Finite Group Databasebeta

← #30638748

Subgroups — #30638748

Order 16256.

312 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 95 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#41×Dic127C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic
4C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127C411yescharacteristic, complemented
10Q8×Dic127C411yescharacteristic, complemented
11D4×Dic127—21no—
12D4×Dic127—21no—
13D4×Dic127C411yescharacteristic, complemented
14D4×Dic127C411yescharacteristic, complemented
15#16861168—21no—
16#16861168C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C2×C4×Dic127—21no—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21Dic127⋊C8—21no—
22Dic127⋊C8—21no—
23#16861136—22no—
24C127⋊#95—21no—
25C2×#41×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×C42×C127C2211yescharacteristic
28#104×C127—22no—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C22×Dic127D411yescharacteristic
34#14523232—21no—
35#14523232—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127C2×C411yescharacteristic
40C4⋊Dic127C2×C411yescharacteristic
41Dic127⋊C4—24no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C4×Dic127C2×C411yescharacteristic
51C4×Dic127C2×C411yescharacteristic
52C4×Dic127D411yescharacteristic
53C4×Dic127D411yescharacteristic
54#41×C127—21no—
55#41×C127—21no—
56#41×C127D412yes—
57#41×C127Q812yes—
58C2×Q8×C127C2×C411yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127C2×C411yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64M16×C127—21no—
65M16×C127—21no—
66C2×C8×C127—21no—
67C42×C127—21no—
68C42×C127—21no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C2×Dic127—42no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—42no—
78C2×Dic127C22⋊C411yescharacteristic
79C2×Dic127C22⋊C411yescharacteristic
80C2×Dic127C22⋊C411yescharacteristic
81C2×Dic127C22⋊C411yescharacteristic
82C127⋊C8—41no—
83C23×C127—21no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—21no—
86Q8×C127C4⋊C412yescomplemented
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—21no—
90D4×C127C4⋊C412yescomplemented
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C22×C411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×Q811yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C8×C127—41no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—22no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127C2×C22⋊C411yescharacteristic
110C22×C127#12111yescharacteristic
111C22×C127C22⋊Q811yescharacteristic
112C4×C127—22no—
113C4×C127—42no—
114C4×C127C2×C4⋊C411yescharacteristic
115C4×C127#11711yescharacteristic
116C4×C127#12111yescharacteristic
117C4×C127C22⋊Q811yescharacteristic
118Dic127—21no—
119Dic127—21no—
120Dic127—21no—
121Dic127—42no—
122Dic127—41no—
123Dic127—41no—
124Dic127#10412yescomplemented
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—42no—
128C2×C127#38011yescharacteristic
129C2×C127C2×#10411yescharacteristic
130C2×C127#42111yescharacteristic
131#1806—1271nomaximal, Sylow
132C127#180611yescharacteristic, complemented, Sylow
133C4×#41—1271no—
134#423—1271no—
135C2×#104—1271no—
136C2×#104D12711yescharacteristic
137C4×C4⋊C4—1271no—
138#339—1271no—
139#339—1271no—
140C2×#41C2×D12711yescharacteristic
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143C4×Q8—1271no—
144C4×Q8—1271no—
145C4×D4—1271no—
146C4×D4—1271no—
147C4×D4—2541no—
148C4×D4—2541no—
149#117—1271no—
150#117—2541no—
151C2×C4⋊C4—2541no—
152C2×C42—1271no—
153C2×C42—1271no—
154C2×C42—2541no—
155C2×C42C2×D12711yescharacteristic
156C4⋊C8—2541no—
157C4⋊C8—2541no—
158#104—22no—
159#104—2542no—
160#95—2541no—
161#41—21no—
162#41—21no—
163#41C22⋊D12712yes—
164#41Dic25412yes—
165C2×Q8C4×D12711yescharacteristic
166C2×D4—21no—
167C2×D4C4×D12711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1271no—
171C22×C4—1271no—
172C22×C4—2541no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176M16—21no—
177M16—21no—
178M16—2541no—
179M16—2541no—
180C2×C8—21no—
181C2×C8—2541no—
182C4⋊C4—1271no—
183C4⋊C4—1271no—
184C4⋊C4—2541no—
185C4⋊C4—2541no—
186C4⋊C4—2544no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C42—21no—
191C42—21no—
192C42—1271no—
193C42—1271no—
194C42—1271no—
195C42—1271no—
196C42—2541no—
197C42—2541no—
198C42—2541no—
199C42—2541no—
200C42—2541no—
201C23—21no—
202C23D4×D12711yescharacteristic
203Q8—21no—
204Q8Dic127⋊C412yescomplemented
205D4—22no—
206D4—22no—
207D4—21no—
208D4Dic127⋊C412yescomplemented
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—1271no—
216C2×C4—1271no—
217C2×C4—1271no—
218C2×C4—1271no—
219C2×C4—5082no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—5082no—
223C2×C4—5081no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2541no—
227C2×C4—2541no—
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C2×Dic25411yescharacteristic
232C2×C4#1452324711yescharacteristic
233C2×C4#1452324711yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22C22⋊C4×D12711yescharacteristic
242C22#1686118111yescharacteristic
243C22#1686118611yescharacteristic
244C4—22no—
245C4—42no—
246C4—1272no—
247C4—5082no—
248C4—5081no—
249C4—5081no—
250C4—2541no—
251C4—2541no—
252C4—2541no—
253C4C2×Dic127⋊C411yescharacteristic
254C4#1686116511yescharacteristic
255C4#1686118111yescharacteristic
256C4#1686118611yescharacteristic
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2254973411yescharacteristic
261C2#104×D12711yescharacteristic
262C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1806
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
4. C4×Dic127⋊C4C2semidirect
5. #22549637C2semidirect
6. #22549636C2semidirect
9. Q8×Dic127C4semidirect
10. Q8×Dic127C4semidirect
13. D4×Dic127C4semidirect
14. D4×Dic127C4semidirect
86. Q8×C127C4⋊C4semidirect
90. D4×C127C4⋊C4semidirect
124. Dic127#104semidirect
132. C127#1806semidirect
204. Q8Dic127⋊C4semidirect
208. D4Dic127⋊C4semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1272
#225498992
C2×#168611362
C4×Dic127⋊C42
#225496372
#225496362
C2×#104×C1272
#1806127