Finite Group Databasebeta

← #30639101

Subgroups — #30639101

Order 16256.

334 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 97 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#22550359C211yesmaximal, characteristic
2C2×#16861136C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4#22549783C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—21no—
10D4×Dic127—22no—
11D4×Dic127—21no—
12#16861168—21no—
13#16861168C2211yescharacteristic
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4×Dic127C2211yescharacteristic
17#16861136—22no—
18#16861133—21no—
19#16861133—21no—
20C127⋊#95—22no—
21C2×#41×C127C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23#137×C127—22no—
24#137×C127C414yescomplemented
25C2×Q16×C127—21no—
26C2×SD16×C127—21no—
27C2×M16×C127C2211yescharacteristic
28C22×Dic127—22no—
29C22×Dic127—21no—
30C22×Dic127D412yes—
31#14523232—21no—
32#14523232—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—22no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39C22⋊Dic127—22no—
40C22⋊Dic127—21no—
41C4×Dic127—22no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127C2×C412yes—
48#41×C127D412yes—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127C2×C412yescomplemented
55C2×Q8×C127D411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127D411yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61Q16×C127—24no—
62SD16×C127—24no—
63M16×C127—21no—
64M16×C127C2×C412yescomplemented
65C2×C8×C127—21no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—22no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—42no—
73C2×Dic127—82no—
74C127⋊C8—41no—
75C23×C127—21no—
76C23×C127C2×D411yescharacteristic
77Q8×C127—21no—
78Q8×C127—22no—
79Q8×C127—22no—
80Q8×C127—41no—
81Q8×C127—41no—
82Q8×C127C22⋊C412yes—
83D4×C127—22no—
84D4×C127—22no—
85D4×C127—21no—
86D4×C127C22⋊C412yes—
87C2×C4×C127—21no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C22×C411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C8×C127—22no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—22no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127C4×D411yescharacteristic
106C22×C127#12011yescharacteristic
107C22×C127#12111yescharacteristic
108C4×C127—22no—
109C4×C127—42no—
110C4×C127C2×C22⋊C411yescharacteristic
111C4×C127C4×D411yescharacteristic
112C4×C127#12111yescharacteristic
113C4×C127#12311yescharacteristic
114Dic127—42no—
115Dic127—42no—
116Dic127—41no—
117Dic127—41no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—42no—
121C2×C127#38611yescharacteristic
122C2×C127#45511yescharacteristic
123C2×C127#45611yescharacteristic
124#1882—1271nomaximal, Sylow
125C127#188211yescharacteristic, complemented, Sylow
126C2×#137D12711yescharacteristic
127#520—1271no—
128C2×#104—1271no—
129#419—1271no—
130#391—1271no—
131#343—1271no—
132#339—1271no—
133C2×#41C2×D12711yescharacteristic
134C22×Q8C2×D12711yescharacteristic
135#137—22no—
136#137Dic12714yescomplemented
137C2×Q16—21no—
138C2×SD16—21no—
139C2×M16—1271no—
140C2×M16C2×D12711yescharacteristic
141C4×Q8—2541no—
142C4×D4—2542no—
143C4×D4—2541no—
144#117—1271no—
145#117—2541no—
146C2×C4⋊C4—1271no—
147C2×C4⋊C4—2541no—
148C2×C42—1271no—
149#104—2542no—
150#103—2541no—
151#103—2541no—
152#95—2542no—
153#41—21no—
154#41—21no—
155#41C2×Dic12712yes—
156#41C22⋊D12712yes—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8C2×Dic12712yescomplemented
163C2×Q8C22⋊D12711yescharacteristic
164C2×D4—21no—
165C2×D4C22⋊D12711yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1272no—
169C22×C4—2542no—
170C22×C4—2541no—
171C22×C4C22×D12711yescharacteristic
172Q16—24no—
173SD16—24no—
174M16—21no—
175M16—2541no—
176M16—2541no—
177M16C2×Dic12712yescomplemented
178C2×C8—21no—
179C2×C8—2541no—
180C4⋊C4—1271no—
181C4⋊C4—1271no—
182C4⋊C4—2542no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C22⋊C4—2542no—
186C22⋊C4—2541no—
187C42—2542no—
188C42—2541no—
189C42—2541no—
190C42—2541no—
191C23—21no—
192C23D4×D12711yescharacteristic
193Q8—21no—
194Q8—22no—
195Q8—22no—
196Q8—41no—
197Q8—41no—
198Q8C22⋊Dic12712yes—
199D4—22no—
200D4—22no—
201D4—21no—
202D4C22⋊Dic12712yes—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—10162no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—5082no—
213C2×C4—2542no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2542no—
217C2×C4C22×Dic12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C2×C22⋊D12711yescharacteristic
220C2×C4C2×C22⋊D12711yescharacteristic
221C2×C4#1452324711yescharacteristic
222C2×C4#1452324711yescharacteristic
223C8—22no—
224C8—5081no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22D4×Dic12711yescharacteristic
231C22#1686117711yescharacteristic
232C22#1686118111yescharacteristic
233C4—22no—
234C4—42no—
235C4—5082no—
236C4—5081no—
237C4—5082no—
238C4—5081no—
239C4C2×C22⋊Dic12711yescharacteristic
240C4D4×Dic12711yescharacteristic
241C4#1686118111yescharacteristic
242C4#1686119011yescharacteristic
243C2—21no—
244C2—21no—
245C2—42no—
246C2#2254975611yescharacteristic
247C2#2255005211yescharacteristic
248C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#137D127
hypocenterC127#1882
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. #22549885C2semidirect
4. #22549783C2semidirect
5. #22549644C2semidirect
6. #22549637C2semidirect
24. #137×C127C4semidirect
54. C2×Q8×C127C2×C4semidirect
64. M16×C127C2×C4semidirect
125. C127#1882semidirect
136. #137Dic127semidirect
162. C2×Q8C2×Dic127semidirect
177. M16C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503592
C2×#168611362
#225498852
#225497832
#225496442
#225496372
C2×#137×C1272
#1882127