Finite Group Databasebeta

← D4×#14523238 #30650478

Subgroups — D4×#14523238 #30650478

Order 16256.

472 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14523238C212yesmaximal, complemented
2D4×Dic254C211yesmaximal, characteristic
3#22550255C211yesmaximal, characteristic, complemented
4#22550115C212yesmaximal, complemented
5#22550086C211yesmaximal, characteristic, complemented
6#22550083C211yesmaximal, characteristic, complemented
7#22550032C212yesmaximal, complemented
8#22550031C212yesmaximal, complemented
9C4×#14523238C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic
11D4×Q8×C127C211yesmaximal, characteristic
12C22×Dic254C2212yes—
13C2×#14523238—28no—
14C2×#14523238C2212yes—
15C2×#14523238C2212yes—
16C2×#14523238C2212yes—
17C2×#14523238C2212yes—
18C2×#14523238C2211yescharacteristic
19C22×C127⋊C8C2212yes—
20#16861209C2211yescharacteristic
21#16861208—21no—
22#16861188C2212yes—
23#16861186—22no—
24C4×Dic254C2211yescharacteristic
25D4×Dic127—21no—
26#16861142C2211yescharacteristic
27C4⋊(C127⋊C8)C2211yescharacteristic
28#16861133C2211yescharacteristic
29#16861133C2212yes—
30#16861132C2211yescharacteristic
31#16861132C2212yes—
32C22⋊(C127⋊C8)C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34C22×Q8×C127C2212yes—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127C2211yescharacteristic
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127C2212yes—
39C4×Q8×C127C2211yescharacteristic
40C4×D4×C127—21no—
41C4×D4×C127C2211yescharacteristic
42C2×Dic254—24no—
43C2×Dic254—22no—
44C2×Dic254—42no—
45C2×Dic254C2312yes—
46C2×Dic254C2311yescharacteristic
47C22×Dic127—22no—
48#14523238—24no—
49#14523238—24no—
50#14523238—24no—
51#14523238—42no—
52#14523238D414yesdirect product factor
53C2×C127⋊C8—24no—
54C2×C127⋊C8C2312yes—
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2311yescharacteristic
57C4⋊Dic127—21no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2311yescharacteristic
60Dic127⋊C4—21no—
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63C4×Dic127—21no—
64C2×Q8×C127—22no—
65C2×Q8×C127—24no—
66C2×Q8×C127—42no—
67C2×Q8×C127C2312yes—
68C2×Q8×C127C2311yescharacteristic
69C2×D4×C127D411yescharacteristic
70C22×C4×C127—22no—
71C22×C4×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127—22no—
76C4⋊C4×C127D411yescharacteristic
77C4⋊C4×C127C2311yescharacteristic
78C22⋊C4×C127—22no—
79C22⋊C4×C127D412yes—
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82Dic254—21no—
83Dic254—22no—
84Dic254—22no—
85Dic254—44no—
86Dic254C2×D412yes—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127—44no—
91C127⋊C8—22no—
92C127⋊C8—21no—
93C127⋊C8C2×D412yes—
94C23×C127C2×D412yes—
95Q8×C127—21no—
96Q8×C127—22no—
97Q8×C127—22no—
98Q8×C127—44no—
99Q8×C127C2×D412yes—
100D4×C127Q1614yescomplemented
101C2×C4×C127—21no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—44no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C22×C127—22no—
112C22×C127—22no—
113C22×C127C2×Q1614yescomplemented
114C22×C127C22×D411yescharacteristic
115C4×C127—22no—
116C4×C127—22no—
117C4×C127—21no—
118C4×C127—42no—
119C4×C127—41no—
120C4×C127C2×Q1612yescomplemented
121C4×C127C22×D411yescharacteristic
122C4×C127H(5,𝔽2)11yescharacteristic
123Dic127—22no—
124Dic127—42no—
125Dic127—41no—
126C2×C127—24no—
127C2×C127D4211yescharacteristic
128C2×C127C22×Q1611yescharacteristic
129C2×C127#57811yescharacteristic
130D4×Q16—1271nomaximal, Sylow
131C127D4×Q1611yescharacteristic, complemented, Sylow
132C22×Q16—1272no—
133D4×Q8—1271no—
134D4×Q8D12711yescharacteristic
135#494—1271no—
136#467—1272no—
137#462—1271no—
138#462—1271no—
139#451—1272no—
140#451—1272no—
141C4×Q16—1271no—
142C8×D4—1271no—
143C22×Q8—1272no—
144C22×Q8C2×D12712yes—
145C2×Q16—1271no—
146C2×Q16—1272no—
147C2×Q16—1272no—
148C2×Q16—1272no—
149C2×Q16—1272no—
150C2×Q16—2548no—
151C22×C8—1272no—
152C4⋊Q8—21no—
153C4⋊Q8—1271no—
154C4⋊Q8—2541no—
155C4⋊Q8C2×D12711yescharacteristic
156C22⋊Q8—22no—
157C22⋊Q8—1272no—
158C22⋊Q8—2542no—
159C22⋊Q8C2×D12712yes—
160C4×Q8—1271no—
161C4×Q8C2×D12711yescharacteristic
162C4×D4—21no—
163C4×D4—2541no—
164C4×D4C2×D12711yescharacteristic
165#107—1271no—
166C4⋊C8—1271no—
167#103—1271no—
168#103—1272no—
169#103—1271no—
170#103—1272no—
171C22⋊C8—1272no—
172C4×C8—1271no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—1271no—
177C2×Q8—1272no—
178C2×Q8—5082no—
179C2×Q8—2544no—
180C2×Q8—2542no—
181C2×Q8C22×D12712yes—
182C2×Q8C22×D12711yescharacteristic
183C2×D4C22⋊D12711yescharacteristic
184C22×C4—22no—
185C22×C4—2542no—
186C22×C4C22×D12712yes—
187Q16—1274no—
188Q16—5082no—
189Q16—2544no—
190Q16—2544no—
191Q16—2544no—
192C2×C8—1271no—
193C2×C8—1271no—
194C2×C8—1272no—
195C2×C8—2544no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—1271no—
201C4⋊C4—2541no—
202C4⋊C4—2542no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4C22⋊D12711yescharacteristic
206C4⋊C4C22×D12711yescharacteristic
207C22⋊C4—22no—
208C22⋊C4—2542no—
209C22⋊C4C22⋊D12712yes—
210C42—21no—
211C42—2541no—
212C42C22×D12711yescharacteristic
213C23C2×C22⋊D12712yes—
214Q8—21no—
215Q8—22no—
216Q8—22no—
217Q8—44no—
218Q8—1272no—
219Q8—5084no—
220Q8—2542no—
221Q8—2542no—
222Q8—2541no—
223Q8D4×D12712yes—
224D4#1452323814yesdirect product factor
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—44no—
231C2×C4—5084no—
232C2×C4—2541no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C2×C22⋊D12712yes—
237C2×C4C2×C22⋊D12711yescharacteristic
238C2×C4C23×D12711yescharacteristic
239C8—1272no—
240C8—2542no—
241C8—2541no—
242C22—22no—
243C22—22no—
244C22C2×#1452323814yescomplemented
245C22C22×C22⋊D12711yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—41no—
250C4—42no—
251C4—5081no—
252C4—5082no—
253C4—2542no—
254C4C2×#1452323812yescomplemented
255C4C2×D4×D12711yescharacteristic
256C4#1686125411yescharacteristic
257C2—24no—
258C2D4×C22⋊D12711yescharacteristic
259C2C22×#1452323811yescharacteristic
260C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD4×Q8×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×C127D4
hypercenterD4×Q8D127
hypocenterC127D4×Q16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
3. #22550255C2semidirect
4. #22550115C2semidirect
5. #22550086C2semidirect
6. #22550083C2semidirect
7. #22550032C2semidirect
8. #22550031C2semidirect
9. C4×#14523238C2semidirect
52. #14523238D4direct (N × H)
100. D4×C127Q16semidirect
113. C22×C127C2×Q16semidirect
120. C4×C127C2×Q16semidirect
131. C127D4×Q16semidirect
224. D4#14523238direct (N × H)
244. C22C2×#14523238semidirect
254. C4C2×#14523238semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
D4×Dic2542
#225502552
#225501152
#225500862
#225500832
#225500322
#225500312
C4×#145232382
D4×C127⋊C82
D4×Q8×C1272
D4×Q16127