Finite Group Databasebeta

← #16831007

Subgroups — #16831007

Order 3712.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745271C211yesmaximal, characteristic, complemented
2C22×#90773C211yesmaximal, characteristic, complemented
3D4×Dic58C211yesmaximal, characteristic, complemented
4#14271809C211yesmaximal, characteristic
5#14271557C211yesmaximal, characteristic, complemented
6#14271409C211yesmaximal, characteristic, complemented
7#14271404C211yesmaximal, characteristic, complemented
8#14271386C211yesmaximal, characteristic, complemented
9#14271369C211yesmaximal, characteristic, complemented
10#14271327C211yesmaximal, characteristic, complemented
11#14271321C211yesmaximal, characteristic, complemented
12#14271316C211yesmaximal, characteristic, complemented
13#14271304C211yesmaximal, characteristic, complemented
14#14271265C211yesmaximal, characteristic, complemented
15C29×#435C211yesmaximal, characteristic, complemented
16C2×#90805C2211yescharacteristic
17C22×D116C2211yescharacteristic
18C22×Dic58C2211yescharacteristic
19C22×Dic58C2211yescharacteristic, complemented
20C2×M16×C29C2211yescharacteristic, complemented
21C22×C8×C29C2211yescharacteristic
22C4×D4×C29C2211yescharacteristic
23#11745299—21no—
24D4×Dic29—21no—
25#11745292C2211yescharacteristic
26#11745286C2211yescharacteristic
27#11745286C2211yescharacteristic
28#11745286C2211yescharacteristic, complemented
29#11745271—24no—
30C2×Dic116C2211yescharacteristic, complemented
31C2×Dic116C2211yescharacteristic, complemented
32C2×Dic116C2211yescharacteristic, complemented
33C2×#90773—24no—
34C2×#90773C2211yescharacteristic
35C2×#90773C2211yescharacteristic
36C2×#90773C2211yescharacteristic
37C2×#90773C2211yescharacteristic
38C2×#90773C2211yescharacteristic
39C2×#90773C2211yescharacteristic
40#11745256—21no—
41#11745250—21no—
42#11745246—21no—
43C22⋊C4×D29—21no—
44#11745240—21no—
45#11745240—21no—
46#11745237C2211yescharacteristic
47C4×D116C2211yescharacteristic
48#11745231C2211yescharacteristic, complemented
49C4×Dic58C2211yescharacteristic, complemented
50C29×C4⋊C8C2211yescharacteristic, complemented
51C29×C22⋊C8C2211yescharacteristic
52C29×C22⋊C8C2211yescharacteristic, complemented
53C29×#97C2211yescharacteristic, complemented
54#11745169C2211yescharacteristic
55#11745169C2211yescharacteristic
56#11745166C2211yescharacteristic, complemented
57#11745164C2211yescharacteristic, complemented
58#11745164C2211yescharacteristic, complemented
59#11745164C2211yescharacteristic, complemented
60#11745164C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×D4×C29D411yescharacteristic
63C22×C4×C29C2311yescharacteristic
64C22×C4×C29C2311yescharacteristic
65C2×C22⋊D29—21no—
66C2×C22⋊D29—21no—
67C22×Dic29—21no—
68C22×Dic29—21no—
69#90805—22no—
70#90805—41no—
71C2×D116—22no—
72C2×D116—21no—
73C2×D116—41no—
74C2×D116C2311yescharacteristic
75C2×D116C2311yescharacteristic
76C2×C4×D29—21no—
77C2×C4×D29—21no—
78C2×Dic58—22no—
79C2×Dic58—22no—
80C2×Dic58—21no—
81C2×Dic58—21no—
82C2×Dic58—41no—
83C2×Dic58—41no—
84C2×Dic58C2311yescharacteristic
85C2×Dic58C2311yescharacteristic
86C2×Dic58C2311yescharacteristic
87C2×Dic58C2311yescharacteristic
88M16×C29—22no—
89C2×C8×C29—22no—
90C2×C8×C29C2311yescharacteristic
91C2×C8×C29C2311yescharacteristic
92C2×C8×C29C2311yescharacteristic
93C2×C8×C29C2311yescharacteristic
94C4⋊C4×C29D411yescharacteristic
95C22⋊C4×C29D411yescharacteristic
96C22⋊C4×C29D411yescharacteristic
97C42×C29C2311yescharacteristic
98C22⋊Dic29—21no—
99C22⋊Dic29—21no—
100C22⋊Dic29—21no—
101#90781—21no—
102#90781—21no—
103#90781—21no—
104#90781—21no—
105C4⋊Dic29—21no—
106C4⋊Dic29—21no—
107C4⋊Dic29C2311yescharacteristic
108C4⋊Dic29C2311yescharacteristic
109#90779—21no—
110#90779—21no—
111#90779—21no—
112#90779—21no—
113C4×Dic29—21no—
114Dic116—22no—
115Dic116—22no—
116Dic116—41no—
117#90773—22no—
118#90773—22no—
119#90773—22no—
120#90773—22no—
121#90773—41no—
122#90773D414yescomplemented
123C23×C29C2×D411yescharacteristic
124C23×C29C2×D411yescharacteristic
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—42no—
128C22×D29—42no—
129C22×D29—81no—
130D4×C29—22no—
131C2×C4×C29—21no—
132C2×C4×C29—21no—
133C2×C4×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29C2×D411yescharacteristic
136C2×C4×C29C2×D411yescharacteristic
137C2×C4×C29C2×D411yescharacteristic
138C2×C4×C29C2×D411yescharacteristic
139C2×C4×C29C2411yescharacteristic
140C22⋊D29—42no—
141C22⋊D29—81no—
142C2×Dic29—21no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—21no—
146C2×Dic29—21no—
147C2×Dic29—21no—
148C2×Dic29—42no—
149C2×Dic29—42no—
150D116—21no—
151D116—21no—
152D116—21no—
153D116—42no—
154D116C2×D412yes—
155C4×D29—42no—
156C4×D29—81no—
157Dic58—21no—
158Dic58—21no—
159Dic58—21no—
160Dic58—21no—
161Dic58—21no—
162Dic58—21no—
163Dic58—21no—
164Dic58—42no—
165Dic58—42no—
166Dic58C2×D412yes—
167C8×C29—21no—
168C8×C29—21no—
169C8×C29—21no—
170C8×C29C2×D412yes—
171#3266—291nomaximal, Sylow
172C22×C29—21no—
173C22×C29—21no—
174C22×C29—21no—
175C22×C29—21no—
176C22×C29—21no—
177C22×C29#13712yescomplemented
178C22×C29C22×D411yescharacteristic
179C2×D29—22no—
180C2×D29—41no—
181C2×D29—41no—
182C2×D29—41no—
183C2×D29—41no—
184C2×D29—41no—
185C2×D29—84no—
186C4×C29—21no—
187C4×C29—21no—
188C4×C29—21no—
189C4×C29—21no—
190C4×C29C22×D411yescharacteristic
191C4×C29H(5,𝔽2)11yescharacteristic
192Dic29—22no—
193Dic29—41no—
194Dic29—41no—
195Dic29—41no—
196Dic29—41no—
197Dic29—41no—
198C2×#137—291no—
199C22×SD16—291no—
200D4×Q8—291no—
201#546—291no—
202#497—291no—
203#468—291no—
204#467—291no—
205#464—291no—
206#461—291no—
207#452—291no—
208#451—291no—
209#450—291no—
210#448—291no—
211#440—291no—
212#435D2911yescharacteristic, complemented
213C2×C29—22no—
214C2×C29—41no—
215C2×C29D4211yescharacteristic
216C2×C29C2×#13711yescharacteristic
217C2×C29#57711yescharacteristic
218D29—42no—
219D29—81no—
220C2×#41—291no—
221C22×Q8—291no—
222C22×Q8—291no—
223C22×D4—291no—
224#137—584no—
225C2×Q16—291no—
226C2×Q16—291no—
227C2×Q16—291no—
228C2×SD16—291no—
229C2×SD16—291no—
230C2×SD16—291no—
231C2×SD16—291no—
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—584no—
235C2×M16C2×D2911yescharacteristic, complemented
236C22×C8C2×D2911yescharacteristic
237C4⋊Q8—291no—
238C4⋊Q8—581no—
239#124—291no—
240#123—581no—
241C22⋊Q8—291no—
242C22⋊Q8—291no—
243C22⋊Q8—291no—
244C22⋊Q8—581no—
245C22⋊Q8—581no—
246#121—291no—
247#121—581no—
248#120—581no—
249C4×Q8—291no—
250C4×D4—291no—
251C4×D4—581no—
252C4×D4C2×D2911yescharacteristic
253C2×C22⋊C4—581no—
254#107—291no—
255C4⋊C8C2×D2911yescharacteristic, complemented
256#103—291no—
257#103—291no—
258#103—291no—
259#103—291no—
260#102—291no—
261#102—291no—
262C22⋊C8C2×D2911yescharacteristic
263C22⋊C8C2×D2911yescharacteristic, complemented
264#97C2×D2911yescharacteristic, complemented
265C29#326611yescharacteristic, complemented, Sylow
266C24—581no—
267#41—582no—
268#41—1161no—
269C2×Q8—291no—
270C2×Q8—291no—
271C2×Q8—291no—
272C2×Q8—291no—
273C2×Q8—582no—
274C2×Q8—582no—
275C2×Q8—581no—
276C2×Q8—581no—
277C2×Q8—1161no—
278C2×Q8—1161no—
279C2×D4—291no—
280C2×D4—291no—
281C2×D4—582no—
282C2×D4—581no—
283C2×D4—581no—
284C2×D4—581no—
285C2×D4—1161no—
286C2×D4D11611yescharacteristic
287C22×C4—581no—
288C22×C4—581no—
289C22×C4—581no—
290C22×C4—581no—
291C22×C4C22×D2911yescharacteristic
292C22×C4C22×D2911yescharacteristic
293Q16—582no—
294Q16—582no—
295Q16—1161no—
296SD16—294no—
297SD16—582no—
298SD16—582no—
299SD16—582no—
300SD16—582no—
301SD16—1161no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D2911yescharacteristic
305C2×C8C22×D2911yescharacteristic
306C2×C8C22×D2911yescharacteristic
307C2×C8C22×D2911yescharacteristic
308C4⋊C4—291no—
309C4⋊C4—291no—
310C4⋊C4—581no—
311C4⋊C4—581no—
312C4⋊C4—581no—
313C4⋊C4—581no—
314C4⋊C4—581no—
315C4⋊C4—581no—
316C4⋊C4D11611yescharacteristic
317C22⋊C4—581no—
318C22⋊C4—581no—
319C22⋊C4—581no—
320C22⋊C4—581no—
321C22⋊C4—581no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4D11611yescharacteristic
325C22⋊C4D11611yescharacteristic
326C42—581no—
327C42C22×D2911yescharacteristic
328C23—581no—
329C23—581no—
330C23—1162no—
331C23—1162no—
332C23—2321no—
333C23C2×D11611yescharacteristic
334C23C2×D11611yescharacteristic
335Q8—292no—
336Q8—581no—
337Q8—581no—
338Q8—581no—
339Q8—581no—
340Q8—581no—
341Q8—581no—
342Q8—581no—
343Q8—1162no—
344Q8—1162no—
345D4—22no—
346D4—292no—
347D4—581no—
348D4—581no—
349D4—581no—
350D4—1162no—
351D4—1162no—
352D4—2321no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—581no—
358C2×C4—581no—
359C2×C4—581no—
360C2×C4—581no—
361C2×C4—581no—
362C2×C4—581no—
363C2×C4—1162no—
364C2×C4—1162no—
365C2×C4—1162no—
366C2×C4—2321no—
367C2×C4C2×D11611yescharacteristic
368C2×C4C2×D11611yescharacteristic
369C2×C4C2×D11611yescharacteristic
370C2×C4C2×D11611yescharacteristic
371C2×C4C23×D2911yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D2912yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—582no—
382C22—1161no—
383C22—1161no—
384C22—1161no—
385C22—1161no—
386C22—1161no—
387C22—2324no—
388C22#1174527112yescomplemented
389C22C22×D11611yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—582no—
395C4—1161no—
396C4—1161no—
397C4—1161no—
398C4—1161no—
399C4—1161no—
400C4C2×D4×D2911yescharacteristic
401C4#1174536411yescharacteristic
402C2—22no—
403C2—41no—
404C2—1162no—
405C2—2321no—
406C2D4×D11611yescharacteristic
407C2C2×#1174527111yescharacteristic
408C2#1427197511yescharacteristic
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#435C2
socleC22×C29C22×D4
prime radical#14271809C2
hypercenter#435D29
hypocenterC29#3266
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#11745271C2semidirect
2. C22×#90773C2semidirect
3. D4×Dic58C2semidirect
5. #14271557C2semidirect
6. #14271409C2semidirect
7. #14271404C2semidirect
8. #14271386C2semidirect
9. #14271369C2semidirect
10. #14271327C2semidirect
11. #14271321C2semidirect
12. #14271316C2semidirect
13. #14271304C2semidirect
14. #14271265C2semidirect
15. C29×#435C2semidirect
19. C22×Dic58C22semidirect
20. C2×M16×C29C22semidirect
28. #11745286C22semidirect
30. C2×Dic116C22semidirect
31. C2×Dic116C22semidirect
32. C2×Dic116C22semidirect
48. #11745231C22semidirect
49. C4×Dic58C22semidirect
50. C29×C4⋊C8C22semidirect
52. C29×C22⋊C8C22semidirect
53. C29×#97C22semidirect
56. #11745166C22semidirect
57. #11745164C22semidirect
58. #11745164C22semidirect
59. #11745164C22semidirect
60. #11745164C22semidirect
122. #90773D4semidirect
177. C22×C29#137semidirect
212. #435D29semidirect
235. C2×M16C2×D29semidirect
255. C4⋊C8C2×D29semidirect
263. C22⋊C8C2×D29semidirect
264. #97C2×D29semidirect
265. C29#3266semidirect
388. C22#11745271semidirect

Maximal subgroups

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

subgroupindex
C2×#117452712
C22×#907732
D4×Dic582
#142718092
#142715572
#142714092
#142714042
#142713862
#142713692
#142713272
#142713212
#142713162
#142713042
#142712652
C29×#4352
#326629