Finite Group Databasebeta

← #30650448

Subgroups — #30650448

Order 16256.

484 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16861229C212yesmaximal, complemented
2#22550531C211yesmaximal, characteristic, complemented
3#22550255C211yesmaximal, characteristic, complemented
4#22550115C212yesmaximal, complemented
5#22550081C211yesmaximal, characteristic, complemented
6#22550078C211yesmaximal, characteristic, complemented
7#22550038C212yesmaximal, complemented
8#22550036C212yesmaximal, complemented
9C4×#14523233C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic, complemented
11#547×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13#16861229—28no—
14C2×#14523238C2212yes—
15C2×#14523238C2212yes—
16C2×#14523236C2212yes—
17C2×#14523235C2212yes—
18C2×#14523233C2211yescharacteristic
19C22×C127⋊C8C2212yes—
20#16861209C2211yescharacteristic
21#16861191—22no—
22#16861188C2212yes—
23#16861180—21no—
24C4×D508C2211yescharacteristic
25C4⋊C4×D127—21no—
26#16861142C2211yescharacteristic
27C4⋊(C127⋊C8)C2211yescharacteristic
28#16861133C2212yes—
29#16861132C2212yes—
30#16861130C2211yescharacteristic
31#16861129C2211yescharacteristic
32C22⋊(C127⋊C8)C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34C2×#41×C127C2212yes—
35C4⋊Q8×C127C2211yescharacteristic
36#123×C127—22no—
37C22⋊Q8×C127C2212yes—
38#121×C127—21no—
39C4×D4×C127C2211yescharacteristic
40C4×D4×C127C2211yescharacteristic
41C2×C4⋊C4×C127—21no—
42#14523248—24no—
43#14523248—42no—
44C2×Dic254C2312yes—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49#14523238—24no—
50#14523238—42no—
51#14523236—24no—
52#14523235—24no—
53#14523233D414yescomplemented
54C2×C127⋊C8—24no—
55C2×C127⋊C8C2312yes—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63(C2×D127)⋊C4—21no—
64#41×C127—24no—
65#41×C127—42no—
66C2×Q8×C127C2312yes—
67C2×D4×C127—22no—
68C2×D4×C127D411yescharacteristic
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—22no—
72C22×C4×C127C2312yes—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—22no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127D411yescharacteristic
77C4⋊C4×C127C2311yescharacteristic
78C22⋊C4×C127—22no—
79C22⋊C4×C127—21no—
80C22⋊C4×C127D412yes—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83Dic254—22no—
84Dic254—22no—
85D508—21no—
86D508C2×D412yes—
87C22⋊D127—44no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C4×D127—44no—
91C4×D127—42no—
92C127⋊C8—22no—
93C127⋊C8—21no—
94C127⋊C8C2×D412yes—
95C23×C127—21no—
96C23×C127C2×D412yes—
97Q8×C127—22no—
98Q8×C127—22no—
99D4×C127—22no—
100D4×C127—21no—
101D4×C127—44no—
102D4×C127C2×D412yes—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—44no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C22×C127—22no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—22no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127C22×D411yescharacteristic
120C2×D127—22no—
121C2×D127—41no—
122C2×D127—41no—
123C4×C127—22no—
124C4×C127—21no—
125C4×C127—42no—
126C4×C127—41no—
127C4×C127#13512yescomplemented
128C4×C127C22×D411yescharacteristic
129C4×C127H(5,𝔽2)11yescharacteristic
130Dic127—42no—
131Dic127—41no—
132C2×C127—42no—
133C2×C127—42no—
134C2×C127D4211yescharacteristic
135C2×C127C2×#13511yescharacteristic
136C2×C127#57811yescharacteristic
137D127—42no—
138#3283—1271nomaximal, Sylow
139C127#328311yescharacteristic, complemented, Sylow
140C2×#135—1272no—
141#547—1271no—
142#547D12711yescharacteristic, complemented
143#494—1271no—
144#467—1272no—
145#461—1271no—
146#461—1271no—
147#452—1272no—
148#452—1272no—
149C4×D8—1271no—
150C8×D4—1271no—
151C2×#41—1272no—
152C2×#41C2×D12712yes—
153#135—2548no—
154C2×Q16—1272no—
155C2×Q16—1272no—
156C2×SD16—1272no—
157C2×SD16—1272no—
158C2×D8—1271no—
159C22×C8—1272no—
160C4⋊Q8—1271no—
161C4⋊Q8C2×D12711yescharacteristic
162#123—22no—
163#123—2542no—
164C22⋊Q8—1272no—
165C22⋊Q8C2×D12712yes—
166#121—21no—
167#121—2541no—
168C4×D4—1271no—
169C4×D4C2×D12711yescharacteristic
170C4×D4C2×D12711yescharacteristic
171C2×C4⋊C4—21no—
172C2×C4⋊C4—2541no—
173#107—1271no—
174C4⋊C8—1271no—
175#103—1272no—
176#103—1272no—
177#102—1271no—
178#102—1271no—
179C22⋊C8—1272no—
180C4×C8—1271no—
181#41—24no—
182#41—42no—
183#41—5082no—
184#41—2544no—
185C2×Q8—1272no—
186C2×Q8C22×D12712yes—
187C2×D4—22no—
188C2×D4—1271no—
189C2×D4—2542no—
190C2×D4C22⋊D12711yescharacteristic
191C2×D4C22×D12711yescharacteristic
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—2541no—
195C22×C4—2542no—
196C22×C4C22×D12712yes—
197Q16—5082no—
198Q16—2544no—
199SD16—2544no—
200SD16—2544no—
201D8—1274no—
202C2×C8—1271no—
203C2×C8—1271no—
204C2×C8—1272no—
205C2×C8—2544no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—1271no—
210C4⋊C4—2542no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4C22⋊D12711yescharacteristic
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—2542no—
218C22⋊C4—2541no—
219C22⋊C4C22⋊D12712yes—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—2541no—
223C23C2×C22⋊D12712yes—
224Q8—22no—
225Q8—22no—
226Q8—2542no—
227Q8—2542no—
228D4—22no—
229D4—21no—
230D4—44no—
231D4—1272no—
232D4—5084no—
233D4—2541no—
234D4D4×D12712yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—5084no—
242C2×C4—5082no—
243C2×C4—2541no—
244C2×C4—2542no—
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4C2×C22⋊D12712yes—
247C2×C4C2×C22⋊D12711yescharacteristic
248C2×C4C23×D12711yescharacteristic
249C8—1272no—
250C8—2542no—
251C8—2541no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—5081no—
259C22—5081no—
260C22—2542no—
261C22C22×C22⋊D12711yescharacteristic
262C4—21no—
263C4—22no—
264C4—41no—
265C4—42no—
266C4—5081no—
267C4—5082no—
268C4#1686122912yescomplemented
269C4C2×D4×D12711yescharacteristic
270C4#1686125411yescharacteristic
271C2—42no—
272C2—42no—
273C2—5082no—
274C2D4×C22⋊D12711yescharacteristic
275C2C2×#1686122911yescharacteristic
276C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3283
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550531C2semidirect
3. #22550255C2semidirect
4. #22550115C2semidirect
5. #22550081C2semidirect
6. #22550078C2semidirect
7. #22550038C2semidirect
8. #22550036C2semidirect
9. C4×#14523233C2semidirect
10. D4×C127⋊C8C2semidirect
11. #547×C127C2semidirect
53. #14523233D4semidirect
127. C4×C127#135semidirect
139. C127#3283semidirect
142. #547D127semidirect
268. C4#16861229semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225505312
#225502552
#225501152
#225500812
#225500782
#225500382
#225500362
C4×#145232332
D4×C127⋊C82
#547×C1272
#3283127