Finite Group Databasebeta

← #30642018

Subgroups — #30642018

Order 16256.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16861174C211yesmaximal, characteristic, complemented
2C2×D4×Dic127C211yesmaximal, characteristic
3#22549816C211yesmaximal, characteristic, complemented
4#22549815C211yesmaximal, characteristic, complemented
5#22549804C212yesmaximal, complemented
6#22549790C211yesmaximal, characteristic
7#22549788C212yesmaximal, complemented
8C22⋊C4×Dic127C212yesmaximal, complemented
9C22⋊C4×Dic127C212yesmaximal, complemented
10#22549721C211yesmaximal, characteristic, complemented
11#394×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C23×Dic127C2212yes—
14#16861174—24no—
15D4×Dic127—24no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4C2211yescharacteristic
18C2×Dic127⋊C4C2211yescharacteristic
19C2×C22⋊Dic127C2212yes—
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2212yes—
22C2×C4×Dic127C2211yescharacteristic
23C2×C4×Dic127C2211yescharacteristic
24C2×C4×Dic127C2212yes—
25C2×C4×Dic127C2211yescharacteristic
26C127⋊#95C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2211yescharacteristic
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C22×D4×C127C2211yescharacteristic
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2212yes—
35C2×C22⋊C4×C127C2212yes—
36#95×C127C2211yescharacteristic
37C2×Dic254—22no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C22×Dic127—24no—
41C22×Dic127—24no—
42C22×Dic127—24no—
43C22×Dic127C2312yes—
44C22×Dic127C2312yes—
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2311yescharacteristic
49C4⋊Dic127—22no—
50Dic127⋊C4—22no—
51Dic127⋊C4—22no—
52C22⋊Dic127—24no—
53C22⋊Dic127—24no—
54C22⋊Dic127—24no—
55C4×Dic127—22no—
56C4×Dic127—22no—
57C4×Dic127—24no—
58C4×Dic127D414yes—
59C24×C127C2312yes—
60C2×D4×C127—24no—
61C2×D4×C127—22no—
62C22×C4×C127C2312yes—
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—22no—
67C22⋊C4×C127—24no—
68C22⋊C4×C127—24no—
69Dic254—28no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—48no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C2×Dic127#4112yes—
86C2×Dic127#4114yes—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127#4114yes—
94C23×C127C2411yescharacteristic
95D4×C127—28no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127C2×D412yes—
105C2×C4×C127#4112yes—
106C22×C127—24no—
107C22×C127—24no—
108C22×C127—24no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127C22×D411yescharacteristic
114C22×C127C2×#4111yescharacteristic
115C22×C127C2×#4112yes—
116C22×C127C2×#4112yes—
117C22×C127C2×#4111yescharacteristic
118C4×C127—22no—
119C4×C127—22no—
120C4×C127—41no—
121C4×C127—42no—
122Dic127—22no—
123Dic127—22no—
124Dic127—41no—
125Dic127—42no—
126Dic127—42no—
127Dic127#12414yes—
128C2×C127—44no—
129C2×C127#52511yescharacteristic
130C2×C127#52511yescharacteristic
131C2×C127C2×#12411yescharacteristic
132C2×C127#52912yes—
133C2×C127#52912yes—
134#2398—1271nomaximal, Sylow
135C127#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—1271no—
137C2×C4×D4—1271no—
138#400—1271no—
139#400—1271no—
140#397—1272no—
141#394—1271no—
142#394D12711yescharacteristic
143#393—1272no—
144C4×C22⋊C4—1272no—
145C4×C22⋊C4—1272no—
146#376—1271no—
147C22×Q8—1271no—
148C22×D4C2×D12711yescharacteristic
149C23×C4—1272no—
150C4×Q8—2544no—
151C4×D4—2544no—
152C2×C4⋊C4—1271no—
153C2×C4⋊C4—1271no—
154C2×C4⋊C4—1271no—
155C2×C4⋊C4C2×D12711yescharacteristic
156C2×C22⋊C4—1272no—
157C2×C22⋊C4—1272no—
158C2×C22⋊C4—1272no—
159C2×C22⋊C4C2×D12712yes—
160C2×C22⋊C4C2×D12712yes—
161C2×C42—1271no—
162C2×C42—1271no—
163C2×C42—1271no—
164C2×C42—1272no—
165#95—1272no—
166#95—1271no—
167#95—1272no—
168#95—1272no—
169#95—1272no—
170#95—1272no—
171#95C2×D12711yescharacteristic
172C24C22×D12712yes—
173C2×Q8—2542no—
174C2×Q8—2542no—
175C2×Q8—2542no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—1271no—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1272no—
182C22×C4—1272no—
183C22×C4—1271no—
184C22×C4—2544no—
185C22×C4—2544no—
186C22×C4—2544no—
187C22×C4C22×D12712yes—
188C22×C4C22×D12711yescharacteristic
189C22×C4C22×D12711yescharacteristic
190C22×C4C22×D12711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—2542no—
193C4⋊C4—2542no—
194C4⋊C4—2542no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—2544no—
198C22⋊C4—2544no—
199C22⋊C4—2544no—
200C42—1274no—
201C42—2544no—
202C42—2542no—
203C42—2542no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1452324714yes—
211C23C23×D12711yescharacteristic
212Q8—2548no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—1272no—
223C2×C4—1274no—
224C2×C4—1272no—
225C2×C4—1272no—
226C2×C4—5088no—
227C2×C4—2541no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2541no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4D4×D12712yes—
240C2×C4#1452324712yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D12711yescharacteristic
249C22#41×D12711yescharacteristic
250C22#41×D12712yes—
251C22C2×#1452324712yes—
252C22C2×#1452324711yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—1274no—
258C4—5081no—
259C4—5082no—
260C4—5082no—
261C4—2542no—
262C4—2542no—
263C2—44no—
264C2#2255038211yescharacteristic
265C2#2255038611yescharacteristic
266C2#124×D12711yescharacteristic
267C2#2255040312yes—
268C2#2255040712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#394×C127C2
prime radicalC22×D4×C127C22
hypercenter#394D127
hypocenterC127#2398
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16861174C2semidirect
3. #22549816C2semidirect
4. #22549815C2semidirect
5. #22549804C2semidirect
7. #22549788C2semidirect
8. C22⋊C4×Dic127C2semidirect
9. C22⋊C4×Dic127C2semidirect
10. #22549721C2semidirect
135. C127#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#168611742
C2×D4×Dic1272
#225498162
#225498152
#225498042
#225497902
#225497882
C22⋊C4×Dic1272
#225497212
#394×C1272
#2398127