Finite Group Databasebeta

← #30637092

Subgroups — #30637092

Order 16256.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#16861187C211yesmaximal, characteristic
2#22549838C211yesmaximal, characteristic
3C22⋊C8×D127C211yesmaximal, characteristic
4#22549602C211yesmaximal, characteristic, complemented
5#22549601C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#324×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C411yescharacteristic
9C22×C4×D127C2211yescharacteristic
10C2×#14523231—21no—
11C2×C8×D127—21no—
12#16861187—21no—
13#16861187—21no—
14#16861187—21no—
15#16861187—21no—
16C2×Dic127⋊C4C411yescharacteristic
17C2×Dic127⋊C4C2211yescharacteristic
18C2×(C2×D127)⋊C4C411yescharacteristic
19C2×(C2×D127)⋊C4C411yescharacteristic
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22(C2×D127)⋊C8—21no—
23(C2×D127)⋊C8—21no—
24C2×C4⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—42no—
31C22×Dic127C2×C411yescharacteristic
32C22×Dic127C2×C411yescharacteristic
33C22×Dic127C2×C411yescharacteristic
34C2×C4×D127—42no—
35C2×C4×D127D411yescharacteristic
36C2×C4×D127D411yescharacteristic
37C2×C4×D127D411yescharacteristic
38C2×C4×D127D411yescharacteristic
39#14523231—42no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C8×D127—42no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51C22×C4×C127C2×C411yescharacteristic
52C22×C4×C127C2×C411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—22no—
61C22×D127—41no—
62C22×D127C22⋊C411yescharacteristic
63C22×D127C22⋊C411yescharacteristic
64Dic254—42no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C2×Dic127C22⋊C411yescharacteristic
73C2×Dic127C22⋊C411yescharacteristic
74C4×D127—22no—
75C4×D127—22no—
76C4×D127—41no—
77C4×D127—41no—
78C127⋊C8—41no—
79C127⋊C8—41no—
80C23×C127C22×C411yescharacteristic
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C8×C127—41no—
90C8×C127—41no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127C2×C22⋊C411yescharacteristic
94C22×C127#13111yescharacteristic
95C22×C127#13111yescharacteristic
96C2×D127—21no—
97C2×D127—21no—
98C2×D127—21no—
99C2×D127—21no—
100C2×D127—21no—
101C2×D127—42no—
102C2×D127—42no—
103C2×D127#10112yes—
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—41no—
107C4×C127—41no—
108Dic127—21no—
109Dic127—41no—
110Dic127—41no—
111Dic127#9912yes—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127#40811yescharacteristic
115C2×C127C2×#9911yescharacteristic
116C2×C127C2×#10111yescharacteristic
117D127—22no—
118D127—41no—
119#1472—1271nomaximal, Sylow
120C127#147211yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—1271no—
122#407—1271no—
123C2×C22⋊C8—1271no—
124#324—1271no—
125#324—1271no—
126#324—1271no—
127#324D12711yescharacteristic, complemented
128C22×Q8—1271no—
129C23×C4—1271no—
130C2×M16—2541no—
131C22×C8—2541no—
132C22⋊Q8—2541no—
133C22⋊Q8—2541no—
134C22⋊Q8—2541no—
135C22⋊Q8—2541no—
136C2×C4⋊C4—1271no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4C2×D12711yescharacteristic
139C2×C22⋊C4—1271no—
140C2×C22⋊C4—1271no—
141C22⋊C8—1271no—
142C22⋊C8—1271no—
143C22⋊C8—2541no—
144C22⋊C8—2541no—
145C22⋊C8C2×D12711yescharacteristic
146C22⋊C8C2×D12711yescharacteristic
147C24—1271no—
148C2×Q8—5082no—
149C2×Q8—2541no—
150C2×Q8—2541no—
151C22×C4—1271no—
152C22×C4—1271no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4—1271no—
158C22×C4—5082no—
159C22×C4C4×D12711yescharacteristic
160C22×C4C4×D12711yescharacteristic
161C22×C4C22×D12711yescharacteristic
162M16—5082no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—5082no—
166C2×C8—2541no—
167C2×C8—2541no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—2541no—
171C4⋊C4—2541no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C23—1271no—
179C23—1271no—
180C23—5081no—
181C23—2542no—
182C23—2541no—
183C23—2541no—
184C23C2×C4×D12711yescharacteristic
185Q8—5082no—
186Q8—5082no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1271no—
194C2×C4—1271no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—2542no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4D4×D12711yescharacteristic
206C2×C4D4×D12711yescharacteristic
207C8—41no—
208C8—41no—
209C8—5081no—
210C8—5081no—
211C22—21no—
212C22—21no—
213C22—1272no—
214C22—5082no—
215C22—5082no—
216C22—2541no—
217C22—2541no—
218C22—2541no—
219C22—2541no—
220C22—2541no—
221C22C22⋊C4×D12711yescharacteristic
222C22#1686121511yescharacteristic
223C22#1686121711yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—1272no—
229C4—5081no—
230C4—5081no—
231C4—2541no—
232C2—21no—
233C2—21no—
234C2—5081no—
235C2—2542no—
236C2#2254984111yescharacteristic
237C2#99×D12711yescharacteristic
238C2#101×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#324D127
hypocenterC127#1472
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #22549602C2semidirect
5. #22549601C2semidirect
6. #22549601C2semidirect
7. #324×C127C2semidirect
120. C127#1472semidirect
127. #324D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611872
#225498382
C22⋊C8×D1272
#225496022
#225496012
#324×C1272
#1472127