Finite Group Databasebeta

← #30648568

Subgroups — #30648568

Order 16256.

362 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 141 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#22550507C211yesmaximal, characteristic
2#22550354C211yesmaximal, characteristic
3#22550185C212yesmaximal, complemented
4#22550181C212yesmaximal, complemented
5#22550169C212yesmaximal, complemented
6#22550159C212yesmaximal, complemented
7#22549916C211yesmaximal, characteristic
8#22549914C211yesmaximal, characteristic
9#22549902C211yesmaximal, characteristic
10#22549885C211yesmaximal, characteristic
11C2×#103×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13#16861208—21no—
14#16861208C2212yes—
15#16861200—21no—
16#16861200C2212yes—
17#16861188—22no—
18Q8×Dic127—21no—
19Q8×Dic127—21no—
20Q8×Dic127C2212yes—
21Q8×Dic127C2212yes—
22#16861168—22no—
23#16861168C2211yescharacteristic
24#16861168C2211yescharacteristic
25#16861168C2211yescharacteristic
26#16861143C2211yescharacteristic
27#16861143C2211yescharacteristic
28#16861142C2212yes—
29#16861141C2211yescharacteristic
30#16861141C2211yescharacteristic
31#16861140C2212yes—
32Dic127⋊C8C2212yes—
33Dic127⋊C8C2212yes—
34#16861133C2212yes—
35#16861133C2212yes—
36C22×Q8×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#103×C127C2212yes—
40#103×C127C2212yes—
41C2×Dic254—22no—
42#14523232—22no—
43C2×C127⋊C8C2312yes—
44C4⋊Dic127—22no—
45C4⋊Dic127—22no—
46C4⋊Dic127—22no—
47C4⋊Dic127D411yescharacteristic
48C4⋊Dic127D411yescharacteristic
49C4⋊Dic127C2311yescharacteristic
50C4⋊Dic127C2311yescharacteristic
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—21no—
56C22⋊Dic127—21no—
57C22⋊Dic127D412yes—
58C4×Dic127—22no—
59C4×Dic127—22no—
60C4×Dic127—21no—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C2×Q8×C127—21no—
64C2×Q8×C127—22no—
65C2×Q8×C127Q814yescomplemented
66C2×Q8×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70C2×C8×C127—22no—
71C2×C8×C127C2311yescharacteristic
72C2×C8×C127C2311yescharacteristic
73C4⋊C4×C127—21no—
74C4⋊C4×C127C2312yes—
75Dic254—82no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127C2×D412yes—
81C2×Dic127C2×D412yes—
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—22no—
90C2×C4×C127—81no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×Q812yes—
93C2×C4×C127#4112yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—21no—
96C8×C127—21no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127C22⋊Q812yes—
100C22×C127C22×D411yescharacteristic
101C4×C127—24no—
102C4×C127—42no—
103C4×C127C22⋊Q812yes—
104C4×C127C22×Q811yescharacteristic
105C4×C127C2×#4111yescharacteristic
106Dic127—22no—
107Dic127—22no—
108Dic127—41no—
109Dic127—41no—
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#519—1271no—
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×#103D12711yescharacteristic
129C22×Q8C2×D12711yescharacteristic
130C2×M16—1271no—
131C22×C8C2×D12711yescharacteristic
132C4⋊Q8—1272no—
133C4⋊Q8—2541no—
134#125—1272no—
135#125—2541no—
136C22⋊Q8—2542no—
137C4×Q8—1272no—
138C4×Q8—1272no—
139C4×Q8—2541no—
140C4×Q8—2541no—
141#117—1271no—
142#117—1271no—
143#117—1271no—
144#117—2542no—
145C2×C4⋊C4C2×D12711yescharacteristic
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#103C2×D12712yes—
157#103C2×D12712yes—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×Q8—2542no—
161C2×Q8Dic25414yescomplemented
162C2×Q8C22×D12712yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D12711yescharacteristic
166M16—2542no—
167C2×C8—22no—
168C2×C8—1272no—
169C2×C8C22×D12711yescharacteristic
170C2×C8C22×D12711yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—1271no—
173C4⋊C4—1271no—
174C4⋊C4—1271no—
175C4⋊C4—1271no—
176C4⋊C4—1271no—
177C4⋊C4—1271no—
178C4⋊C4—2542no—
179C4⋊C4—2542no—
180C4⋊C4—2542no—
181C4⋊C4—2542no—
182C4⋊C4—2542no—
183C4⋊C4C22×D12712yes—
184C22⋊C4—1272no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C42—1272no—
188C42—2542no—
189C42—2542no—
190C42—2541no—
191C42—2541no—
192C23D4×D12711yescharacteristic
193Q8—24no—
194Q8—22no—
195Q8—10162no—
196Q8C2×Dic25414yes—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—81no—
201C2×C4—1272no—
202C2×C4—1272no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4—2541no—
207C2×C4D4×D12711yescharacteristic
208C2×C4C2×Dic25412yes—
209C2×C4#1452324712yes—
210C2×C4C23×D12711yescharacteristic
211C8—21no—
212C8—21no—
213C8—2542no—
214C22—21no—
215C22—21no—
216C22#1686118612yes—
217C22C2×D4×D12711yescharacteristic
218C4—24no—
219C4—42no—
220C4—5081no—
221C4—5081no—
222C4—5081no—
223C4—5081no—
224C4—2542no—
225C4—2542no—
226C4#1686118612yes—
227C4C22×Dic25411yescharacteristic
228C4C2×#1452324711yescharacteristic
229C2—22no—
230C2C2×#1686118611yescharacteristic
231C2#2255068211yescharacteristic
232C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127#3082
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #22550185C2semidirect
4. #22550181C2semidirect
5. #22550169C2semidirect
6. #22550159C2semidirect
65. 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
#225505072
#225503542
#225501852
#225501812
#225501692
#225501592
#225499162
#225499142
#225499022
#225498852
C2×#103×C1272
#3082127