Finite Group Databasebeta

← #30639947

Subgroups — #30639947

Order 16256.

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

normal subgroups only

This group has 278 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#22550450C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic
4#22549875C211yesmaximal, characteristic
5C2×#16861128C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#391×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861207—21no—
12#16861198—21no—
13#16861192—22no—
14#16861185—21no—
15#16861185—21no—
16#16861168C2211yescharacteristic
17#16861142—21no—
18#16861140—21no—
19#16861137—22no—
20#16861132—21no—
21#16861129—21no—
22#16861128—22no—
23C22×Q8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#95×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42C4⋊Dic127D411yescharacteristic
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—22no—
47(C2×D127)⋊C4—22no—
48C4×Dic127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127D412yes—
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C42×C127—21no—
62C42×C127—21no—
63C22×D127—21no—
64Dic254—21no—
65Dic254—21no—
66D508—21no—
67D508—21no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C4×D127—41no—
74C4×D127—41no—
75C127⋊C8—22no—
76C127⋊C8—41no—
77C23×C127C2×D411yescharacteristic
78Q8×C127—41no—
79Q8×C127—41no—
80Q8×C127—41no—
81Q8×C127—41no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—42no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127#12011yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12311yescharacteristic
101C2×D127—41no—
102C2×D127—41no—
103C2×D127—41no—
104C4×C127—42no—
105C4×C127—42no—
106C4×C127—81no—
107C4×C127#12111yescharacteristic
108C4×C127#12111yescharacteristic
109C4×C127#12311yescharacteristic
110C4×C127#12411yescharacteristic
111Dic127—41no—
112Dic127—41no—
113Dic127—41no—
114C2×C127—21no—
115C2×C127—21no—
116C2×C127#39411yescharacteristic
117C2×C127#45411yescharacteristic
118C2×C127#45611yescharacteristic
119D127—81no—
120#2038—1271nomaximal, Sylow
121C127#203811yescharacteristic, complemented, Sylow
122#536—1271no—
123#428—1271no—
124C2×#104—1271no—
125#417—1271no—
126C2×#101—1271no—
127#391D12711yescharacteristic, complemented
128#339—1271no—
129C2×#41—1271no—
130C22×Q8C2×D12711yescharacteristic
131C2×M16—1271no—
132C2×M16—1271no—
133C4⋊Q8—2541no—
134#124—2541no—
135#123—2542no—
136C22⋊Q8—2541no—
137C22⋊Q8—2541no—
138#117—1271no—
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C42C2×D12711yescharacteristic
141#107—2541no—
142#106—2541no—
143#104—2542no—
144#103—2541no—
145#102—2541no—
146#101—2542no—
147#95—22no—
148#41—2541no—
149#41—2541no—
150#41—2541no—
151#41—2541no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—1271no—
159C2×D4—1271no—
160C2×D4—2541no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—2541no—
164C22×C4C22⋊D12712yes—
165C22×C4C22×D12711yescharacteristic
166M16—1272no—
167M16—2541no—
168M16—2541no—
169M16—2541no—
170C2×C8—2541no—
171C2×C8—2541no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1271no—
175C4⋊C4—1271no—
176C4⋊C4—2542no—
177C4⋊C4—2542no—
178C22⋊C4—2542no—
179C22⋊C4—2542no—
180C22⋊C4—2541no—
181C42—21no—
182C42—21no—
183C42—2541no—
184C23—2541no—
185C23C2×C22⋊D12711yescharacteristic
186Q8—41no—
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—2541no—
191Q8—2541no—
192D4—5081no—
193D4—5081no—
194D4—2541no—
195D4—2541no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C2×C4D4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4#1452324711yescharacteristic
213C2×C4#1452324811yescharacteristic
214C2×C4#1452324811yescharacteristic
215C8—5081no—
216C8—2542no—
217C22—21no—
218C22—21no—
219C22—5081no—
220C22—5081no—
221C22—5081no—
222C22#1686117711yescharacteristic
223C22#1686118111yescharacteristic
224C22#1686119211yescharacteristic
225C4—42no—
226C4—42no—
227C4—81no—
228C4—5081no—
229C4—5081no—
230C4—5081no—
231C4#1686118211yescharacteristic
232C4#1686118211yescharacteristic
233C4#1686119111yescharacteristic
234C4#1686119511yescharacteristic
235C2—21no—
236C2—21no—
237C2—10161no—
238C2#2254979111yescharacteristic
239C2#2255004811yescharacteristic
240C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC22×C127#120
prime radicalC2×#14523248C22
hypercenter#391D127
hypocenterC127#2038
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #22549916C2semidirect
5. C2×#16861128C2semidirect
6. #22549636C2semidirect
7. #391×C127C2semidirect
121. C127#2038semidirect
127. #391D127semidirect

Maximal subgroups

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

subgroupindex
#225504502
#225499162
C2×#168611372
#225498752
C2×#168611282
#225496362
#391×C1272
#2038127