Finite Group Databasebeta

← #16831011

Subgroups — #16831011

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745274C211yesmaximal, characteristic, complemented
2#14271812C211yesmaximal, characteristic, complemented
3#14271807C211yesmaximal, characteristic, complemented
4#14271554C211yesmaximal, characteristic, complemented
5#14271421C211yesmaximal, characteristic, complemented
6#14271389C211yesmaximal, characteristic, complemented
7#14271376C211yesmaximal, characteristic, complemented
8#14271366C211yesmaximal, characteristic, complemented
9#14271325C211yesmaximal, characteristic, complemented
10#14271322C211yesmaximal, characteristic, complemented
11#14271314C211yesmaximal, characteristic, complemented
12#14271305C211yesmaximal, characteristic, complemented
13#14271275C211yesmaximal, characteristic, complemented
14#14271232C211yesmaximal, characteristic, complemented
15C2×C29×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C22×D4×C29C2211yescharacteristic
18C2×#90807C2211yescharacteristic
19C2×#90807C2211yescharacteristic, complemented
20C29×AGL(1,ℤ/8)—24no—
21C2×SD16×C29C2211yescharacteristic
22C2×SD16×C29C2211yescharacteristic
23C2×D8×C29C2211yescharacteristic
24C2×D8×C29C2211yescharacteristic
25C2×M16×C29C2211yescharacteristic
26#11745315—21no—
27C2×C22⋊Dic29—21no—
28#11745308C2211yescharacteristic, complemented
29#11745306C2211yescharacteristic, complemented
30C2×#90785C2211yescharacteristic, complemented
31#11745301—21no—
32#11745301—21no—
33#11745300C2211yescharacteristic
34#11745298C2211yescharacteristic
35#11745297—21no—
36#11745297—21no—
37D4×Dic29C2211yescharacteristic
38D4×Dic29C2211yescharacteristic, complemented
39C2×#90783C2211yescharacteristic
40C2×#90783C2211yescharacteristic
41C2×#90783C2211yescharacteristic, complemented
42#11745291—21no—
43C4×C22⋊D29C2211yescharacteristic
44#11745286C2211yescharacteristic
45#11745286C2211yescharacteristic
46C2×#90779—21no—
47#11745274—24no—
48C2×Dic116C2211yescharacteristic, complemented
49C2×C8×D29C2211yescharacteristic, complemented
50#11745183C2211yescharacteristic, complemented
51#11745180C2211yescharacteristic
52#11745180C2211yescharacteristic
53#11745180C2211yescharacteristic, complemented
54C22⋊#6597C2211yescharacteristic
55#11745168C2211yescharacteristic, complemented
56#11745165C2211yescharacteristic, complemented
57#11745164C2211yescharacteristic, complemented
58#11745164C2211yescharacteristic, complemented
59#11745163C2211yescharacteristic, complemented
60#11745162C2211yescharacteristic, complemented
61C24×C29—21no—
62#41×C29—22no—
63#41×C29—41no—
64C2×Q8×C29C2311yescharacteristic
65C2×D4×C29—22no—
66C2×D4×C29—21no—
67C2×D4×C29—21no—
68C2×D4×C29—41no—
69C2×D4×C29C2311yescharacteristic
70C2×D4×C29C2311yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C22×C4×C29—21no—
73C22×C4×C29C2311yescharacteristic
74C2×C22⋊D29—21no—
75C2×C22⋊D29—21no—
76C2×C22⋊D29—21no—
77C2×C22⋊D29D411yescharacteristic
78C22×Dic29—21no—
79C22×Dic29—21no—
80C22×Dic29—21no—
81C22×Dic29—21no—
82#90807—22no—
83#90807—22no—
84#90807—41no—
85#90807—41no—
86C2×C4×D29C2311yescharacteristic
87C2×Dic58C2311yescharacteristic
88C2×Dic58C2311yescharacteristic
89SD16×C29—22no—
90SD16×C29—22no—
91D8×C29—22no—
92D8×C29D414yescomplemented
93M16×C29—22no—
94C2×C8×C29C2311yescharacteristic
95C2×C8×C29C2311yescharacteristic
96C22⋊Dic29—21no—
97C22⋊Dic29—21no—
98C22⋊Dic29—21no—
99C22⋊Dic29—21no—
100C22⋊Dic29—21no—
101C22⋊Dic29—21no—
102C22⋊Dic29—21no—
103C22⋊Dic29D411yescharacteristic
104#90785—41no—
105#90783—22no—
106#90783—22no—
107#90783—41no—
108#90781—21no—
109#90781D411yescharacteristic
110C4⋊Dic29C2311yescharacteristic
111C4⋊Dic29C2311yescharacteristic
112#90779—21no—
113#90779—21no—
114#90779—21no—
115#90779—21no—
116#90779—21no—
117#90779D411yescharacteristic
118C4×Dic29C2311yescharacteristic
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121Dic116—22no—
122C8×D29—22no—
123C23×C29—21no—
124C23×C29—21no—
125C23×C29—21no—
126C23×C29—42no—
127C23×C29—42no—
128C23×C29—81no—
129C23×C29C2×D411yescharacteristic
130C22×D29C2×D411yescharacteristic
131Q8×C29—21no—
132Q8×C29—21no—
133D4×C29—21no—
134D4×C29—21no—
135D4×C29—21no—
136D4×C29—21no—
137D4×C29—42no—
138D4×C29—42no—
139D4×C29C2×D412yes—
140D4×C29C2×D412yes—
141C2×C4×C29—21no—
142C2×C4×C29—21no—
143C2×C4×C29—21no—
144C2×C4×C29—42no—
145C2×C4×C29C2×D411yescharacteristic
146C2×C4×C29C2411yescharacteristic
147C22⋊D29—41no—
148C22⋊D29—42no—
149C22⋊D29—42no—
150C22⋊D29—81no—
151C2×Dic29—21no—
152C2×Dic29—21no—
153C2×Dic29—21no—
154C2×Dic29—21no—
155C2×Dic29—42no—
156C2×Dic29—42no—
157C2×Dic29—42no—
158C2×Dic29—81no—
159C2×Dic29C2×D411yescharacteristic
160C2×Dic29C2×D411yescharacteristic
161C2×Dic29C2×D411yescharacteristic
162C4×D29—21no—
163C4×D29—21no—
164Dic58—21no—
165Dic58—21no—
166Dic58—21no—
167Dic58—21no—
168C8×C29—21no—
169C8×C29C2×D412yes—
170#6597—21no—
171#6597—21no—
172#3267—291nomaximal, Sylow
173C22×C29—22no—
174C22×C29—22no—
175C22×C29—21no—
176C22×C29—21no—
177C22×C29—21no—
178C22×C29—41no—
179C22×C29—41no—
180C22×C29—41no—
181C22×C29—41no—
182C22×C29—41no—
183C22×C29—41no—
184C22×C29—41no—
185C22×C29—84no—
186C22×C29C22×D411yescharacteristic
187C2×D29—21no—
188C2×D29—21no—
189C2×D29—21no—
190C4×C29—21no—
191C4×C29—41no—
192C4×C29C22×D411yescharacteristic
193C4×C29H(5,𝔽2)11yescharacteristic
194Dic29—21no—
195Dic29—21no—
196Dic29—21no—
197Dic29—41no—
198Dic29—41no—
199Dic29—41no—
200Dic29—41no—
201C2×AGL(1,ℤ/8)D2911yescharacteristic, complemented
202C2×#135—291no—
203#547—291no—
204#546—291no—
205#497—291no—
206#470—291no—
207#465—291no—
208#463—291no—
209#461—291no—
210#452—291no—
211#452—291no—
212#450—291no—
213#449—291no—
214#442—291no—
215#435—291no—
216C2×C29—41no—
217C2×C29—42no—
218C2×C29—42no—
219C2×C29—81no—
220C2×C29D4211yescharacteristic
221C2×C29#57511yescharacteristic
222C2×C29#57711yescharacteristic
223D29—41no—
224C2×#41—291no—
225C2×#41—291no—
226C2×#41C2×D2911yescharacteristic
227C22×D4C2×D2911yescharacteristic
228AGL(1,ℤ/8)—24no—
229#135—584no—
230C2×Q16—291no—
231C2×Q16—291no—
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—291no—
235C2×SD16C2×D2911yescharacteristic
236C2×SD16C2×D2911yescharacteristic
237C2×D8C2×D2911yescharacteristic
238C2×D8C2×D2911yescharacteristic
239C2×M16C2×D2911yescharacteristic
240C22×C8—291no—
241C4⋊Q8—291no—
242#124—291no—
243#123—581no—
244#123—581no—
245#123—581no—
246C22⋊Q8—291no—
247C22⋊Q8—291no—
248C22⋊Q8—291no—
249#121—291no—
250#121—581no—
251#121—581no—
252#120—581no—
253C4×D4—291no—
254C4×D4—291no—
255C4×D4—291no—
256C2×C4⋊C4—581no—
257C2×C22⋊C4—581no—
258#106—291no—
259C4⋊C8—291no—
260#103—291no—
261#103—291no—
262#103—291no—
263#102—291no—
264#102—291no—
265#102—291no—
266C22⋊C8—291no—
267C22⋊C8—291no—
268#97—291no—
269C29#326711yescharacteristic, complemented, Sylow
270C24—21no—
271#41—22no—
272#41—41no—
273#41—582no—
274#41—582no—
275#41—1161no—
276#41—1161no—
277C2×Q8—291no—
278C2×Q8—291no—
279C2×Q8C22×D2911yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—291no—
285C2×D4—581no—
286C2×D4—581no—
287C2×D4—581no—
288C2×D4C22×D2911yescharacteristic
289C2×D4C22×D2911yescharacteristic
290C2×D4C22×D2911yescharacteristic
291C22×C4—21no—
292C22×C4—291no—
293C22×C4—581no—
294C22×C4—581no—
295C22×C4—581no—
296C22×C4—581no—
297C22×C4C22×D2911yescharacteristic
298Q16—582no—
299Q16—1161no—
300SD16—22no—
301SD16—22no—
302SD16—582no—
303SD16—582no—
304SD16—1161no—
305D8—22no—
306D8C22⋊D2914yescomplemented
307M16—22no—
308C2×C8—291no—
309C2×C8—291no—
310C2×C8—582no—
311C2×C8C22×D2911yescharacteristic
312C2×C8C22×D2911yescharacteristic
313C4⋊C4—291no—
314C4⋊C4—291no—
315C4⋊C4—291no—
316C4⋊C4—581no—
317C4⋊C4—581no—
318C4⋊C4—581no—
319C4⋊C4—581no—
320C4⋊C4—581no—
321C22⋊C4—291no—
322C22⋊C4—291no—
323C22⋊C4—581no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C22⋊C4—581no—
327C22⋊C4—581no—
328C22⋊C4—581no—
329C22⋊C4—581no—
330C22⋊C4—581no—
331C42—291no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—291no—
339C23D4×D2911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—581no—
343Q8—581no—
344Q8—581no—
345Q8—581no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—1161no—
353D4—1162no—
354D4—1162no—
355D4—2321no—
356D4C2×C22⋊D2912yes—
357D4C2×C22⋊D2912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—291no—
363C2×C4—291no—
364C2×C4—291no—
365C2×C4—581no—
366C2×C4—581no—
367C2×C4—581no—
368C2×C4—581no—
369C2×C4—581no—
370C2×C4—581no—
371C2×C4—1162no—
372C2×C4—1162no—
373C2×C4—1162no—
374C2×C4—2321no—
375C2×C4D4×D2911yescharacteristic
376C2×C4C23×D2911yescharacteristic
377C8—21no—
378C8—581no—
379C8—581no—
380C8C2×C22⋊D2912yes—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—581no—
395C22—581no—
396C22—581no—
397C22C2×D4×D2911yescharacteristic
398C4—21no—
399C4—41no—
400C4—581no—
401C4—581no—
402C4—581no—
403C4—1161no—
404C4—1161no—
405C4—1161no—
406C4—1161no—
407C4#1174535911yescharacteristic
408C4C22×C22⋊D2911yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—1161no—
414C2D4×C22⋊D2911yescharacteristic
415C2#1427195911yescharacteristic
416C2#1427197311yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×AGL(1,ℤ/8)C2
socleC22×C29C22×D4
hypercenterC2×AGL(1,ℤ/8)D29
hypocenterC29#3267
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11745274C2semidirect
2. #14271812C2semidirect
3. #14271807C2semidirect
4. #14271554C2semidirect
5. #14271421C2semidirect
6. #14271389C2semidirect
7. #14271376C2semidirect
8. #14271366C2semidirect
9. #14271325C2semidirect
10. #14271322C2semidirect
11. #14271314C2semidirect
12. #14271305C2semidirect
13. #14271275C2semidirect
14. #14271232C2semidirect
15. C2×C29×AGL(1,ℤ/8)C2semidirect
19. C2×#90807C22semidirect
28. #11745308C22semidirect
29. #11745306C22semidirect
30. C2×#90785C22semidirect
38. D4×Dic29C22semidirect
41. C2×#90783C22semidirect
48. C2×Dic116C22semidirect
49. C2×C8×D29C22semidirect
50. #11745183C22semidirect
53. #11745180C22semidirect
55. #11745168C22semidirect
56. #11745165C22semidirect
57. #11745164C22semidirect
58. #11745164C22semidirect
59. #11745163C22semidirect
60. #11745162C22semidirect
92. D8×C29D4semidirect
201. C2×AGL(1,ℤ/8)D29semidirect
269. C29#3267semidirect
306. D8C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452742
#142718122
#142718072
#142715542
#142714212
#142713892
#142713762
#142713662
#142713252
#142713222
#142713142
#142713052
#142712752
#142712322
C2×C29×AGL(1,ℤ/8)2
#326729