Finite Group Databasebeta

← #30639865

Subgroups — #30639865

Order 16256.

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

normal subgroups only

This group has 326 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×#16861208C211yesmaximal, characteristic, complemented
2C2×#16861131C211yesmaximal, characteristic
3C2×#16861129C211yesmaximal, characteristic
4#22549777C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×Dic254—21no—
9C22×D508C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861208—22no—
12#16861208—22no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17C2×C4×Dic127C2211yescharacteristic
18#16861131—22no—
19#16861129—22no—
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C23×D127—21no—
24C2×Dic254—24no—
25C2×Dic254—42no—
26C2×Dic254—42no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508Q812yescomplemented
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C22×Dic127D411yescharacteristic
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127D412yes—
41Dic127⋊C4—24no—
42(C2×D127)⋊C4—42no—
43(C2×D127)⋊C4—42no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127D412yes—
56C22×D127—22no—
57C22×D127—41no—
58C22×D127—41no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—41no—
63Dic254—48no—
64D508—22no—
65D508—22no—
66D508—41no—
67D508—41no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—22no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127D812yes—
76C2×Dic127SD1612yes—
77C127⋊C8—41no—
78C23×C127C2×D411yescharacteristic
79C2×C4×C127—22no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×Q811yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C8×C127—41no—
92C22×C127#12011yescharacteristic
93C22×C127C22⋊Q811yescharacteristic
94C22×C127#12411yescharacteristic
95C22×C127C2×D811yescharacteristic
96C22×C127C2×SD1611yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C22×C127#13711yescharacteristic
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C2×D127—81no—
103C2×D127—81no—
104C2×D127—81no—
105C2×D127—81no—
106C4×C127—42no—
107C4×C127—81no—
108C4×C127C22⋊Q811yescharacteristic
109C4×C127C22⋊Q811yescharacteristic
110C4×C127#12411yescharacteristic
111C4×C127#12411yescharacteristic
112Dic127—22no—
113Dic127—22no—
114Dic127—42no—
115C2×C127#39311yescharacteristic
116C2×C127#44711yescharacteristic
117C2×C127#44811yescharacteristic
118C2×C127#47411yescharacteristic
119C2×C127#47611yescharacteristic
120C2×C127#48611yescharacteristic
121C2×C127#48911yescharacteristic
122D127—82no—
123#2023—1271nomaximal, Sylow
124C127#202311yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—1271no—
126C2×#102—1271no—
127C2×#102—1271no—
128#390—1271no—
129#340—1271no—
130#340D12711yescharacteristic, complemented
131#336—1271no—
132C22×Q8—2541no—
133C22×D4—1271no—
134C22×C8—1271no—
135C22×C8C2×D12711yescharacteristic
136C4⋊Q8—2542no—
137C4⋊Q8—2542no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4—2541no—
140C2×C4⋊C4C2×D12711yescharacteristic
141C2×C4⋊C4C2×D12711yescharacteristic
142C2×C22⋊C4—2541no—
143C2×C22⋊C4—2541no—
144C2×C42—1271no—
145#102—2542no—
146#102—2542no—
147C24—2541no—
148C2×Q8—5082no—
149C2×Q8—5082no—
150C2×Q8—2544no—
151C2×D4—1272no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—1271no—
160C22×C4—1271no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2541no—
167C2×C8—2541no—
168C2×C8—2541no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1272no—
173C4⋊C4—2544no—
174C4⋊C4—2541no—
175C4⋊C4D50812yes—
176C22⋊C4—5082no—
177C22⋊C4—5082no—
178C42—2541no—
179C42—2541no—
180C23—5081no—
181C23—5081no—
182C23—5081no—
183C23—5081no—
184C23—5081no—
185C23—5081no—
186C23—2542no—
187C23D4×D12711yescharacteristic
188Q8—5088no—
189D4—5081no—
190D4—5081no—
191D4—2542no—
192D4—2542no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1272no—
200C2×C4—1272no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—2542no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4D4×D12711yescharacteristic
209C2×C4C2×D50811yescharacteristic
210C2×C4Q8×D12711yescharacteristic
211C2×C4#1452324711yescharacteristic
212C2×C4#1452324811yescharacteristic
213C2×C4#1452324811yescharacteristic
214C8—41no—
215C8—5081no—
216C22—10161no—
217C22—10161no—
218C22—10161no—
219C22—10161no—
220C22—5082no—
221C22—5082no—
222C22—5082no—
223C22#1686117911yescharacteristic
224C22#1686118711yescharacteristic
225C22#1686119511yescharacteristic
226C22D8×D12711yescharacteristic
227C22SD16×D12711yescharacteristic
228C22#1686123611yescharacteristic
229C22#1686124111yescharacteristic
230C4—42no—
231C4—81no—
232C4—5082no—
233C4—2542no—
234C4—2542no—
235C4#1686118711yescharacteristic
236C4#1686118911yescharacteristic
237C4#1686119511yescharacteristic
238C4#1686119711yescharacteristic
239C2—10162no—
240C2#2254978911yescharacteristic
241C2#2255001111yescharacteristic
242C2#2255001611yescharacteristic
243C2#2255015511yescharacteristic
244C2#2255016511yescharacteristic
245C2#2255022111yescharacteristic
246C2#2255023711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#340D127
hypocenterC127#2023
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#16861208C2semidirect
5. #22549638C2semidirect
6. #22549630C2semidirect
7. #340×C127C2semidirect
32. C2×D508Q8semidirect
124. C127#2023semidirect
130. #340D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612082
C2×#168611312
C2×#168611292
#225497772
#225496382
#225496302
#340×C1272
#2023127