Finite Group Databasebeta

← #30638782

Subgroups — #30638782

Order 16256.

256 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16861216C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549912C211yesmaximal, characteristic, complemented
4#22549893C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic
8#16861216—21no—
9#16861216—21no—
10#16861216C412yescomplemented
11#16861216C412yescomplemented
12C2×#14523232—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8—21no—
16C22×C127⋊C8C2211yescharacteristic
17#16861168C2211yescharacteristic
18#16861144—21no—
19#16861144—21no—
20#16861142—21no—
21#16861140—21no—
22#16861136—22no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26#104×C127—22no—
27#14523232—22no—
28#14523232—22no—
29#14523232—21no—
30#14523232—21no—
31#14523232—21no—
32#14523232C2×C412yes—
33C2×C127⋊C8—22no—
34C2×C127⋊C8—22no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8C2×C411yescharacteristic
38C2×C127⋊C8C2×C411yescharacteristic
39C2×C127⋊C8D411yescharacteristic
40C2×C127⋊C8D411yescharacteristic
41C2×C127⋊C8D411yescharacteristic
42C2×C127⋊C8D411yescharacteristic
43C4⋊Dic127—21no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127C2×C412yes—
49#41×C127D412yes—
50C2×Q8×C127Q811yescharacteristic
51C2×D4×C127—21no—
52C2×D4×C127Q811yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55M16×C127—21no—
56M16×C127—21no—
57C2×C8×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C42×C127—21no—
61C2×Dic127—41no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64C127⋊C8C22⋊C411yescharacteristic
65C127⋊C8C22⋊C411yescharacteristic
66C127⋊C8C22⋊C411yescharacteristic
67C127⋊C8C22⋊C411yescharacteristic
68C23×C127—21no—
69C23×C127#4111yescharacteristic
70Q8×C127—21no—
71Q8×C127C4⋊C412yes—
72D4×C127—22no—
73D4×C127—22no—
74D4×C127—21no—
75D4×C127C4⋊C412yes—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—41no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—41no—
87C22×C127—22no—
88C22×C127—41no—
89C22×C127—42no—
90C22×C127#11711yescharacteristic
91C22×C127#12111yescharacteristic
92C22×C127C22⋊Q811yescharacteristic
93C4×C127—22no—
94C4×C127—81no—
95C4×C127C2×C22⋊C411yescharacteristic
96C4×C127C2×C4⋊C411yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127C22⋊Q811yescharacteristic
99Dic127—81no—
100C2×C127—21no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127—42no—
104C2×C127#38011yescharacteristic
105#1814—1271nomaximal, Sylow
106C127#181411yescharacteristic, complemented, Sylow
107C2×#131—1271no—
108C2×#108—1271no—
109#427—1271no—
110#421—1271no—
111#421D12711yescharacteristic
112#344—1271no—
113#344—1271no—
114C2×#41C2×D12711yescharacteristic
115#131—1272no—
116#131—1272no—
117#131—2541no—
118#131—2541no—
119C2×M16—1271no—
120C2×M16—1271no—
121C2×M16—2541no—
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—22no—
132#104—2542no—
133#41—21no—
134#41—21no—
135#41C4×D12712yes—
136#41C22⋊D12712yes—
137C2×Q8Dic25411yescharacteristic
138C2×D4—21no—
139C2×D4Dic25411yescharacteristic
140C22×C4—21no—
141C22×C4C22×D12711yescharacteristic
142M16—21no—
143M16—21no—
144M16—1272no—
145M16—2542no—
146M16—2542no—
147M16—2541no—
148M16—2541no—
149M16—2541no—
150C2×C8—21no—
151C2×C8—1271no—
152C2×C8—1271no—
153C2×C8—1271no—
154C2×C8—1271no—
155C2×C8—1271no—
156C2×C8—1271no—
157C2×C8—2542no—
158C2×C8—2542no—
159C2×C8—2541no—
160C2×C8—2541no—
161C4⋊C4—21no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—2541no—
165C42—21no—
166C42—2541no—
167C23—21no—
168C23#1452324711yescharacteristic
169Q8—21no—
170Q8Dic127⋊C412yes—
171D4—22no—
172D4—22no—
173D4—21no—
174D4Dic127⋊C412yes—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—5081no—
180C2×C4C2×C4×D12711yescharacteristic
181C2×C4D4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C2×C4C2×C22⋊D12711yescharacteristic
184C2×C4C2×Dic25411yescharacteristic
185C2×C4#1452324711yescharacteristic
186C8—41no—
187C8—1271no—
188C8—1271no—
189C8—1271no—
190C8—1271no—
191C8—5081no—
192C8—2542no—
193C22—22no—
194C22—41no—
195C22—42no—
196C22#1686116511yescharacteristic
197C22#1686118111yescharacteristic
198C22#1686118611yescharacteristic
199C4—22no—
200C4—81no—
201C4—10161no—
202C4C22⋊C4×D12711yescharacteristic
203C4C2×Dic127⋊C411yescharacteristic
204C4#1686118111yescharacteristic
205C4#1686118611yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—42no—
210C2#2254973411yescharacteristic
subgroupG/H
centerC4#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#380
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1814
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549912C2semidirect
5. #22549647C2semidirect
6. #22549646C2semidirect
10. #16861216C4semidirect
11. #16861216C4semidirect
106. C127#1814semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
C2×#168611442
#225499122
#225498932
#225496472
#225496462
#421×C1272
#1814127