Finite Group Databasebeta

← #30639914

Subgroups — #30639914

Order 16256.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861184C211yesmaximal, characteristic
2C2×#16861131C211yesmaximal, characteristic, complemented
3C2×#16861129C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
5#22549733C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#107×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic
10C22×C4×D127C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861184—22no—
13#16861184—22no—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17#16861131—22no—
18#16861129—22no—
19(C2×D127)⋊C8—22no—
20C127⋊#95—21no—
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#107×C127—22no—
25C23×D127—21no—
26C23×D127D411yescharacteristic
27C2×D508—24no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—42no—
34C2×D508—42no—
35C2×D508D412yes—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C4×D127—21no—
40C2×C4×D127—21no—
41C2×C4×D127—21no—
42C2×C4×D127—21no—
43C2×C4×D127—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49(C2×D127)⋊C4—24no—
50(C2×D127)⋊C4—42no—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127D412yescomplemented
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127D412yes—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—22no—
68C22×D127—41no—
69C22×D127—41no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—41no—
73C22×D127—41no—
74C22×D127D812yes—
75D508—22no—
76D508—22no—
77D508—48no—
78D508—41no—
79D508—41no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—41no—
84C2×Dic127—41no—
85C2×Dic127—41no—
86C4×D127—41no—
87C4×D127—41no—
88C4×D127—41no—
89C4×D127—41no—
90C127⋊C8—41no—
91C23×C127C2×D411yescharacteristic
92C2×C4×C127—22no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C8×C127—22no—
105C22×C127#12011yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12311yescharacteristic
108C22×C127C2×D811yescharacteristic
109C22×C127#13511yescharacteristic
110C22×C127AGL(1,ℤ/8)11yescharacteristic
111C22×C127#13711yescharacteristic
112C2×D127—22no—
113C2×D127—22no—
114C2×D127—22no—
115C2×D127—41no—
116C2×D127—41no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—42no—
122C2×D127—81no—
123C2×D127—81no—
124C2×D127—81no—
125C2×D127—81no—
126C4×C127—42no—
127C4×C127—81no—
128C4×C127#12111yescharacteristic
129C4×C127#12111yescharacteristic
130C4×C127#12311yescharacteristic
131C4×C127#12411yescharacteristic
132Dic127—41no—
133Dic127—81no—
134C2×C127#39411yescharacteristic
135C2×C127#44711yescharacteristic
136C2×C127#45211yescharacteristic
137C2×C127#46611yescharacteristic
138C2×C127#46811yescharacteristic
139C2×C127#48011yescharacteristic
140C2×C127#48211yescharacteristic
141D127—42no—
142D127—82no—
143#2033—1271nomaximal, Sylow
144C127#203311yescharacteristic, complemented, Sylow
145C2×#121—1271no—
146C2×#107D12711yescharacteristic, complemented
147C2×#102—1271no—
148C2×#102—1271no—
149C2×C22⋊C8—1271no—
150#380—1271no—
151#340—1271no—
152C22×D4—1271no—
153C22×D4—2541no—
154C23×C4—1271no—
155C22×C8—1271no—
156C22×C8C2×D12711yescharacteristic
157#121—2542no—
158#121—2542no—
159C2×C4⋊C4—1271no—
160C2×C4⋊C4C2×D12711yescharacteristic
161C2×C4⋊C4C2×D12711yescharacteristic
162C2×C22⋊C4—2541no—
163C2×C22⋊C4—2541no—
164#107—22no—
165#102—2542no—
166#102—2542no—
167C22⋊C8—2542no—
168#95—2541no—
169C24—1271no—
170C24—2541no—
171C2×D4—1272no—
172C2×D4—5082no—
173C2×D4—5082no—
174C2×D4—2544no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—1271no—
183C22×C4—2541no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4—2541no—
187C22×C4—2541no—
188C22×C4—2541no—
189C22×C4—2541no—
190C22×C4C22×D12711yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—2541no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8D50812yescomplemented
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2541no—
201C4⋊C4—2541no—
202C4⋊C4C22⋊D12712yes—
203C22⋊C4—5082no—
204C22⋊C4—2544no—
205C23—1272no—
206C23—5081no—
207C23—5081no—
208C23—5081no—
209C23—5081no—
210C23—5081no—
211C23—5081no—
212C23—2542no—
213C23—2541no—
214C23—2541no—
215C23—2541no—
216C23—2541no—
217C23—2541no—
218C23—2541no—
219C23D4×D12711yescharacteristic
220D4—5088no—
221D4—5081no—
222D4—5081no—
223D4—2542no—
224D4—2542no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—5081no—
232C2×C4—5081no—
233C2×C4—5081no—
234C2×C4—5081no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—5081no—
238C2×C4—2541no—
239C2×C4—2541no—
240C2×C4—2541no—
241C2×C4D4×D12711yescharacteristic
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C2×D50811yescharacteristic
244C2×C4#1452324811yescharacteristic
245C2×C4#1452324911yescharacteristic
246C2×C4#1452324911yescharacteristic
247C8—22no—
248C8—5081no—
249C22—10161no—
250C22—10161no—
251C22—10161no—
252C22—10161no—
253C22—5082no—
254C22—5082no—
255C22—5082no—
256C22—5081no—
257C22—5081no—
258C22—5081no—
259C22—5081no—
260C22—2542no—
261C22—2542no—
262C22—2542no—
263C22#1686117711yescharacteristic
264C22#1686118411yescharacteristic
265C22#1686119411yescharacteristic
266C22D8×D12711yescharacteristic
267C22#1686123211yescharacteristic
268C22#1686123311yescharacteristic
269C22#1686124111yescharacteristic
270C4—42no—
271C4—81no—
272C4—10161no—
273C4—5081no—
274C4#1686118311yescharacteristic
275C4#1686118411yescharacteristic
276C4#1686119411yescharacteristic
277C4#1686119811yescharacteristic
278C2—10162no—
279C2—5082no—
280C2#2254979311yescharacteristic
281C2#2255000811yescharacteristic
282C2#2255003711yescharacteristic
283C2#2255011311yescharacteristic
284C2#2255012311yescharacteristic
285C2#2255019111yescharacteristic
286C2#2255020311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#107×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861184C2
hypercenterC2×#107D127
hypocenterC127#2033
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#16861131C2semidirect
3. C2×#16861129C2semidirect
4. C2×(C2×D127)⋊C8C2semidirect
5. #22549733C2semidirect
6. #22549638C2semidirect
7. C2×#107×C127C2semidirect
56. C2×C8×C127D4semidirect
144. C127#2033semidirect
146. C2×#107D127semidirect
196. C2×C8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168611842
C2×#168611312
C2×#168611292
C2×(C2×D127)⋊C82
#225497332
#225496382
C2×#107×C1272
#2033127