Finite Group Databasebeta

← #16831299

Subgroups — #16831299

Order 3712.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D116C211yesmaximal, characteristic, complemented
2#14271511C211yesmaximal, characteristic, complemented
3#14271476C212yesmaximal, complemented
4#14271440C211yesmaximal, characteristic, complemented
5#14271397C212yesmaximal, complemented
6#14271358C211yesmaximal, characteristic, complemented
7#14271296C212yesmaximal, complemented
8C4×#90782C211yesmaximal, characteristic, complemented
9D4×#6597C211yesmaximal, characteristic, complemented
10C2×#11745157C212yesmaximal, complemented
11C29×#547C211yesmaximal, characteristic, complemented
12C2×#41×C29—21no—
13C2×D4×D29—21no—
14C22×D116C2212yes—
15C29×C4⋊Q8C2211yescharacteristic, complemented
16C29×#123—22no—
17C29×C22⋊Q8—21no—
18C29×#121C2212yescomplemented
19C4×D4×C29C2211yescharacteristic
20C4×D4×C29C2211yescharacteristic, complemented
21C2×C4⋊C4×C29C2212yescomplemented
22C2×#90782C2211yescharacteristic
23C2×#90782C2212yescomplemented
24#11745292C2212yes—
25C22×#6597C2212yescomplemented
26#11745257—21no—
27#11745244—22no—
28#11745236C2211yescharacteristic
29C4×D116C2211yescharacteristic
30#11745180C2211yescharacteristic, complemented
31C22⋊#6597C2212yescomplemented
32#11745157C2211yescharacteristic
33#11745157C2212yes—
34#11745157C2212yescomplemented
35#11745157C2212yescomplemented
36#11745157C2212yescomplemented
37#11745155C2211yescharacteristic, complemented
38#11745155C2212yescomplemented
39C4⋊#6597C2211yescharacteristic, complemented
40C4×#6597C2211yescharacteristic, complemented
41C23×D29—22no—
42#41×C29—44no—
43C2×Q8×C29—21no—
44C2×D4×C29—22no—
45C2×D4×C29D411yescharacteristic
46C2×D4×C29C2311yescharacteristic
47C22×C4×C29—22no—
48C22×C4×C29—21no—
49C22×C4×C29C2312yes—
50C2×C22⋊D29—22no—
51D4×D29—44no—
52C2×D116—24no—
53C2×D116—22no—
54C2×D116—21no—
55C2×D116—21no—
56C2×D116—42no—
57C2×D116C2312yes—
58C2×D116C2311yescharacteristic
59C2×C4×D29—21no—
60C4⋊C4×C29—22no—
61C4⋊C4×C29—21no—
62C4⋊C4×C29D411yescharacteristic
63C4⋊C4×C29C2311yescharacteristic
64C4⋊C4×C29C2312yes—
65C22⋊C4×C29—21no—
66C22⋊C4×C29—22no—
67C22⋊C4×C29D412yes—
68C42×C29C2311yescharacteristic
69#90782—22no—
70#90782—42no—
71#90781—22no—
72#90781—21no—
73C4⋊Dic29C2311yescharacteristic
74C2×#6597—42no—
75C2×#6597C2311yescharacteristic
76C2×#6597C2312yes—
77C2×#6597C2311yescharacteristic
78C23×C29—21no—
79C23×C29C2×D412yes—
80C22×D29—21no—
81C22×D29—22no—
82C22×D29—44no—
83C22×D29—44no—
84C22×D29—82no—
85Q8×C29—42no—
86D4×C29—21no—
87D4×C29—21no—
88D4×C29—44no—
89D4×C29D814yescomplemented
90C2×C4×C29—22no—
91C2×C4×C29—22no—
92C2×C4×C29—21no—
93C2×C4×C29—22no—
94C2×C4×C29—44no—
95C2×C4×C29—42no—
96C2×C4×C29C2×D411yescharacteristic
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2411yescharacteristic
100C22⋊D29—44no—
101C2×Dic29—21no—
102D116—22no—
103D116—21no—
104D116—22no—
105D116—44no—
106D116—42no—
107D116—81no—
108D116#4112yescomplemented
109C4×D29—42no—
110#6597—21no—
111#6597—21no—
112#6597—22no—
113#3294—291nomaximal, Sylow
114C22×C29—22no—
115C22×C29—22no—
116C22×C29—41no—
117C22×C29—41no—
118C22×C29—41no—
119C22×C29C2×D814yes—
120C22×C29C22×D411yescharacteristic
121C2×D29—22no—
122C2×D29—41no—
123C2×D29—41no—
124C2×D29—42no—
125C2×D29—42no—
126C2×D29—42no—
127C2×D29—88no—
128C4×C29—22no—
129C4×C29—21no—
130C4×C29—41no—
131C4×C29—42no—
132C4×C29C2×D812yes—
133C4×C29C2×#4111yescharacteristic
134C4×C29H(5,𝔽2)11yescharacteristic
135Dic29—41no—
136#547D2911yescharacteristic, complemented
137D42—291no—
138#486—291no—
139#480—292no—
140#474—291no—
141#466—292no—
142#459—291no—
143#447—292no—
144C4×D8—291no—
145C8×D4—291no—
146C2×#102—292no—
147C2×C29—24no—
148C2×C29—81no—
149C2×C29#54611yescharacteristic
150C2×C29C22×D811yescharacteristic
151C2×C29#57711yescharacteristic
152D29—42no—
153D29—82no—
154C2×#41—21no—
155C22×D4—292no—
156C22×D4—581no—
157C2×D8—291no—
158C2×D8—292no—
159C22×C8—292no—
160C4⋊Q8C2×D2911yescharacteristic, complemented
161#127—291no—
162#123—22no—
163C22⋊Q8—21no—
164#121—292no—
165#121—581no—
166#121C2×D2912yescomplemented
167#120—582no—
168C4×D4—291no—
169C4×D4C2×D2911yescharacteristic
170C4×D4C2×D2911yescharacteristic, complemented
171C2×C4⋊C4C2×D2912yescomplemented
172#107—291no—
173#107—292no—
174C4⋊C8—291no—
175#102—291no—
176#102—291no—
177#102—292no—
178#102—292no—
179#102—292no—
180#102—292no—
181C22⋊C8—292no—
182C4×C8—291no—
183C29#329411yescharacteristic, complemented, Sylow
184C24—582no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—291no—
189C2×D4—292no—
190C2×D4—584no—
191C2×D4—582no—
192C2×D4—582no—
193C2×D4—581no—
194C2×D4—581no—
195C2×D4—1162no—
196C2×D4—1164no—
197C2×D4C22⋊D2911yescharacteristic
198C2×D4C22×D2911yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—581no—
202C22×C4C22×D2912yes—
203D8—582no—
204D8—1162no—
205C2×C8—291no—
206C2×C8—291no—
207C2×C8—292no—
208C2×C8—1162no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—291no—
212C4⋊C4C22⋊D2911yescharacteristic
213C4⋊C4C22×D2911yescharacteristic
214C4⋊C4C22×D2912yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—581no—
218C22⋊C4—582no—
219C22⋊C4C22⋊D2912yes—
220C42C22×D2911yescharacteristic
221C23—21no—
222C23—581no—
223C23—582no—
224C23—1164no—
225C23—1164no—
226C23—2322no—
227C23C2×C22⋊D2912yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—292no—
233D4—582no—
234D4—581no—
235D4—582no—
236D4—1164no—
237D4—1164no—
238D4—1162no—
239D4—2321no—
240D4#9078214yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—581no—
248C2×C4—1162no—
249C2×C4C2×C22⋊D2911yescharacteristic
250C2×C4C2×C22⋊D2912yes—
251C2×C4C2×C22⋊D2911yescharacteristic
252C2×C4C23×D2911yescharacteristic
253C8—581no—
254C8—581no—
255C8—582no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—582no—
262C22—1161no—
263C22—1161no—
264C22—1162no—
265C22—1162no—
266C22—1162no—
267C22—2328no—
268C22C2×#9078214yes—
269C22C22×C22⋊D2911yescharacteristic
270C4—22no—
271C4—21no—
272C4—41no—
273C4—42no—
274C4—1161no—
275C4C2×#9078212yes—
276C4#41×D2911yescharacteristic
277C4#1174536411yescharacteristic
278C2—24no—
279C2—81no—
280C2—1162no—
281C2—2322no—
282C2#1427180111yescharacteristic
283C2C22×#9078211yescharacteristic
284C2#1427197111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#547C2
socleC22×C29C22×D4
hypercenter#547D29
hypocenterC29#3294
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D116C2semidirect
2. #14271511C2semidirect
3. #14271476C2semidirect
4. #14271440C2semidirect
5. #14271397C2semidirect
6. #14271358C2semidirect
7. #14271296C2semidirect
8. C4×#90782C2semidirect
9. D4×#6597C2semidirect
10. C2×#11745157C2semidirect
11. C29×#547C2semidirect
15. C29×C4⋊Q8C22semidirect
18. C29×#121C22semidirect
20. C4×D4×C29C22semidirect
21. C2×C4⋊C4×C29C22semidirect
23. C2×#90782C22semidirect
25. C22×#6597C22semidirect
30. #11745180C22semidirect
31. C22⋊#6597C22semidirect
34. #11745157C22semidirect
35. #11745157C22semidirect
36. #11745157C22semidirect
37. #11745155C22semidirect
38. #11745155C22semidirect
39. C4⋊#6597C22semidirect
40. C4×#6597C22semidirect
89. D4×C29D8semidirect
108. D116#41semidirect
136. #547D29semidirect
160. C4⋊Q8C2×D29semidirect
166. #121C2×D29semidirect
170. C4×D4C2×D29semidirect
171. C2×C4⋊C4C2×D29semidirect
183. C29#3294semidirect
240. D4#90782semidirect

Maximal subgroups

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

subgroupindex
D4×D1162
#142715112
#142714762
#142714402
#142713972
#142713582
#142712962
C4×#907822
D4×#65972
C2×#117451572
C29×#5472
#329429