Finite Group Databasebeta

← #30648572

Subgroups — #30648572

Order 16256.

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

normal subgroups only

This group has 362 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
1#22550504C211yesmaximal, characteristic
2#22550183C212yesmaximal, complemented
3#22550183C212yesmaximal, complemented
4#22550171C212yesmaximal, complemented
5#22550161C212yesmaximal, complemented
6#22549919C211yesmaximal, characteristic
7#22549912C211yesmaximal, characteristic
8#22549903C211yesmaximal, characteristic
9#22549885C211yesmaximal, characteristic
10C2×#16861133C211yesmaximal, characteristic
11#519×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861209C2212yes—
15#16861208—21no—
16#16861201C2212yes—
17#16861200—21no—
18#16861186—22no—
19#16861168C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21#16861142C2211yescharacteristic
22#16861142C2211yescharacteristic
23#16861142C2212yes—
24#16861140C2211yescharacteristic
25#16861140C2212yes—
26#16861140C2211yescharacteristic
27C4⋊(C127⋊C8)C2212yes—
28C4⋊(C127⋊C8)C2212yes—
29#16861133C2212yes—
30#16861133C2212yes—
31#16861133C2212yes—
32#16861133C2212yes—
33C22×Q8×C127C2211yescharacteristic
34C4×Q8×C127—21no—
35C4×Q8×C127—21no—
36C4×Q8×C127C2212yes—
37C4×Q8×C127C2212yes—
38#117×C127—22no—
39#117×C127C2211yescharacteristic
40#117×C127C2211yescharacteristic
41C2×Dic254—22no—
42C22×Dic127—21no—
43#14523232—22no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127C2311yescharacteristic
51C4⋊Dic127C2312yes—
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—21no—
56C4×Dic127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—22no—
59C2×Q8×C127Q814yescomplemented
60C2×Q8×C127C2312yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63C4⋊C4×C127—22no—
64C4⋊C4×C127—22no—
65C4⋊C4×C127D411yescharacteristic
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127C2311yescharacteristic
68C4⋊C4×C127C2311yescharacteristic
69C22⋊C4×C127—21no—
70C22⋊C4×C127D412yes—
71C42×C127—22no—
72C42×C127—22no—
73C42×C127—21no—
74C42×C127C2312yes—
75Dic254—82no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C2×Dic127—81no—
80C127⋊C8—21no—
81C127⋊C8—21no—
82C127⋊C8—22no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—24no—
85Q8×C127—22no—
86Q8×C127C2×Q814yes—
87C2×C4×C127—24no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×Q812yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C22×C127—21no—
98C22×C127—21no—
99C22×C127C22⋊Q812yes—
100C22×C127C22×D411yescharacteristic
101C4×C127—24no—
102C4×C127—22no—
103C4×C127—22no—
104C4×C127—41no—
105C4×C127—41no—
106C4×C127C22⋊Q812yes—
107C4×C127C22×Q811yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—42no—
110Dic127—41no—
111Dic127—41no—
112C2×C127—22no—
113C2×C127C2×C22⋊Q811yescharacteristic
114C2×C127#57711yescharacteristic
115C2×C127#57811yescharacteristic
116#3082—1271nomaximal, Sylow
117C127#308211yescharacteristic, complemented, Sylow
118#544—1271no—
119#519D12711yescharacteristic
120#479—1272no—
121#479—1272no—
122#477—1272no—
123#475—1272no—
124#428—1271no—
125#427—1271no—
126#424—1271no—
127#419—1271no—
128C2×#103—1271no—
129C22×Q8C2×D12711yescharacteristic
130C2×M16—1271no—
131C22×C8—1271no—
132C4⋊Q8—1272no—
133C4⋊Q8—2541no—
134#125—1272no—
135#125—2541no—
136C22⋊Q8—2542no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8C2×D12712yes—
140C4×Q8C2×D12712yes—
141#117—22no—
142#117—1271no—
143#117C2×D12711yescharacteristic
144#117C2×D12711yescharacteristic
145C2×C4⋊C4—1271no—
146#107—1271no—
147#107—1271no—
148#107—1272no—
149#106—1271no—
150#106—1271no—
151#106—1272no—
152C4⋊C8—1272no—
153C4⋊C8—1272no—
154#103—1272no—
155#103—1272no—
156#103—1272no—
157#103—1272no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×Q8—2542no—
161C2×Q8Dic25414yescomplemented
162C2×Q8C22×D12712yes—
163C22×C4—21no—
164C22×C4—2541no—
165C22×C4C22×D12711yescharacteristic
166M16—2542no—
167C2×C8—1271no—
168C2×C8—1272no—
169C2×C8—1271no—
170C2×C8—2542no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—1272no—
174C4⋊C4—1271no—
175C4⋊C4—1271no—
176C4⋊C4—2542no—
177C4⋊C4—2542no—
178C4⋊C4—2541no—
179C4⋊C4—2542no—
180C4⋊C4C22⋊D12711yescharacteristic
181C4⋊C4C22⋊D12711yescharacteristic
182C4⋊C4C22×D12711yescharacteristic
183C4⋊C4C22×D12711yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—2541no—
186C22⋊C4C22⋊D12712yes—
187C42—22no—
188C42—22no—
189C42—21no—
190C42—2541no—
191C42C22×D12712yes—
192C23C2×C22⋊D12711yescharacteristic
193Q8—24no—
194Q8—22no—
195Q8—10162no—
196Q8C2×Dic25414yes—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—10161no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2542no—
205C2×C4C2×C22⋊D12712yes—
206C2×C4C2×C22⋊D12712yes—
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4C2×Dic25412yes—
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—2541no—
212C8—2541no—
213C8—2542no—
214C22—21no—
215C22—21no—
216C22#1686118812yes—
217C22C22×C22⋊D12711yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—41no—
222C4—41no—
223C4—5082no—
224C4—5081no—
225C4—5081no—
226C4#1686118812yes—
227C4C22×Dic25411yescharacteristic
228C4C2×#1452324811yescharacteristic
229C2—22no—
230C2C2×#1686118811yescharacteristic
231C2#2255068411yescharacteristic
232C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#519×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#519D127
hypocenterC127#3082
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #22550183C2semidirect
3. #22550183C2semidirect
4. #22550171C2semidirect
5. #22550161C2semidirect
59. C2×Q8×C127Q8semidirect
117. C127#3082semidirect
161. C2×Q8Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225501832
#225501712
#225501612
#225499192
#225499122
#225499032
#225498852
C2×#168611332
#519×C1272
#3082127