Finite Group Databasebeta

← #16831033

Subgroups — #16831033

Order 3712.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745311C211yesmaximal, characteristic, complemented
2C2×#11745312C211yesmaximal, characteristic, complemented
3#14271809C211yesmaximal, characteristic, complemented
4#14271550C211yesmaximal, characteristic, complemented
5#14271407C211yesmaximal, characteristic, complemented
6#14271397C211yesmaximal, characteristic, complemented
7#14271385C211yesmaximal, characteristic, complemented
8#14271355C211yesmaximal, characteristic, complemented
9#14271325C211yesmaximal, characteristic, complemented
10#14271308C211yesmaximal, characteristic, complemented
11#14271306C211yesmaximal, characteristic, complemented
12#14271296C211yesmaximal, characteristic, complemented
13#14271261C211yesmaximal, characteristic, complemented
14#14271235C211yesmaximal, characteristic, complemented
15C29×#550C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C2×#41×C29C2211yescharacteristic, complemented
18C2×#90805C2211yescharacteristic
19C22×D116C2211yescharacteristic
20C29×#127—21no—
21C29×#127C2211yescharacteristic, complemented
22C29×#121—21no—
23C29×#121—21no—
24C29×#121C2211yescharacteristic
25C29×#121C2211yescharacteristic, complemented
26C4×Q8×C29C2211yescharacteristic, complemented
27C4×D4×C29—21no—
28C4×D4×C29C2211yescharacteristic
29#11745312—24no—
30#11745311—24no—
31C2×#90785C2211yescharacteristic, complemented
32C2×#90784C2211yescharacteristic
33C2×#90784C2211yescharacteristic
34#11745299—21no—
35C2×#90783C2211yescharacteristic, complemented
36C2×#90783C2211yescharacteristic, complemented
37C2×#90782C2211yescharacteristic
38C2×#90782C2211yescharacteristic
39C2×#90782C2211yescharacteristic
40C2×#90782C2211yescharacteristic
41#11745292C2211yescharacteristic
42#11745286C2211yescharacteristic
43C2×#90777C2211yescharacteristic
44C22×#6597C2211yescharacteristic, complemented
45#11745256—21no—
46#11745246—21no—
47C22⋊C4×D29—21no—
48#11745237C2211yescharacteristic
49C4×D116C2211yescharacteristic
50#11745183C2211yescharacteristic, complemented
51#11745180C2211yescharacteristic, complemented
52#11745180C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic
54C22⋊#6597C2211yescharacteristic, complemented
55#11745158C2211yescharacteristic, complemented
56#11745157C2211yescharacteristic
57#11745157C2211yescharacteristic
58#11745155C2211yescharacteristic, complemented
59C4⋊#6597C2211yescharacteristic, complemented
60#11745151C2211yescharacteristic, complemented
61C23×D29—21no—
62#41×C29—22no—
63#41×C29—22no—
64#41×C29—41no—
65#41×C29—41no—
66C2×Q8×C29C2311yescharacteristic
67C2×D4×C29—21no—
68C2×D4×C29—21no—
69C2×D4×C29—21no—
70C2×D4×C29—21no—
71C2×D4×C29—21no—
72C2×D4×C29—21no—
73C2×D4×C29D411yescharacteristic
74C2×D4×C29C2311yescharacteristic
75C2×D4×C29C2311yescharacteristic
76C22×C4×C29—21no—
77C22×C4×C29—21no—
78C22×C4×C29C2311yescharacteristic
79C22×C4×C29C2311yescharacteristic
80C2×C22⋊D29—21no—
81C2×C22⋊D29—21no—
82#90805—22no—
83#90805—41no—
84C2×D116—22no—
85C2×D116—21no—
86C2×D116—41no—
87C2×D116C2311yescharacteristic
88C2×D116C2311yescharacteristic
89C2×C4×D29—21no—
90C2×C4×D29—21no—
91C2×Dic58C2311yescharacteristic
92C4⋊C4×C29—21no—
93C4⋊C4×C29D411yescharacteristic
94C4⋊C4×C29C2311yescharacteristic
95C22⋊C4×C29—21no—
96C22⋊C4×C29—21no—
97C22⋊C4×C29D411yescharacteristic
98C22⋊C4×C29D411yescharacteristic
99C42×C29—21no—
100C42×C29C2311yescharacteristic
101C22⋊Dic29—21no—
102#90785—22no—
103#90784—22no—
104#90784D414yescomplemented
105#90783—22no—
106#90783—41no—
107#90782—22no—
108#90782—22no—
109#90782—22no—
110#90782—41no—
111#90781—21no—
112#90781—21no—
113#90781—21no—
114#90781—21no—
115C4⋊Dic29C2311yescharacteristic
116#90779—21no—
117#90777—22no—
118C2×#6597—22no—
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122C2×#6597C2311yescharacteristic
123C23×C29—21no—
124C23×C29—21no—
125C23×C29C2×D411yescharacteristic
126C23×C29C2×D411yescharacteristic
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—42no—
130C22×D29—42no—
131C22×D29—81no—
132Q8×C29—21no—
133Q8×C29C2×D412yes—
134D4×C29—21no—
135D4×C29—21no—
136D4×C29—21no—
137D4×C29—21no—
138D4×C29—41no—
139D4×C29—42no—
140D4×C29—42no—
141D4×C29—42no—
142D4×C29—81no—
143D4×C29—81no—
144D4×C29—81no—
145C2×C4×C29—21no—
146C2×C4×C29—21no—
147C2×C4×C29—21no—
148C2×C4×C29—21no—
149C2×C4×C29—21no—
150C2×C4×C29—21no—
151C2×C4×C29—42no—
152C2×C4×C29—42no—
153C2×C4×C29C2×D411yescharacteristic
154C2×C4×C29C2×D411yescharacteristic
155C2×C4×C29C2×D411yescharacteristic
156C2×C4×C29C2×D411yescharacteristic
157C2×C4×C29C2411yescharacteristic
158C22⋊D29—42no—
159C22⋊D29—81no—
160C2×Dic29—21no—
161C2×Dic29—21no—
162D116—21no—
163D116—21no—
164D116—21no—
165D116—42no—
166D116C2×D412yes—
167C4×D29—42no—
168C4×D29—81no—
169Dic58—21no—
170Dic58—21no—
171#6597—21no—
172#6597—21no—
173#6597—21no—
174#6597C2×D412yes—
175#3268—291nomaximal, Sylow
176C22×C29—21no—
177C22×C29—21no—
178C22×C29—21no—
179C22×C29—21no—
180C22×C29—21no—
181C22×C29—21no—
182C22×C29—41no—
183C22×C29—41no—
184C22×C29—41no—
185C22×C29—41no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29C22×D411yescharacteristic
189C2×D29—22no—
190C2×D29—41no—
191C2×D29—41no—
192C2×D29—41no—
193C2×D29—41no—
194C2×D29—41no—
195C2×D29—84no—
196C4×C29—21no—
197C4×C29—21no—
198C4×C29—22no—
199C4×C29—21no—
200C4×C29—21no—
201C4×C29—41no—
202C4×C29C22×D411yescharacteristic
203C4×C29H(5,𝔽2)11yescharacteristic
204Dic29—41no—
205Dic29—41no—
206C2×AGL(1,ℤ/8)—291no—
207C2×#135—291no—
208#550D2911yescharacteristic, complemented
209#546—291no—
210#496—291no—
211#468—291no—
212#466—291no—
213#464—291no—
214#459—291no—
215#452—291no—
216#449—291no—
217#449—291no—
218#447—291no—
219#440—291no—
220#435—291no—
221C2×C29—41no—
222C2×C29—41no—
223C2×C29—81no—
224C2×C29—81no—
225C2×C29D4211yescharacteristic
226C2×C29#57511yescharacteristic
227C2×C29#57611yescharacteristic
228D29—42no—
229D29—81no—
230C2×#41—291no—
231C2×#41C2×D2911yescharacteristic
232C2×#41C2×D2911yescharacteristic, complemented
233C22×D4—291no—
234AGL(1,ℤ/8)—584no—
235#135—584no—
236C2×Q16—291no—
237C2×SD16—291no—
238C2×SD16—291no—
239C2×SD16—291no—
240C2×SD16—291no—
241C2×D8—291no—
242C2×D8—291no—
243C2×D8—291no—
244C2×D8—291no—
245C2×M16—291no—
246C22×C8—291no—
247#127—21no—
248#127C2×D2911yescharacteristic, complemented
249#124—291no—
250#123—581no—
251C22⋊Q8—291no—
252#121—21no—
253#121—21no—
254#121—291no—
255#121—581no—
256#121C2×D2911yescharacteristic
257#121C2×D2911yescharacteristic, complemented
258#120—581no—
259C4×Q8C2×D2911yescharacteristic, complemented
260C4×D4—21no—
261C4×D4—291no—
262C4×D4C2×D2911yescharacteristic
263C2×C22⋊C4—581no—
264#107—291no—
265C4⋊C8—291no—
266#103—291no—
267#103—291no—
268#102—291no—
269#102—291no—
270#102—291no—
271#102—291no—
272C22⋊C8—291no—
273C22⋊C8—291no—
274#97—291no—
275C29#326811yescharacteristic, complemented, Sylow
276C24—581no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—582no—
282#41—1161no—
283C2×Q8—291no—
284C2×Q8C22×D2911yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—291no—
292C2×D4—291no—
293C2×D4—582no—
294C2×D4—581no—
295C2×D4—581no—
296C2×D4—581no—
297C2×D4—1161no—
298C2×D4C22⋊D2911yescharacteristic
299C2×D4C22×D2911yescharacteristic
300C2×D4C22×D2911yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—581no—
304C22×C4—581no—
305C22×C4C22×D2911yescharacteristic
306C22×C4C22×D2911yescharacteristic
307Q16—582no—
308SD16—294no—
309SD16—582no—
310SD16—582no—
311SD16—1161no—
312D8—582no—
313D8—582no—
314D8—582no—
315D8—1161no—
316M16—582no—
317C2×C8—291no—
318C2×C8—291no—
319C2×C8—291no—
320C2×C8—291no—
321C2×C8—582no—
322C4⋊C4—21no—
323C4⋊C4—291no—
324C4⋊C4—581no—
325C4⋊C4C22⋊D2911yescharacteristic
326C4⋊C4C22×D2911yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—581no—
330C22⋊C4—581no—
331C22⋊C4—581no—
332C22⋊C4—581no—
333C22⋊C4—581no—
334C22⋊C4C22⋊D2911yescharacteristic
335C22⋊C4C22⋊D2911yescharacteristic
336C42—21no—
337C42C22×D2911yescharacteristic
338C23—21no—
339C23—21no—
340C23—581no—
341C23—581no—
342C23—1162no—
343C23—1162no—
344C23—2321no—
345C23C2×C22⋊D2911yescharacteristic
346C23C2×C22⋊D2911yescharacteristic
347Q8—21no—
348Q8—581no—
349Q8—581no—
350Q8D4×D2912yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—292no—
363D4—581no—
364D4—581no—
365D4—581no—
366D4—1162no—
367D4—1162no—
368D4—2321no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—581no—
378C2×C4—581no—
379C2×C4—1162no—
380C2×C4—2321no—
381C2×C4C2×C22⋊D2911yescharacteristic
382C2×C4C2×C22⋊D2911yescharacteristic
383C2×C4C2×C22⋊D2911yescharacteristic
384C2×C4C2×C22⋊D2911yescharacteristic
385C2×C4C23×D2911yescharacteristic
386C8—292no—
387C8—581no—
388C8—581no—
389C8—581no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—582no—
403C22—1161no—
404C22—1161no—
405C22—1161no—
406C22—1161no—
407C22—1161no—
408C22—2324no—
409C22C22×C22⋊D2911yescharacteristic
410C4—21no—
411C4—21no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—1161no—
417C4—1161no—
418C4C2×D4×D2911yescharacteristic
419C4#1174535911yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—1162no—
425C2—2321no—
426C2D4×C22⋊D2911yescharacteristic
427C2#1427196111yescharacteristic
428C2#1427196611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#550C2
socleC22×C29C22×D4
hypercenter#550D29
hypocenterC29#3268
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. C2×#11745312C2semidirect
3. #14271809C2semidirect
4. #14271550C2semidirect
5. #14271407C2semidirect
6. #14271397C2semidirect
7. #14271385C2semidirect
8. #14271355C2semidirect
9. #14271325C2semidirect
10. #14271308C2semidirect
11. #14271306C2semidirect
12. #14271296C2semidirect
13. #14271261C2semidirect
14. #14271235C2semidirect
15. C29×#550C2semidirect
17. C2×#41×C29C22semidirect
21. C29×#127C22semidirect
25. C29×#121C22semidirect
26. C4×Q8×C29C22semidirect
31. C2×#90785C22semidirect
35. C2×#90783C22semidirect
36. C2×#90783C22semidirect
44. C22×#6597C22semidirect
50. #11745183C22semidirect
51. #11745180C22semidirect
52. #11745180C22semidirect
54. C22⋊#6597C22semidirect
55. #11745158C22semidirect
58. #11745155C22semidirect
59. C4⋊#6597C22semidirect
60. #11745151C22semidirect
104. #90784D4semidirect
208. #550D29semidirect
232. C2×#41C2×D29semidirect
248. #127C2×D29semidirect
257. #121C2×D29semidirect
259. C4×Q8C2×D29semidirect
275. C29#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#117453112
C2×#117453122
#142718092
#142715502
#142714072
#142713972
#142713852
#142713552
#142713252
#142713082
#142713062
#142712962
#142712612
#142712352
C29×#5502
#326829