Finite Group Databasebeta

← #30639940

Subgroups — #30639940

Order 16256.

326 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#22550443C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549875C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861205—21no—
12#16861196—21no—
13#16861180—21no—
14#16861180—21no—
15#16861178—22no—
16#16861168C2211yescharacteristic
17#16861142—21no—
18#16861140—21no—
19#16861137—22no—
20#16861132—21no—
21#16861129—21no—
22#16861126—22no—
23C22×D4×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127—22no—
26C2×C42×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127—22no—
35C2×C22⋊D127—21no—
36C2×C4×D127—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yescomplemented
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C4⋊Dic127D411yescharacteristic
44C4⋊Dic127D411yescharacteristic
45C22⋊Dic127—22no—
46C22⋊Dic127—22no—
47C22⋊Dic127—21no—
48C4×Dic127—21no—
49C24×C127D412yes—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C22⋊C4×C127—44no—
62C42×C127—21no—
63C42×C127—21no—
64C22×D127—21no—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—82no—
72C22⋊D127—82no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C4×D127—41no—
77C4×D127—41no—
78C127⋊C8—22no—
79C127⋊C8—41no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127C2×D411yescharacteristic
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89D4×C127—41no—
90C2×C4×C127—22no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—42no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—44no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—82no—
108C22×C127#12011yescharacteristic
109C22×C127#12111yescharacteristic
110C22×C127#12311yescharacteristic
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—41no—
114C4×C127—42no—
115C4×C127—81no—
116C4×C127#12111yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127#12311yescharacteristic
119C4×C127#12411yescharacteristic
120Dic127—41no—
121Dic127—41no—
122Dic127—41no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—82no—
126C2×C127#39411yescharacteristic
127C2×C127#45311yescharacteristic
128C2×C127#45511yescharacteristic
129D127—81no—
130#2037—1271nomaximal, Sylow
131C127#203711yescharacteristic, complemented, Sylow
132#535—1271no—
133#428—1271no—
134C2×#104—1271no—
135#417—1271no—
136C2×#100—1271no—
137#390D12711yescharacteristic, complemented
138#339—1271no—
139C2×#41—1271no—
140C22×D4C2×D12711yescharacteristic
141C2×M16—1271no—
142C2×M16—1271no—
143#127—2541no—
144#124—2541no—
145#121—2541no—
146#121—2541no—
147#120—2542no—
148#117—1271no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—22no—
151C2×C42C2×D12711yescharacteristic
152#107—2541no—
153#106—2541no—
154#104—2542no—
155#103—2541no—
156#102—2541no—
157#100—2542no—
158C24C22⋊D12712yes—
159#41—2541no—
160#41—2541no—
161#41—2541no—
162#41—2541no—
163C2×Q8—1271no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—1271no—
171C2×D4—2542no—
172C2×D4—2542no—
173C2×D4—2541no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178M16—1272no—
179M16—2541no—
180M16—2541no—
181M16—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—1271no—
187C4⋊C4—1271no—
188C22⋊C4—44no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2541no—
192C42—21no—
193C42—21no—
194C42—2541no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—42no—
199C23—42no—
200C23—2541no—
201C23C2×C22⋊D12711yescharacteristic
202Q8—2541no—
203Q8—2541no—
204D4—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208D4—10162no—
209D4—10162no—
210D4—5081no—
211D4—5081no—
212D4—2541no—
213D4—2541no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4D4×D12711yescharacteristic
227C2×C4#1452324711yescharacteristic
228C2×C4#1452324811yescharacteristic
229C2×C4#1452324811yescharacteristic
230C8—5081no—
231C8—2542no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—82no—
239C22—5081no—
240C22—5081no—
241C22—5081no—
242C22#1686117711yescharacteristic
243C22#1686118111yescharacteristic
244C22#1686119211yescharacteristic
245C4—42no—
246C4—81no—
247C4—5081no—
248C4—5081no—
249C4—5081no—
250C4#1686118211yescharacteristic
251C4#1686118211yescharacteristic
252C4#1686119111yescharacteristic
253C4#1686119511yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—10161no—
258C2#2254979111yescharacteristic
259C2#2255004311yescharacteristic
260C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC22×C127#120
prime radical#22550443C2
hypercenter#390D127
hypocenterC127#2037
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #22549916C2semidirect
3. C2×#16861137C2semidirect
4. #22549875C2semidirect
5. C2×#16861126C2semidirect
6. #22549636C2semidirect
7. #390×C127C2semidirect
40. #14523232D4semidirect
131. C127#2037semidirect
137. #390D127semidirect

Maximal subgroups

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

subgroupindex
#225504432
#225499162
C2×#168611372
#225498752
C2×#168611262
#225496362
#390×C1272
#2037127