Finite Group Databasebeta

← #30638783

Subgroups — #30638783

Order 16256.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861217C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549912C211yesmaximal, characteristic, complemented
4#22549894C211yesmaximal, characteristic
5#22549892C211yesmaximal, characteristic
6#22549648C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—21no—
10#16861217—21no—
11#16861217C412yescomplemented
12#16861217C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523231—21no—
15C22×C127⋊C8C2211yescharacteristic
16C2×C8×D127—21no—
17#16861168C2211yescharacteristic
18#16861144—21no—
19#16861144—21no—
20#16861142—21no—
21#16861140—21no—
22#16861137—22no—
23#16861135—22no—
24C2×M16×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26#117×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248C2×C412yes—
30#14523248D412yes—
31C2×Dic254Q811yescharacteristic
32C2×D508Q811yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523232—21no—
36#14523232—21no—
37#14523231—22no—
38#14523231—22no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C411yescharacteristic
41C2×C127⋊C8C2×C411yescharacteristic
42C2×C127⋊C8D411yescharacteristic
43C2×C127⋊C8D411yescharacteristic
44C2×C127⋊C8D411yescharacteristic
45C2×C127⋊C8D411yescharacteristic
46C8×D127—22no—
47C8×D127—22no—
48C4⋊Dic127—21no—
49C22⋊Dic127—21no—
50C4×Dic127—21no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127—21no—
54M16×C127—21no—
55M16×C127C2×C412yes—
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C42×C127—21no—
61C22×D127—21no—
62Dic254—21no—
63Dic254C4⋊C412yes—
64D508—21no—
65D508C4⋊C412yes—
66C22⋊D127—22no—
67C22⋊D127—22no—
68C2×Dic127—21no—
69C2×Dic127—41no—
70C4×D127—22no—
71C4×D127—22no—
72C127⋊C8—41no—
73C127⋊C8C22⋊C411yescharacteristic
74C127⋊C8C22⋊C411yescharacteristic
75C127⋊C8C22⋊C411yescharacteristic
76C127⋊C8C22⋊C411yescharacteristic
77C23×C127#4111yescharacteristic
78C2×C4×C127—41no—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×Q811yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—22no—
86C8×C127—41no—
87C22×C127—41no—
88C22×C127#11711yescharacteristic
89C22×C127#12111yescharacteristic
90C22×C127C22⋊Q811yescharacteristic
91C2×D127—22no—
92C2×D127—42no—
93C4×C127—81no—
94C4×C127C2×C22⋊C411yescharacteristic
95C4×C127C2×C4⋊C411yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127C22⋊Q811yescharacteristic
98Dic127—22no—
99Dic127—81no—
100C2×C127—21no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127#38011yescharacteristic
104D127—42no—
105#1814—1271nomaximal, Sylow
106C127#181411yescharacteristic, complemented, Sylow
107C2×#131—1271no—
108C2×#108—1271no—
109#427—1271no—
110#421—1271no—
111#421—1271no—
112#344—1271no—
113#344D12711yescharacteristic, complemented
114C2×#41—1271no—
115#131—1272no—
116#131—1272no—
117#131—2541no—
118#131—2541no—
119C2×M16—1271no—
120C2×M16—2541no—
121C2×M16C2×D12711yescharacteristic
122C2×M16C2×D12711yescharacteristic
123C22×C8—1271no—
124C22×C8—2541no—
125#117—1271no—
126#117C2×D12711yescharacteristic
127#108—2541no—
128#108—2541no—
129#107—2541no—
130#106—2541no—
131#104—2542no—
132#104—2542no—
133#41—1272no—
134#41—1272no—
135#41—2541no—
136#41—2541no—
137C2×Q8—1271no—
138C2×D4—1271no—
139C2×D4—2541no—
140C22×C4—2541no—
141C22×C4C22×D12711yescharacteristic
142M16—21no—
143M16—21no—
144M16—21no—
145M16—2542no—
146M16—2542no—
147M16—2541no—
148M16—2541no—
149M16C4×D12712yes—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—1271no—
153C2×C8—1271no—
154C2×C8—1271no—
155C2×C8—1271no—
156C2×C8—1271no—
157C2×C8—1271no—
158C2×C8—2542no—
159C2×C8—2542no—
160C2×C8—2541no—
161C4⋊C4—21no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—2541no—
165C42—21no—
166C42—2541no—
167C23—2541no—
168C23#1452324811yescharacteristic
169Q8—1272no—
170Q8—2541no—
171D4—1272no—
172D4—2542no—
173D4—2542no—
174D4—2541no—
175C2×C4—41no—
176C2×C4—5081no—
177C2×C4—2541no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4C2×C4×D12711yescharacteristic
181C2×C4D4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C2×C4Q8×D12711yescharacteristic
185C2×C4#1452324811yescharacteristic
186C8—22no—
187C8—41no—
188C8—1271no—
189C8—1271no—
190C8—1271no—
191C8—1271no—
192C8—5081no—
193C22—41no—
194C22—5082no—
195C22—2542no—
196C22#1686116611yescharacteristic
197C22#1686118211yescharacteristic
198C22#1686118711yescharacteristic
199C4—81no—
200C4—10161no—
201C4—2542no—
202C4C22⋊C4×D12711yescharacteristic
203C4C4⋊C4×D12711yescharacteristic
204C4#1686118211yescharacteristic
205C4#1686118711yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—5082no—
210C2#2254973511yescharacteristic
subgroupG/H
centerC4#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#380
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#344D127
hypocenterC127#1814
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549912C2semidirect
6. #22549648C2semidirect
7. #344×C127C2semidirect
11. #16861217C4semidirect
12. #16861217C4semidirect
106. C127#1814semidirect
113. #344D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612172
C2×#168611442
#225499122
#225498942
#225498922
#225496482
#344×C1272
#1814127