Finite Group Databasebeta

← #30637300

Subgroups — #30637300

Order 16256.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D127C211yesmaximal, characteristic
2C22⋊C8×D127C211yesmaximal, characteristic
3#22549735C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic, complemented
5#22549613C211yesmaximal, characteristic, complemented
6#22549612C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127C411yescharacteristic, complemented
9C22×C4×D127C2211yescharacteristic
10C2×C8×D127—21no—
11#16861189C411yescharacteristic, complemented
12#16861188C2211yescharacteristic
13#16861187—21no—
14#16861186—21no—
15#16861185C411yescharacteristic, complemented
16C2×Dic127⋊C4—21no—
17C4⋊C4×D127—21no—
18C4⋊C4×D127C411yescharacteristic, complemented
19C2×(C2×D127)⋊C4—21no—
20C22⋊C4×D127—21no—
21C22⋊(C127⋊C8)C2211yescharacteristic
22(C2×D127)⋊C8—21no—
23C127⋊#95C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254C2×C411yescharacteristic
30Q8×D127—42no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—42no—
36C2×C4×D127C2×C411yescharacteristic
37C2×C4×D127C2×C411yescharacteristic
38C2×C4×D127D411yescharacteristic
39C2×C4×D127D411yescharacteristic
40C2×C127⋊C8—21no—
41C8×D127—42no—
42C4⋊Dic127—21no—
43C4⋊Dic127C2×C411yescharacteristic
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47Dic127⋊C4—42no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—42no—
52C2×Q8×C127C2×C411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2×C411yescharacteristic
58C22⋊C4×C127—21no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127C22⋊C411yescharacteristic
62C22×D127C22⋊C411yescharacteristic
63C22×D127SD1612yes—
64C22×D127Q1612yes—
65Dic254—42no—
66Dic254—41no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127C22⋊C411yescharacteristic
74C2×Dic127C22⋊C411yescharacteristic
75C4×D127—22no—
76C4×D127—22no—
77C4×D127—41no—
78C4×D127—41no—
79C4×D127—42no—
80C4×D127—42no—
81C127⋊C8—41no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—41no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C22×C411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—41no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127C2×SD1611yescharacteristic
97C22×C127C2×Q1611yescharacteristic
98C2×D127—21no—
99C2×D127—21no—
100C2×D127—21no—
101C2×D127—21no—
102C2×D127—42no—
103C2×D127—42no—
104C2×D127#10312yes—
105C2×D127#10312yes—
106C4×C127—21no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—81no—
111Dic127—21no—
112Dic127—41no—
113Dic127—41no—
114Dic127—81no—
115Dic127#9912yes—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127C2×#9911yescharacteristic
119C2×C127C2×#10311yescharacteristic
120C2×C127#42111yescharacteristic
121D127—22no—
122D127—22no—
123#1520—1271nomaximal, Sylow
124C127#152011yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—1271no—
126C2×C22⋊C8—1271no—
127#380—1271no—
128#328—1271no—
129#328—1271no—
130#328—1271no—
131#328D12711yescharacteristic, complemented
132C22×Q8—1271no—
133C23×C4—1271no—
134C22×C8—2541no—
135C22⋊Q8—1271no—
136C22⋊Q8—1271no—
137C22⋊Q8—1271no—
138C22⋊Q8—2541no—
139C22⋊Q8—2541no—
140C22⋊Q8C2×D12711yescharacteristic
141C2×C4⋊C4—1271no—
142C2×C4⋊C4—2541no—
143C2×C4⋊C4—2541no—
144C2×C22⋊C4—2541no—
145C2×C22⋊C4—2541no—
146C22⋊C8—1271no—
147C22⋊C8—2541no—
148C22⋊C8C2×D12711yescharacteristic
149#95—1271no—
150#95C2×D12711yescharacteristic
151C24—1271no—
152C2×Q8—1271no—
153C2×Q8—5082no—
154C2×Q8—2541no—
155C2×Q8C4×D12711yescharacteristic
156C22×C4—21no—
157C22×C4—1271no—
158C22×C4—1271no—
159C22×C4—1271no—
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—5082no—
163C22×C4—2541no—
164C22×C4—2541no—
165C22×C4—2541no—
166C22×C4C22×D12711yescharacteristic
167C2×C8—21no—
168C2×C8—5082no—
169C2×C8—2541no—
170C4⋊C4—21no—
171C4⋊C4—1271no—
172C4⋊C4—5082no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4C4×D12711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—5082no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C23—1272no—
184C23—1272no—
185C23—1271no—
186C23—1271no—
187C23—5081no—
188C23—5081no—
189C23D4×D12711yescharacteristic
190Q8—41no—
191Q8—5081no—
192Q8—5082no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1271no—
200C2×C4—1271no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—5082no—
204C2×C4—5082no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4—2542no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4C2×C4×D12711yescharacteristic
214C2×C4D4×D12711yescharacteristic
215C8—41no—
216C8—5081no—
217C22—21no—
218C22—21no—
219C22—1272no—
220C22—1272no—
221C22—5082no—
222C22—5082no—
223C22—2541no—
224C22—2541no—
225C22—2541no—
226C22—2541no—
227C22C22⋊C4×D12711yescharacteristic
228C22SD16×D12711yescharacteristic
229C22Q16×D12711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—1272no—
236C4—10161no—
237C4—5081no—
238C4—5081no—
239C4—2541no—
240C2—21no—
241C2—21no—
242C2—2542no—
243C2—2542no—
244C2#99×D12711yescharacteristic
245C2#103×D12711yescharacteristic
246C2#2254989111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenter#328D127
hypocenterC127#1520
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #22549614C2semidirect
5. #22549613C2semidirect
6. #22549612C2semidirect
7. #328×C127C2semidirect
8. C2×Q8×D127C4semidirect
11. #16861189C4semidirect
15. #16861185C4semidirect
18. C4⋊C4×D127C4semidirect
124. C127#1520semidirect
131. #328D127semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1272
C22⋊C8×D1272
#225497352
#225496142
#225496132
#225496122
#328×C1272
#1520127