Finite Group Databasebeta

← #30677776

Subgroups — #30677776

Order 16768.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22553967C211yesmaximal, characteristic, complemented
2D4×D524C211yesmaximal, characteristic, complemented
3#22553659C211yesmaximal, characteristic, complemented
4#22553627C212yesmaximal, complemented
5#22553592C211yesmaximal, characteristic, complemented
6#22553547C212yesmaximal, complemented
7#22553507C211yesmaximal, characteristic, complemented
8#22553444C212yesmaximal, complemented
9C8×D524C211yesmaximal, characteristic, complemented
10#102×D131C212yesmaximal, complemented
11C4×D8×C131C211yesmaximal, characteristic, complemented
12C2×#14523513—21no—
13C22×D524—21no—
14C2×D4×D131C2212yes—
15C2×#14523498C2212yescomplemented
16C2×C8×D131C2212yescomplemented
17#16862246C2211yescharacteristic, complemented
18#16862243C2211yescharacteristic
19#16862228—22no—
20#16862225—21no—
21#16862221C2212yes—
22#16862220—21no—
23#16862217C2212yescomplemented
24#16862216—22no—
25C4×D524C2211yescharacteristic
26C4⋊C4×D131C2212yescomplemented
27#16862179C2212yescomplemented
28C4⋊(C131⋊C8)C2211yescharacteristic, complemented
29#16862168C2212yescomplemented
30#16862168C2212yescomplemented
31#16862167C2212yes—
32#16862166C2212yescomplemented
33(C2×D131)⋊C8C2212yescomplemented
34C2×D8×C131C2211yescharacteristic
35C4×D4×C131C2211yescharacteristic
36C4×D4×C131C2211yescharacteristic, complemented
37#107×C131C2211yescharacteristic, complemented
38#102×C131C2211yescharacteristic
39#102×C131C2211yescharacteristic, complemented
40C4×C8×C131C2211yescharacteristic, complemented
41C23×D131—22no—
42#14523513—44no—
43C2×Dic262—21no—
44C2×D524—22no—
45C2×D524—21no—
46C2×D524—21no—
47C2×D524—44no—
48C2×D524D411yescharacteristic
49C2×D524C2312yes—
50C2×C22⋊D131—22no—
51C2×C22⋊D131—22no—
52D4×D131—24no—
53D4×D131—42no—
54C2×C4×D131—22no—
55C2×C4×D131C2312yes—
56#14523498—42no—
57C2×C131⋊C8C2312yes—
58C8×D131—42no—
59C4⋊Dic131—21no—
60C4⋊Dic131D411yescharacteristic
61C4⋊Dic131C2312yes—
62Dic131⋊C4—22no—
63C22⋊Dic131—22no—
64(C2×D131)⋊C4—22no—
65(C2×D131)⋊C4D412yes—
66C2×D4×C131C2311yescharacteristic
67C2×D4×C131C2311yescharacteristic
68C22×C4×C131—21no—
69C22×C4×C131—21no—
70D8×C131—22no—
71C2×C8×C131C2311yescharacteristic
72C2×C8×C131C2311yescharacteristic
73C4⋊C4×C131C2311yescharacteristic
74C4⋊C4×C131C2311yescharacteristic
75C22⋊C4×C131—21no—
76C22⋊C4×C131—21no—
77C42×C131C2311yescharacteristic
78C22×D131—22no—
79C22×D131—44no—
80C22×D131—44no—
81C22×D131—82no—
82C22×D131C2×D412yes—
83Dic262—42no—
84D524—22no—
85D524—22no—
86D524—44no—
87D524—42no—
88D524—81no—
89D524D814yescomplemented
90C22⋊D131—44no—
91C22⋊D131—44no—
92C2×Dic131—22no—
93C2×Dic131C2×D412yes—
94C4×D131—22no—
95C4×D131—22no—
96C4×D131—44no—
97C131⋊C8—22no—
98C23×C131—21no—
99C23×C131—21no—
100D4×C131—21no—
101D4×C131—21no—
102D4×C131—21no—
103D4×C131#4112yescomplemented
104C2×C4×C131—21no—
105C2×C4×C131—21no—
106C2×C4×C131—42no—
107C2×C4×C131—42no—
108C2×C4×C131C2×D411yescharacteristic
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2411yescharacteristic
111C8×C131—21no—
112C8×C131—21no—
113C22×C131—22no—
114C22×C131—41no—
115C22×C131—41no—
116C22×C131—41no—
117C22×C131—41no—
118C22×C131—41no—
119C22×C131C22×D411yescharacteristic
120C2×D131—22no—
121C2×D131—22no—
122C2×D131—42no—
123C2×D131—42no—
124C2×D131—42no—
125C2×D131—88no—
126C2×D131C2×D814yes—
127C4×C131—21no—
128C4×C131—41no—
129C4×C131—41no—
130C4×C131C2×D812yes—
131C4×C131C2×#4111yescharacteristic
132C4×C131H(5,𝔽2)11yescharacteristic
133Dic131—22no—
134Dic131—42no—
135C2×C131—42no—
136C2×C131—81no—
137C2×C131#54611yescharacteristic
138C2×C131C22×D811yescharacteristic
139C2×C131#57711yescharacteristic
140D131—24no—
141D131—82no—
142C131#329411yescharacteristic, complemented, Sylow
143#3294—1311nomaximal, Sylow
144#547—1311no—
145D42—1311no—
146#486—1311no—
147#480—1312no—
148#474—1311no—
149#466—1312no—
150#459—1311no—
151#447—1312no—
152C4×D8D13111yescharacteristic, complemented
153C8×D4—1311no—
154C2×#102—1312no—
155C2×#41—2621no—
156C22×D4—2621no—
157C22×D4—1312no—
158C2×D8—1312no—
159C2×D8C2×D13111yescharacteristic
160C22×C8—1312no—
161C4⋊Q8—1311no—
162#127—1311no—
163#123—2622no—
164C22⋊Q8—2621no—
165#121—2621no—
166#121—1312no—
167#121—1312no—
168#120—2622no—
169C4×D4—1311no—
170C4×D4C2×D13111yescharacteristic
171C4×D4C2×D13111yescharacteristic, complemented
172C2×C4⋊C4—1312no—
173#107—1312no—
174#107C2×D13111yescharacteristic, complemented
175C4⋊C8—1311no—
176#102—1312no—
177#102—1312no—
178#102—1312no—
179#102—1312no—
180#102C2×D13111yescharacteristic
181#102C2×D13111yescharacteristic, complemented
182C22⋊C8—1312no—
183C4×C8C2×D13111yescharacteristic, complemented
184C24—2622no—
185#41—5244no—
186C2×Q8—2621no—
187C2×D4—2622no—
188C2×D4—2624no—
189C2×D4—2622no—
190C2×D4—2622no—
191C2×D4—2621no—
192C2×D4—2621no—
193C2×D4—5244no—
194C2×D4—5242no—
195C2×D4—1311no—
196C2×D4—1312no—
197C2×D4C22×D13111yescharacteristic
198C2×D4C22×D13111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—2622no—
202C22×C4—1312no—
203D8—22no—
204D8—5242no—
205C2×C8—5242no—
206C2×C8—1312no—
207C2×C8C22×D13111yescharacteristic
208C2×C8C22×D13111yescharacteristic
209C4⋊C4—2622no—
210C4⋊C4—2621no—
211C4⋊C4—1312no—
212C4⋊C4—1311no—
213C4⋊C4C22×D13111yescharacteristic
214C4⋊C4C22×D13111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2622no—
218C22⋊C4—2622no—
219C22⋊C4—1312no—
220C42C22×D13111yescharacteristic
221C23—21no—
222C23—21no—
223C23—2622no—
224C23—5244no—
225C23—5244no—
226C23—10482no—
227C23—1312no—
228Q8—5242no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—2622no—
233D4—2622no—
234D4—5244no—
235D4—5244no—
236D4—5244no—
237D4—5242no—
238D4—10481no—
239D4—1314no—
240D4#1452351312yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2622no—
248C2×C4—5244no—
249C2×C4—1312no—
250C2×C4D4×D13111yescharacteristic
251C2×C4D4×D13111yescharacteristic
252C2×C4C23×D13111yescharacteristic
253C8—21no—
254C8—21no—
255C8—2622no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—2622no—
263C22—2622no—
264C22—5242no—
265C22—5242no—
266C22—5242no—
267C22—10488no—
268C22—1314no—
269C22C2×D4×D13111yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—2622no—
274C4—5242no—
275C4D8×D13112yes—
276C4C2×#1452351311yescharacteristic
277C4#1686229211yescharacteristic
278C2—42no—
279C2—2624no—
280C2—81no—
281C2—10482no—
282C2#2255395811yescharacteristic
283C2C2×D8×D13111yescharacteristic
284C2#2255411911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC4×D8×C131C2
socleC22×C131C22×D4
hypercenterC4×D8D131
hypocenterC131#3294
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22553967C2semidirect
2. D4×D524C2semidirect
3. #22553659C2semidirect
4. #22553627C2semidirect
5. #22553592C2semidirect
6. #22553547C2semidirect
7. #22553507C2semidirect
8. #22553444C2semidirect
9. C8×D524C2semidirect
10. #102×D131C2semidirect
11. C4×D8×C131C2semidirect
15. C2×#14523498C22semidirect
16. C2×C8×D131C22semidirect
17. #16862246C22semidirect
23. #16862217C22semidirect
26. C4⋊C4×D131C22semidirect
27. #16862179C22semidirect
28. C4⋊(C131⋊C8)C22semidirect
29. #16862168C22semidirect
30. #16862168C22semidirect
32. #16862166C22semidirect
33. (C2×D131)⋊C8C22semidirect
36. C4×D4×C131C22semidirect
37. #107×C131C22semidirect
39. #102×C131C22semidirect
40. C4×C8×C131C22semidirect
89. D524D8semidirect
103. D4×C131#41semidirect
142. C131#3294semidirect
152. C4×D8D131semidirect
171. C4×D4C2×D131semidirect
174. #107C2×D131semidirect
181. #102C2×D131semidirect
183. C4×C8C2×D131semidirect
240. D4#14523513semidirect

Maximal subgroups

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

subgroupindex
#225539672
D4×D5242
#225536592
#225536272
#225535922
#225535472
#225535072
#225534442
C8×D5242
#102×D1312
C4×D8×C1312
#3294131