Finite Group Databasebeta

← #30639910

Subgroups — #30639910

Order 16256.

368 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861180C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
5#22549733C211yesmaximal, characteristic, complemented
6#22549639C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C22×C4×D127C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861180—22no—
12#16861180—22no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×C22⋊Dic127—21no—
16C2×(C2×D127)⋊C4—21no—
17#16861142—22no—
18#16861130—22no—
19(C2×D127)⋊C8—22no—
20C127⋊#95—21no—
21C22×D4×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#102×C127—22no—
25C23×D127D411yescharacteristic
26C2×C22⋊D127—24no—
27C2×C22⋊D127—42no—
28C2×C22⋊D127—42no—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8D412yescomplemented
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127D412yes—
45C22⋊Dic127—24no—
46(C2×D127)⋊C4—42no—
47C24×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127D412yes—
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127D812yes—
68C22⋊D127—48no—
69C2×Dic127—21no—
70C2×Dic127—22no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C4×D127—41no—
79C4×D127—41no—
80C4×D127—41no—
81C4×D127—41no—
82C127⋊C8—22no—
83C23×C127—22no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127—41no—
90C23×C127C2×D411yescharacteristic
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—41no—
94D4×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—41no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—81no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127—81no—
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—22no—
120C2×D127—22no—
121C2×D127—22no—
122C2×D127—41no—
123C2×D127—41no—
124C2×D127—41no—
125C2×D127—41no—
126C4×C127—81no—
127C4×C127#12111yescharacteristic
128C4×C127#12111yescharacteristic
129C4×C127#12311yescharacteristic
130C4×C127#12411yescharacteristic
131Dic127—41no—
132Dic127—42no—
133Dic127—81no—
134C2×C127—82no—
135C2×C127#39411yescharacteristic
136C2×C127#44711yescharacteristic
137C2×C127#45211yescharacteristic
138C2×C127#46611yescharacteristic
139C2×C127#46811yescharacteristic
140C2×C127#48011yescharacteristic
141C2×C127#48211yescharacteristic
142D127—42no—
143#2033—1271nomaximal, Sylow
144C127#203311yescharacteristic, complemented, Sylow
145C2×#121—1271no—
146C2×#107—1271no—
147C2×#102—1271no—
148C2×#102D12711yescharacteristic, complemented
149C2×C22⋊C8—1271no—
150#380—1271no—
151#340—1271no—
152C22×D4—2541no—
153C22×D4C2×D12711yescharacteristic
154C23×C4—1271no—
155C22×C8—1271no—
156C22×C8C2×D12711yescharacteristic
157#121—2542no—
158#121—2542no—
159C2×C4⋊C4—1271no—
160C2×C4⋊C4—1271no—
161C2×C4⋊C4C2×D12711yescharacteristic
162C2×C22⋊C4—2541no—
163C2×C22⋊C4—2541no—
164#107—2542no—
165#102—22no—
166#102—2542no—
167C22⋊C8—2542no—
168#95—2541no—
169C24—21no—
170C24—1271no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—5082no—
177C2×D4—5082no—
178C2×D4—2544no—
179C2×D4C22⋊D12712yes—
180C22×C4—21no—
181C22×C4—1271no—
182C22×C4—2541no—
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—21no—
194C2×C8—1272no—
195C2×C8—2541no—
196C2×C8—2541no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—1272no—
200C4⋊C4—2541no—
201C4⋊C4—2541no—
202C4⋊C4—2541no—
203C22⋊C4—5082no—
204C22⋊C4—2544no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—1272no—
213C23—2541no—
214C23—2541no—
215C23—2541no—
216C23—2541no—
217C23—2541no—
218C23—2541no—
219C23D4×D12711yescharacteristic
220D4—22no—
221D4—22no—
222D4—41no—
223D4—41no—
224D4—5088no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—5081no—
229C2×C4—5081no—
230C2×C4—5081no—
231C2×C4—5081no—
232C2×C4—5081no—
233C2×C4—5081no—
234C2×C4—5081no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C2×C22⋊D12711yescharacteristic
244C2×C4#1452324711yescharacteristic
245C2×C4#1452324811yescharacteristic
246C2×C4#1452324811yescharacteristic
247C8—41no—
248C8—2542no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—5081no—
257C22—5081no—
258C22—5081no—
259C22—5081no—
260C22—2542no—
261C22—2542no—
262C22—2542no—
263C22#1686117711yescharacteristic
264C22#1686118211yescharacteristic
265C22#1686119111yescharacteristic
266C22D8×D12711yescharacteristic
267C22#1686123011yescharacteristic
268C22#1686123311yescharacteristic
269C22#1686123811yescharacteristic
270C4—81no—
271C4—10161no—
272C4—5081no—
273C4—5082no—
274C4#1686118111yescharacteristic
275C4#1686118211yescharacteristic
276C4#1686119211yescharacteristic
277C4#1686119511yescharacteristic
278C2—82no—
279C2—5082no—
280C2#2254979111yescharacteristic
281C2#2255000811yescharacteristic
282C2#2255003511yescharacteristic
283C2#2255011111yescharacteristic
284C2#2255012111yescharacteristic
285C2#2255018811yescharacteristic
286C2#2255019911yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC2×#102D127
hypocenterC127#2033
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861142C2semidirect
3. C2×#16861130C2semidirect
4. C2×(C2×D127)⋊C8C2semidirect
5. #22549733C2semidirect
6. #22549639C2semidirect
7. C2×#102×C127C2semidirect
40. C2×C127⋊C8D4semidirect
144. C127#2033semidirect
148. C2×#102D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611802
C2×#168611422
C2×#168611302
C2×(C2×D127)⋊C82
#225497332
#225496392
C2×#102×C1272
#2033127