Finite Group Databasebeta

← #30648025

Subgroups — #30648025

Order 16256.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523238C211yesmaximal, characteristic, complemented
3#22550447C211yesmaximal, characteristic, complemented
4#22550038C214yesmaximal, complemented
5#22550032C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9C2×2−1+4×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861239—24no—
12C2×#14523238—24no—
13C2×#14523238C2212yes—
14C2×#14523238C2214yes—
15C2×#14523236C2212yes—
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18#16861207—21no—
19#16861196—21no—
20#16861190—22no—
21#16861188C2214yes—
22#16861168C2211yescharacteristic
23#16861133C2212yes—
24#16861133C2214yes—
25#16861130C2212yes—
26C22⋊(C127⋊C8)C2214yes—
272−1+4×C127—24no—
282−1+4×C127—24no—
29C2×#41×C127—22no—
30C2×#41×C127—21no—
31C2×#41×C127C2212yes—
32C2×#41×C127C2211yescharacteristic
33C2×#41×C127C2211yescharacteristic
34C22×Q8×C127—21no—
35C22×Q8×C127C2212yes—
36C22×Q8×C127C2211yescharacteristic
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254C2312yes—
41C22×Dic127—21no—
42#14523238—28no—
43#14523238—24no—
44#14523236—24no—
45#14523232—22no—
46C2×C127⋊C8—22no—
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2312yes—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—24no—
51C22⋊Dic127—24no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54#41×C127—216no—
55#41×C127—28no—
56#41×C127—24no—
57#41×C127—24no—
58#41×C127—22no—
59#41×C127—22no—
60#41×C127—22no—
61#41×C127D414yes—
62C2×Q8×C127—28no—
63C2×Q8×C127—24no—
64C2×Q8×C127—24no—
65C2×Q8×C127—22no—
66C2×Q8×C127—21no—
67C2×Q8×C127—22no—
68C2×Q8×C127D414yes—
69C2×Q8×C127D411yescharacteristic
70C2×Q8×C127C2311yescharacteristic
71C2×Q8×C127C2312yes—
72C2×D4×C127—22no—
73C2×D4×C127—21no—
74C2×D4×C127D412yes—
75C2×D4×C127D411yescharacteristic
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—22no—
78C22×C4×C127—21no—
79C22×C4×C127—22no—
80C22×C4×C127—21no—
81C22×C4×C127C2312yes—
82C22×C4×C127C2311yescharacteristic
83Dic254—22no—
84Dic254—22no—
85Dic254—42no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—42no—
89C127⋊C8—22no—
90C127⋊C8—22no—
91C23×C127—21no—
92C23×C127C2×D412yes—
93C23×C127C2×D411yescharacteristic
94Q8×C127—28no—
95Q8×C127—24no—
96Q8×C127—22no—
97Q8×C127—21no—
98Q8×C127—22no—
99Q8×C127C2×D412yes—
100Q8×C127C2×D414yes—
101D4×C127—28no—
102D4×C127—24no—
103D4×C127—24no—
104D4×C127—22no—
105D4×C127—21no—
106D4×C127C2×D412yes—
107C2×C4×C127—28no—
108C2×C4×C127—28no—
109C2×C4×C127—24no—
110C2×C4×C127—22no—
111C2×C4×C127—21no—
112C2×C4×C127—24no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2411yescharacteristic
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—24no—
124C22×C127—42no—
125C22×C127#12012yes—
126C22×C127C22×D411yescharacteristic
127C4×C127—24no—
128C4×C127—22no—
129C4×C127—22no—
130C4×C127#12012yes—
131C4×C127C22×D411yescharacteristic
132C4×C127C22×D411yescharacteristic
133Dic127—42no—
134Dic127—42no—
135C2×C127—22no—
136C2×C127—42no—
137C2×C127—42no—
138C2×C127C2×#12011yescharacteristic
139C2×C127#57812yes—
140#3011—1271nomaximal, Sylow
141C127#301111yescharacteristic, complemented, Sylow
142C2×2−1+4D12711yescharacteristic
143C2×#137—1271no—
144C22×Q16—1271no—
145#536—1271no—
146#452—1274no—
147#451—1274no—
148#419—1271no—
149#416—1271no—
150#408—1271no—
1512−1+4—24no—
1522−1+4—24no—
153C2×#41—22no—
154C2×#41—21no—
155C2×#41C2×D12712yes—
156C2×#41C2×D12711yescharacteristic
157C2×#41C2×D12711yescharacteristic
158C22×Q8—21no—
159C22×Q8—1271no—
160C22×Q8C2×D12712yes—
161C22×Q8C2×D12711yescharacteristic
162#137—2544no—
163C2×Q16—1274no—
164C2×Q16—1272no—
165C2×Q16—2544no—
166C2×SD16—1272no—
167C2×M16—1271no—
168C22×C8—1271no—
169C4⋊Q8—2541no—
170#124—2541no—
171#123—2542no—
172C22⋊Q8—1274no—
173#117—1271no—
174#103—1272no—
175#103—1274no—
176#102—1272no—
177C22⋊C8—1274no—
178#41—216no—
179#41—28no—
180#41—24no—
181#41—24no—
182#41—22no—
183#41—22no—
184#41—22no—
185#41C22⋊D12714yes—
186C2×Q8—28no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—22no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1272no—
193C2×Q8—2542no—
194C2×Q8—2542no—
195C2×Q8—2541no—
196C2×Q8C22⋊D12714yes—
197C2×Q8C22⋊D12711yescharacteristic
198C2×Q8C22×D12711yescharacteristic
199C2×Q8C22×D12712yes—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4C22⋊D12712yes—
203C2×D4C22⋊D12711yescharacteristic
204C2×D4C22×D12711yescharacteristic
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—2541no—
210C22×C4C22×D12712yes—
211C22×C4C22×D12711yescharacteristic
212Q16—2548no—
213Q16—2544no—
214SD16—2544no—
215M16—2542no—
216C2×C8—1272no—
217C2×C8—1272no—
218C2×C8—2542no—
219C4⋊C4—1272no—
220C4⋊C4—2544no—
221C22⋊C4—2541no—
222C22⋊C4—2544no—
223C42—2541no—
224C23—21no—
225C23C2×C22⋊D12712yes—
226C23C2×C22⋊D12711yescharacteristic
227Q8—28no—
228Q8—21no—
229Q8—22no—
230Q8—24no—
231Q8—22no—
232Q8—5082no—
233Q8—2542no—
234Q8—2542no—
235Q8C2×C22⋊D12712yes—
236Q8C2×C22⋊D12714yes—
237D4—28no—
238D4—24no—
239D4—24no—
240D4—21no—
241D4—22no—
242D4C2×C22⋊D12712yes—
243C2×C4—28no—
244C2×C4—28no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—5082no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4C2×C22⋊D12712yes—
255C2×C4C2×C22⋊D12712yes—
256C2×C4C2×C22⋊D12712yes—
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4C23×D12711yescharacteristic
259C8—2542no—
260C8—2542no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—24no—
265C22—42no—
266C22#1686117812yes—
267C22C22×C22⋊D12711yescharacteristic
268C4—24no—
269C4—22no—
270C4—22no—
271C4—5082no—
272C4—5082no—
273C4#1686117812yes—
274C4C22×C22⋊D12711yescharacteristic
275C4C22×C22⋊D12711yescharacteristic
276C2—22no—
277C2—42no—
278C2—42no—
279C2C2×#1686117811yescharacteristic
280C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×2−1+4×C127C2
socleC22×C127C22×D4
hypercenterC2×2−1+4D127
hypocenterC127#3011
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523238C2semidirect
3. #22550447C2semidirect
4. #22550038C2semidirect
5. #22550032C2semidirect
6. #22549885C2semidirect
7. #22549872C2semidirect
8. #22549843C2semidirect
141. C127#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232382
#225504472
#225500382
#225500322
#225498852
#225498722
#225498432
C2×2−1+4×C1272
#3011127