristic
465C4C22×C22⋊D2911yescharacteristic
466C2—22no—
467C2—42no—
468C2—81no—
469C2—582no—
470C2—1161no—
471C2—1162no—
472C2C2×#1174529911yescharacteristic
473C2#135×D2911yescharacteristic
474C2#1427196011yescharacteristic
subgroupG/H
centerC2×C4D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×#135C2
socleC22×C29C22×D4
hypercenterC2×#135D29
hypocenterC29#3002
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#41×D29C2semidirect
2. C2×#11745312C2semidirect
3. #14271670C2semidirect
4. #14271324C2semidirect
5. #14271322C2semidirect
6. #14271317C2semidirect
7. #14271312C2semidirect
8. #14271307C2semidirect
9. #14271305C2semidirect
10. #14271300C2semidirect
11. #14271295C2semidirect
12. #14271160C2semidirect
13. #14271159C2semidirect
14. C2×#11745168C2semidirect
15. C2×C29×#135C2semidirect
17. C2×#41×C29C22semidirect
31. C2×Q16×C29C22semidirect
32. C2×SD16×C29C22semidirect
33. C2×SD16×C29C22semidirect
34. C2×D8×C29C22semidirect
35. C22×C8×C29C22semidirect
38. #11745308C22semidirect
39. C2×#90785C22semidirect
40. C2×#90784C22semidirect
42. #11745300C22semidirect
45. C2×#90783C22semidirect
46. C2×#90782C22semidirect
49. #11745288C22semidirect
50. C22×#6597C22semidirect
51. #11745183C22semidirect
52. #11745183C22semidirect
53. #11745180C22semidirect
54. #11745180C22semidirect
55. #11745169C22semidirect
56. #11745169C22semidirect
59. #11745168C22semidirect
60. #11745168C22semidirect
61. #11745164C22semidirect
62. #11745164C22semidirect
68. #41×C29D4semidirect
97. #90805D4semidirect
218. C2×D29#135semidirect
231. C2×#135D29semidirect
261. C2×#41C2×D29semidirect
270. C2×Q16C2×D29semidirect
273. C2×SD16C2×D29semidirect
274. C2×SD16C2×D29semidirect
276. C2×D8C2×D29semidirect
278. C22×C8C2×D29semidirect
300. C29#3002semidirect
314. #41C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D292
C2×#117453122
#142716702
#142713242
#142713222
#142713172
#142713122
#142713072
#142713052
#142713002
#142712952
#142711602
#142711592
C2×#117451682
C2×C29×#1352
#300229