Finite Group Databasebeta

← #30639912

Subgroups — #30639912

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×#16861183C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic, complemented
3C2×#16861129C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7#380×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—21no—
10C22×C127⋊C8C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861183—22no—
13#16861183—22no—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—21no—
16#16861142—22no—
17#16861129—22no—
18#16861129—22no—
19C22⋊(C127⋊C8)—22no—
20C23×C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127—21no—
24#95×C127—21no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C2×C22⋊D127—24no—
33C2×C22⋊D127—42no—
34C2×C22⋊D127—42no—
35C22×Dic127—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8D412yescomplemented
42C4⋊Dic127—21no—
43C4⋊Dic127D412yes—
44(C2×D127)⋊C4—24no—
45C24×C127D411yescharacteristic
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127D411yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127—21no—
60C22⋊C4×C127—42no—
61C22×D127—22no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68D508—22no—
69D508—22no—
70D508—41no—
71D508—41no—
72C22⋊D127—48no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C127⋊C8—22no—
77C127⋊C8—41no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—21no—
84C23×C127D812yes—
85C23×C127C2×D411yescharacteristic
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C22×C127—22no—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127#12011yescharacteristic
113C22×C127#12111yescharacteristic
114C22×C127#12311yescharacteristic
115C22×C127C2×D811yescharacteristic
116C22×C127#13511yescharacteristic
117C22×C127AGL(1,ℤ/8)11yescharacteristic
118C22×C127#13711yescharacteristic
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—42no—
122C2×D127—81no—
123C2×D127—81no—
124C2×D127—81no—
125C2×D127—81no—
126C4×C127—41no—
127C4×C127—81no—
128C4×C127—81no—
129C4×C127#12111yescharacteristic
130C4×C127#12111yescharacteristic
131C4×C127#12311yescharacteristic
132C4×C127#12411yescharacteristic
133Dic127—42no—
134C2×C127—42no—
135C2×C127#39411yescharacteristic
136C2×C127#44711yescharacteristic
137C2×C127#45211yescharacteristic
138C2×C127#46611yescharacteristic
139C2×C127#46811yescharacteristic
140C2×C127#48011yescharacteristic
141C2×C127#48211yescharacteristic
142D127—82no—
143#2033—1271nomaximal, Sylow
144C127#203311yescharacteristic, complemented, Sylow
145C2×#121—1271no—
146C2×#107—1271no—
147C2×#102—1271no—
148C2×#102—1271no—
149C2×C22⋊C8—1271no—
150#380D12711yescharacteristic, complemented
151#340—1271no—
152C22×D4—1271no—
153C22×D4—2541no—
154C23×C4C2×D12711yescharacteristic
155C22×C8—1271no—
156C22×C8—1271no—
157#121—2542no—
158#121—2542no—
159C2×C4⋊C4—1271no—
160C2×C4⋊C4C2×D12711yescharacteristic
161C2×C4⋊C4C2×D12711yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—2541no—
164#107—2542no—
165#102—2542no—
166#102—2542no—
167C22⋊C8—2542no—
168#95—21no—
169C24—2541no—
170C24C22⋊D12711yescharacteristic
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—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—2541no—
189C22×C4C22⋊D12711yescharacteristic
190C22×C4C22×D12711yescharacteristic
191C2×C8—1272no—
192C2×C8—2541no—
193C2×C8—2541no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8—2541no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—1272no—
202C4⋊C4—2541no—
203C22⋊C4—42no—
204C22⋊C4—2544no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—5081no—
212C23—5081no—
213C23—5081no—
214C23—5081no—
215C23—5081no—
216C23—5081no—
217C23—2542no—
218C23#1452323312yes—
219C23C2×C22⋊D12711yescharacteristic
220D4—5088no—
221D4—5081no—
222D4—5081no—
223D4—2542no—
224D4—2542no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4#1452324711yescharacteristic
245C2×C4#1452324811yescharacteristic
246C2×C4#1452324811yescharacteristic
247C8—5081no—
248C8—2542no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—10161no—
257C22—10161no—
258C22—10161no—
259C22—10161no—
260C22—5082no—
261C22—5082no—
262C22—5082no—
263C22#1686117711yescharacteristic
264C22#1686118111yescharacteristic
265C22#1686119211yescharacteristic
266C22C2×#1452323311yescharacteristic
267C22#1686122911yescharacteristic
268C22#1686123411yescharacteristic
269C22#1686124011yescharacteristic
270C4—41no—
271C4—81no—
272C4—81no—
273C4—5082no—
274C4#1686118211yescharacteristic
275C4#1686118211yescharacteristic
276C4#1686119111yescharacteristic
277C4#1686119511yescharacteristic
278C2—42no—
279C2—10162no—
280C2#2254979111yescharacteristic
281C2#2255000911yescharacteristic
282C2#2255003611yescharacteristic
283C2#2255011011yescharacteristic
284C2#2255012411yescharacteristic
285C2#2255018911yescharacteristic
286C2#2255020011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#380×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenter#380D127
hypocenterC127#2033
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16861142C2semidirect
3. C2×#16861129C2semidirect
4. C2×#16861129C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
6. #22549638C2semidirect
7. #380×C127C2semidirect
41. C2×C127⋊C8D4semidirect
144. C127#2033semidirect
150. #380D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611422
C2×#168611292
C2×C22⋊(C127⋊C8)2
#225496382
#380×C1272
#2033127