Finite Group Databasebeta

← #30637088

Subgroups — #30637088

Order 16256.

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

normal subgroups only

This group has 322 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×#16861182C211yesmaximal, characteristic
2#22549838C211yesmaximal, characteristic, complemented
3C22⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#22549599C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#323×C127C211yesmaximal, characteristic, complemented
8C22×D508C411yescharacteristic
9C22×C22⋊D127C411yescharacteristic
10C22×C22⋊D127C411yescharacteristic
11C22×C4×D127C2211yescharacteristic
12C2×#14523231—21no—
13C2×C8×D127—21no—
14#16861182—21no—
15#16861182—21no—
16#16861182—21no—
17#16861182—21no—
18C2×Dic127⋊C4C411yescharacteristic
19C2×(C2×D127)⋊C4C2211yescharacteristic
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22(C2×D127)⋊C8—21no—
23(C2×D127)⋊C8—21no—
24C2×C22⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C23×D127C2×C411yescharacteristic
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—42no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—42no—
37C2×C22⋊D127—42no—
38C22×Dic127C2×C411yescharacteristic
39C22×Dic127C2×C411yescharacteristic
40C2×C4×D127—42no—
41C2×C4×D127D411yescharacteristic
42C2×C4×D127D411yescharacteristic
43C2×C4×D127D411yescharacteristic
44C2×C4×D127D411yescharacteristic
45#14523231—42no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8—21no—
48C8×D127—42no—
49Dic127⋊C4—21no—
50Dic127⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C24×C127C2×C411yescharacteristic
54C22×C4×C127C2×C411yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22×D127—21no—
61C22×D127—21no—
62C22×D127—22no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127C22⋊C411yescharacteristic
70C22×D127C22⋊C411yescharacteristic
71D508—42no—
72D508—42no—
73C22⋊D127—42no—
74C22⋊D127—42no—
75C22⋊D127—42no—
76C22⋊D127—42no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C2×Dic127—41no—
80C2×Dic127C22⋊C411yescharacteristic
81C2×Dic127C22⋊C411yescharacteristic
82C4×D127—22no—
83C4×D127—22no—
84C4×D127—41no—
85C4×D127—41no—
86C127⋊C8—41no—
87C127⋊C8—41no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127C22×C411yescharacteristic
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C8×C127—41no—
100C8×C127—41no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—81no—
107C22×C127—81no—
108C22×C127C2×C22⋊C411yescharacteristic
109C22×C127#13111yescharacteristic
110C22×C127#13111yescharacteristic
111C2×D127—21no—
112C2×D127—21no—
113C2×D127—21no—
114C2×D127—21no—
115C2×D127—21no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—41no—
121C2×D127—81no—
122C2×D127—81no—
123C2×D127#10012yes—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—41no—
127Dic127—21no—
128Dic127—41no—
129Dic127#9912yes—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—81no—
133C2×C127#40811yescharacteristic
134C2×C127C2×#9911yescharacteristic
135C2×C127C2×#10011yescharacteristic
136D127—22no—
137D127—41no—
138D127—81no—
139#1471—1271nomaximal, Sylow
140C127#147111yescharacteristic, complemented, Sylow
141C2×#121—1271no—
142#407—1271no—
143C2×C22⋊C8—1271no—
144#323—1271no—
145#323—1271no—
146#323—1271no—
147#323D12711yescharacteristic, complemented
148C22×D4—1271no—
149C22×D4—1271no—
150C22×D4—1271no—
151C23×C4—1271no—
152C2×M16—2541no—
153C22×C8—2541no—
154#121—2541no—
155#121—2541no—
156#121—2541no—
157#121—2541no—
158C2×C4⋊C4—1271no—
159C2×C22⋊C4—1271no—
160C2×C22⋊C4C2×D12711yescharacteristic
161C22⋊C8—1271no—
162C22⋊C8—1271no—
163C22⋊C8—2541no—
164C22⋊C8—2541no—
165C22⋊C8C2×D12711yescharacteristic
166C22⋊C8C2×D12711yescharacteristic
167C24—1271no—
168C24—1271no—
169C24C4×D12711yescharacteristic
170C2×D4—5082no—
171C2×D4—5082no—
172C2×D4—5082no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1271no—
182C22×C4—1271no—
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—5082no—
186C22×C4C4×D12711yescharacteristic
187C22×C4C22×D12711yescharacteristic
188M16—5082no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—5082no—
192C2×C8—2541no—
193C2×C8—2541no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—2541no—
199C22⋊C4—2541no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—1271no—
206C23—1271no—
207C23—5081no—
208C23—5081no—
209C23—5081no—
210C23—2542no—
211C23—2541no—
212C23—2541no—
213C23—2541no—
214C23—2541no—
215C23—2541no—
216C23C2×C4×D12711yescharacteristic
217D4—5082no—
218D4—5082no—
219D4—5082no—
220D4—5082no—
221D4—5082no—
222D4—5082no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—1271no—
227C2×C4—1271no—
228C2×C4—5081no—
229C2×C4—5081no—
230C2×C4—5081no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4D4×D12711yescharacteristic
236C2×C4D4×D12711yescharacteristic
237C8—41no—
238C8—41no—
239C8—5081no—
240C8—5081no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—1272no—
249C22—10161no—
250C22—10161no—
251C22—5082no—
252C22—5082no—
253C22—5081no—
254C22—5081no—
255C22—5081no—
256C22—2541no—
257C22—2541no—
258C22—2541no—
259C22—2541no—
260C22—2541no—
261C22C22⋊C4×D12711yescharacteristic
262C22#1686121511yescharacteristic
263C22#1686121711yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—1272no—
268C4—5081no—
269C4—2541no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—10161no—
274C2—5081no—
275C2—2542no—
276C2#2254984111yescharacteristic
277C2#99×D12711yescharacteristic
278C2#100×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861182C2
cosocleC22×C4×C127C23
hypercenter#323D127
hypocenterC127#1471
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22549838C2semidirect
3. C22⋊C8×D127C2semidirect
4. #22549600C2semidirect
5. #22549599C2semidirect
6. #22549599C2semidirect
7. #323×C127C2semidirect
140. C127#1471semidirect
147. #323D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611822
#225498382
C22⋊C8×D1272
#225496002
#225495992
#323×C1272
#1471127