Finite Group Databasebeta

← #30651444

Subgroups — #30651444

Order 16256.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D508C211yesmaximal, characteristic, complemented
2#22550235C212yesmaximal, complemented
3#22550226C211yesmaximal, characteristic, complemented
4#22550171C212yesmaximal, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550088C212yesmaximal
7#22550078C211yesmaximal, characteristic, complemented
8#22550007C211yesmaximal, characteristic, complemented
9#22549974C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic
11#556×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127—21no—
13C2×#14523235C2212yes—
14C2×#14523233C2211yescharacteristic
15#16861209C2211yescharacteristic
16#16861209C2212yes—
17#16861189—21no—
18#16861189—22no—
19C4×D508C2211yescharacteristic
20C4×D508C2212yes—
21#16861142C2212yes—
22#16861142C2211yescharacteristic
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2212yes—
26#16861133C2212yes—
27#16861133C2212yes—
28#16861130C2212yes—
29#16861130C2211yescharacteristic
30#16861129C2211yescharacteristic
31#16861120C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#126×C127—22no—
35#124×C127C2212yes—
36C22⋊Q8×C127—21no—
37C4×Q8×C127C2212yes—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40#117×C127—21no—
41C2×Dic254—22no—
42C2×Dic254—21no—
43Q8×D127—44no—
44C2×D508C2311yescharacteristic
45C2×C4×D127—22no—
46C2×C4×D127—21no—
47#14523235—24no—
48#14523233—22no—
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2312yes—
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127—21no—
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2311yescharacteristic
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—21no—
58C2×Q8×C127D411yescharacteristic
59C2×Q8×C127C2312yes—
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—21no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127D411yescharacteristic
66C4⋊C4×C127D412yes—
67C4⋊C4×C127C2311yescharacteristic
68C22⋊C4×C127—22no—
69C22⋊C4×C127—21no—
70C22⋊C4×C127—22no—
71C42×C127—22no—
72C42×C127C2311yescharacteristic
73C42×C127C2312yes—
74C22×D127—21no—
75Dic254—44no—
76Dic254—42no—
77D508—21no—
78D508#4112yes—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C4×D127—44no—
82C4×D127—42no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C127⋊C8—21no—
86C23×C127—21no—
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127—22no—
90D4×C127—21no—
91D4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—22no—
108C4×C127—22no—
109C4×C127—21no—
110C4×C127—42no—
111C4×C127—41no—
112C4×C127AGL(1,ℤ/8)12yes—
113C4×C127C2×#4111yescharacteristic
114C4×C1272−1+411yescharacteristic
115Dic127—42no—
116Dic127—41no—
117C2×C127—81no—
118C2×C127#54811yescharacteristic
119C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C127#57811yescharacteristic
121D127—42no—
122#3367—1271nomaximal, Sylow
123C127#336711yescharacteristic, complemented, Sylow
124#556D12711yescharacteristic, complemented
125D4×Q8—1271no—
126#489—1272no—
127#487—1271no—
128#477—1272no—
129#474—1271no—
130#463—1272no—
131#461—1271no—
132#446—1271no—
133#440—1272no—
134C4×D8—1271no—
135C22×Q8—2541no—
136C2×SD16—1272no—
137C2×D8—1271no—
138C4⋊Q8—1271no—
139C4⋊Q8—1272no—
140C4⋊Q8C2×D12711yescharacteristic
141#126—22no—
142#124C2×D12712yes—
143C22⋊Q8—21no—
144C22⋊Q8—2541no—
145C22⋊Q8—2542no—
146C4×Q8C2×D12712yes—
147C4×Q8C2×D12711yescharacteristic
148C4×D4—1271no—
149C4×D4—1272no—
150C4×D4C2×D12711yescharacteristic
151#117—21no—
152#107—1271no—
153#107—1272no—
154C4⋊C8—1271no—
155C4⋊C8—1272no—
156#103—1272no—
157#103—1272no—
158#103—1272no—
159#102—1271no—
160#102—1272no—
161#102—1271no—
162#97—1272no—
163C4×C8—1271no—
164C2×Q8—5084no—
165C2×Q8—2542no—
166C2×Q8—2541no—
167C2×Q8C22⋊D12711yescharacteristic
168C2×Q8C22×D12712yes—
169C2×D4—1271no—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2542no—
173C22×C4—2541no—
174SD16—2544no—
175D8—2542no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8—1272no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—1271no—
183C4⋊C4—1272no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4C22⋊D12711yescharacteristic
187C4⋊C4C22⋊D12712yes—
188C4⋊C4C22×D12711yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2542no—
193C22⋊C4—2541no—
194C42—22no—
195C42C22×D12711yescharacteristic
196C42C22×D12712yes—
197C23—21no—
198C23—2541no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—5084no—
203Q8—5082no—
204D4—21no—
205D4—21no—
206D4—1272no—
207D4—2541no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—5084no—
212C2×C4—5082no—
213C2×C4—2542no—
214C2×C4—2541no—
215C2×C4C2×C22⋊D12712yes—
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4C2×C22⋊D12712yes—
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—2541no—
221C8—2542no—
222C8—2541no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—5081no—
227C22—5081no—
228C22—2542no—
229C22C22×C22⋊D12711yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—42no—
234C4—41no—
235C4—5082no—
236C4—5081no—
237C4#1686123412yes—
238C4#41×D12711yescharacteristic
239C4#1686125611yescharacteristic
240C2—81no—
241C2—5082no—
242C2#2255053311yescharacteristic
243C2C2×#1686123411yescharacteristic
244C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#556×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523233C22
hypercenter#556D127
hypocenterC127#3367
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D508C2semidirect
2. #22550235C2semidirect
3. #22550226C2semidirect
4. #22550171C2semidirect
5. #22550156C2semidirect
7. #22550078C2semidirect
8. #22550007C2semidirect
9. #22549974C2semidirect
11. #556×C127C2semidirect
123. C127#3367semidirect
124. #556D127semidirect

Maximal subgroups

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

subgroupindex
Q8×D5082
#225502352
#225502262
#225501712
#225501562
#225500882
#225500782
#225500072
#225499742
C4×#145232332
#556×C1272
#3367127