Finite Group Databasebeta

← #30677768

Subgroups — #30677768

Order 16768.

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×D524C211yesmaximal, characteristic, complemented
2#22553660C211yesmaximal, characteristic, complemented
3#22553625C212yesmaximal, complemented
4#22553589C211yesmaximal, characteristic, complemented
5#22553546C212yesmaximal, complemented
6#22553507C211yesmaximal, characteristic, complemented
7#22553445C212yesmaximal, complemented
8C4×#14523498C211yesmaximal, characteristic, complemented
9D4×C131⋊C8C211yesmaximal, characteristic, complemented
10C2×#16862166C212yesmaximal, complemented
11#547×C131C211yesmaximal, characteristic, complemented
12C22×D524C2212yes—
13C2×D4×D131—21no—
14C2×#14523498C2211yescharacteristic
15C2×#14523498C2212yescomplemented
16C22×C131⋊C8C2212yescomplemented
17#16862243C2211yescharacteristic
18#16862221—21no—
19#16862220C2212yes—
20#16862216—22no—
21C4×D524C2211yescharacteristic
22#16862179C2211yescharacteristic, complemented
23#16862179C2212yescomplemented
24C4⋊(C131⋊C8)C2211yescharacteristic, complemented
25#16862167C2211yescharacteristic, complemented
26#16862166C2211yescharacteristic
27#16862166C2212yes—
28#16862166C2212yescomplemented
29#16862166C2212yescomplemented
30#16862166C2212yescomplemented
31C22⋊(C131⋊C8)C2212yescomplemented
32C4×C131⋊C8C2211yescharacteristic, complemented
33C2×#41×C131—21no—
34C4⋊Q8×C131C2211yescharacteristic, complemented
35#123×C131—22no—
36C22⋊Q8×C131—21no—
37#121×C131C2212yescomplemented
38C4×D4×C131C2211yescharacteristic
39C4×D4×C131C2211yescharacteristic, complemented
40C2×C4⋊C4×C131C2212yescomplemented
41C23×D131—22no—
42C2×D524—24no—
43C2×D524—22no—
44C2×D524—21no—
45C2×D524—21no—
46C2×D524—42no—
47C2×D524C2312yes—
48C2×D524C2311yescharacteristic
49C2×C22⋊D131—22no—
50D4×D131—44no—
51C2×C4×D131—21no—
52#14523498—22no—
53#14523498—42no—
54C2×C131⋊C8—42no—
55C2×C131⋊C8C2311yescharacteristic
56C2×C131⋊C8C2312yes—
57C2×C131⋊C8C2311yescharacteristic
58C4⋊Dic131C2311yescharacteristic
59(C2×D131)⋊C4—21no—
60(C2×D131)⋊C4—22no—
61#41×C131—44no—
62C2×Q8×C131—21no—
63C2×D4×C131—22no—
64C2×D4×C131D411yescharacteristic
65C2×D4×C131C2311yescharacteristic
66C22×C4×C131—22no—
67C22×C4×C131—21no—
68C22×C4×C131C2312yes—
69C4⋊C4×C131—22no—
70C4⋊C4×C131—21no—
71C4⋊C4×C131D411yescharacteristic
72C4⋊C4×C131C2311yescharacteristic
73C4⋊C4×C131C2312yes—
74C22⋊C4×C131—21no—
75C22⋊C4×C131—22no—
76C22⋊C4×C131D412yes—
77C42×C131C2311yescharacteristic
78C22×D131—21no—
79C22×D131—22no—
80C22×D131—44no—
81C22×D131—44no—
82C22×D131—82no—
83D524—22no—
84D524—21no—
85D524—22no—
86D524—44no—
87D524—42no—
88D524—81no—
89D524#4112yescomplemented
90C22⋊D131—44no—
91C2×Dic131—21no—
92C4×D131—42no—
93C131⋊C8—21no—
94C131⋊C8—21no—
95C131⋊C8—22no—
96C23×C131—21no—
97C23×C131C2×D412yes—
98Q8×C131—42no—
99D4×C131—21no—
100D4×C131—21no—
101D4×C131—44no—
102D4×C131D814yescomplemented
103C2×C4×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—21no—
106C2×C4×C131—22no—
107C2×C4×C131—44no—
108C2×C4×C131—42no—
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2×D412yes—
111C2×C4×C131C2×D411yescharacteristic
112C2×C4×C131C2411yescharacteristic
113C22×C131—22no—
114C22×C131—22no—
115C22×C131—41no—
116C22×C131—41no—
117C22×C131—41no—
118C22×C131C2×D814yes—
119C22×C131C22×D411yescharacteristic
120C2×D131—22no—
121C2×D131—41no—
122C2×D131—41no—
123C2×D131—42no—
124C2×D131—42no—
125C2×D131—42no—
126C2×D131—88no—
127C4×C131—22no—
128C4×C131—21no—
129C4×C131—41no—
130C4×C131—42no—
131C4×C131C2×D812yes—
132C4×C131C2×#4111yescharacteristic
133C4×C131H(5,𝔽2)11yescharacteristic
134Dic131—41no—
135C2×C131—24no—
136C2×C131—81no—
137C2×C131#54611yescharacteristic
138C2×C131C22×D811yescharacteristic
139C2×C131#57711yescharacteristic
140D131—42no—
141D131—82no—
142C131#329411yescharacteristic, complemented, Sylow
143#3294—1311nomaximal, Sylow
144#547D13111yescharacteristic, complemented
145D42—1311no—
146#486—1311no—
147#480—1312no—
148#474—1311no—
149#466—1312no—
150#459—1311no—
151#447—1312no—
152C4×D8—1311no—
153C8×D4—1311no—
154C2×#102—1312no—
155C2×#41—21no—
156C22×D4—2621no—
157C22×D4—1312no—
158C2×D8—1311no—
159C2×D8—1312no—
160C22×C8—1312no—
161C4⋊Q8C2×D13111yescharacteristic, complemented
162#127—1311no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2621no—
166#121—1312no—
167#121C2×D13112yescomplemented
168#120—2622no—
169C4×D4—1311no—
170C4×D4C2×D13111yescharacteristic
171C4×D4C2×D13111yescharacteristic, complemented
172C2×C4⋊C4C2×D13112yescomplemented
173#107—1311no—
174#107—1312no—
175C4⋊C8—1311no—
176#102—1311no—
177#102—1311no—
178#102—1312no—
179#102—1312no—
180#102—1312no—
181#102—1312no—
182C22⋊C8—1312no—
183C4×C8—1311no—
184C24—2622no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2624no—
189C2×D4—2622no—
190C2×D4—2622no—
191C2×D4—2621no—
192C2×D4—2621no—
193C2×D4—5242no—
194C2×D4—5244no—
195C2×D4—1312no—
196C2×D4—1311no—
197C2×D4C22⋊D13111yescharacteristic
198C2×D4C22×D13111yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2621no—
202C22×C4C22×D13112yes—
203D8—2622no—
204D8—5242no—
205C2×C8—5242no—
206C2×C8—1311no—
207C2×C8—1311no—
208C2×C8—1312no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1311no—
212C4⋊C4C22⋊D13111yescharacteristic
213C4⋊C4C22×D13111yescharacteristic
214C4⋊C4C22×D13112yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2621no—
218C22⋊C4—2622no—
219C22⋊C4C22⋊D13112yes—
220C42C22×D13111yescharacteristic
221C23—21no—
222C23—2621no—
223C23—2622no—
224C23—5244no—
225C23—5244no—
226C23—10482no—
227C23C2×C22⋊D13112yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2622no—
233D4—2621no—
234D4—2622no—
235D4—5244no—
236D4—5244no—
237D4—5242no—
238D4—10481no—
239D4—1312no—
240D4#1452349814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—2621no—
248C2×C4—5242no—
249C2×C4C2×C22⋊D13111yescharacteristic
250C2×C4C2×C22⋊D13112yes—
251C2×C4C2×C22⋊D13111yescharacteristic
252C2×C4C23×D13111yescharacteristic
253C8—2621no—
254C8—2621no—
255C8—2622no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—2622no—
262C22—5241no—
263C22—5241no—
264C22—5242no—
265C22—5242no—
266C22—5242no—
267C22—10488no—
268C22C2×#1452349814yes—
269C22C22×C22⋊D13111yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—5241no—
275C4C2×#1452349812yes—
276C4#41×D13111yescharacteristic
277C4#1686229211yescharacteristic
278C2—24no—
279C2—81no—
280C2—5242no—
281C2—10482no—
282C2#2255395011yescharacteristic
283C2C22×#1452349811yescharacteristic
284C2#2255412011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#547×C131C2
socleC22×C131C22×D4
hypercenter#547D131
hypocenterC131#3294
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D524C2semidirect
2. #22553660C2semidirect
3. #22553625C2semidirect
4. #22553589C2semidirect
5. #22553546C2semidirect
6. #22553507C2semidirect
7. #22553445C2semidirect
8. C4×#14523498C2semidirect
9. D4×C131⋊C8C2semidirect
10. C2×#16862166C2semidirect
11. #547×C131C2semidirect
15. C2×#14523498C22semidirect
16. C22×C131⋊C8C22semidirect
22. #16862179C22semidirect
23. #16862179C22semidirect
24. C4⋊(C131⋊C8)C22semidirect
25. #16862167C22semidirect
28. #16862166C22semidirect
29. #16862166C22semidirect
30. #16862166C22semidirect
31. C22⋊(C131⋊C8)C22semidirect
32. C4×C131⋊C8C22semidirect
34. C4⋊Q8×C131C22semidirect
37. #121×C131C22semidirect
39. C4×D4×C131C22semidirect
40. C2×C4⋊C4×C131C22semidirect
89. D524#41semidirect
102. D4×C131D8semidirect
142. C131#3294semidirect
144. #547D131semidirect
161. C4⋊Q8C2×D131semidirect
167. #121C2×D131semidirect
171. C4×D4C2×D131semidirect
172. C2×C4⋊C4C2×D131semidirect
240. D4#14523498semidirect

Maximal subgroups

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

subgroupindex
D4×D5242
#225536602
#225536252
#225535892
#225535462
#225535072
#225534452
C4×#145234982
D4×C131⋊C82
C2×#168621662
#547×C1312
#3294131