Finite Group Databasebeta

← #30639375

Subgroups — #30639375

Order 16256.

224 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861144C211yesmaximal, characteristic, complemented
2C2×#16861136C211yesmaximal, characteristic
3#22549872C211yesmaximal, characteristic
4#22549831C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#408×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861168C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861144—22no—
13#16861136—22no—
14#16861133C411yescharacteristic, complemented
15#16861133C411yescharacteristic, complemented
16#16861130C411yescharacteristic, complemented
17#16861130C411yescharacteristic, complemented
18#16861119—21no—
19C8×Dic127—21no—
20C2×#41×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C22⋊C8×C127—21no—
24C22⋊C8×C127—21no—
25C22×Dic127—21no—
26#14523232—21no—
27#14523232D412yes—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8C2×C411yescharacteristic
32C2×C127⋊C8C2×C411yescharacteristic
33C4⋊Dic127C2×C411yescharacteristic
34C4⋊Dic127C2×C411yescharacteristic
35C22⋊Dic127—21no—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C4×Dic127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127—21no—
43C2×Q8×C127C2×C411yescharacteristic
44C2×D4×C127—21no—
45C2×D4×C127C2×C411yescharacteristic
46C22×C4×C127—21no—
47C22×C4×C127C2311yescharacteristic
48M16×C127—21no—
49M16×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127D411yescharacteristic
54C2×C8×C127D411yescharacteristic
55C2×Dic127—22no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C127⋊C8—21no—
61C127⋊C8—21no—
62C127⋊C8—22no—
63C23×C127—21no—
64C23×C127#4111yescharacteristic
65Q8×C127—21no—
66Q8×C127—21no—
67D4×C127—21no—
68D4×C127—21no—
69D4×C127—41no—
70D4×C127—41no—
71C2×C4×C127—21no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C22×C411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C8×C127—21no—
81C8×C127—21no—
82C8×C127—41no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—41no—
86C22×C127—41no—
87C22×C127—41no—
88C22×C127C4×D411yescharacteristic
89C22×C127#12411yescharacteristic
90C22×C127#12611yescharacteristic
91C4×C127—41no—
92C4×C127#11711yescharacteristic
93C4×C127C4×D411yescharacteristic
94C4×C127#12111yescharacteristic
95C4×C127#12311yescharacteristic
96Dic127—42no—
97Dic127—41no—
98Dic127—41no—
99C2×C127—21no—
100C2×C127—21no—
101C2×C127—81no—
102C2×C127#38811yescharacteristic
103C2×C127#44311yescharacteristic
104C2×C127#44411yescharacteristic
105#1929—1271nomaximal, Sylow
106C127#192911yescharacteristic, complemented, Sylow
107C2×#108—1271no—
108C2×#104—1271no—
109#416—1271no—
110#408D12711yescharacteristic
111#405—1271no—
112#339—1271no—
113#336—1271no—
114C2×#41C2×D12711yescharacteristic
115C2×M16—1271no—
116C2×M16C2×D12711yescharacteristic
117C22×C8—1271no—
118C22×C8C2×D12711yescharacteristic
119#117—1271no—
120C2×C42—1271no—
121#108—2542no—
122#104—2542no—
123#103—1271no—
124#103—1271no—
125#102—1271no—
126#102—1271no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129#97—2541no—
130C4×C8—2541no—
131#41—21no—
132#41—21no—
133#41—21no—
134#41—21no—
135C2×Q8C4×D12711yescharacteristic
136C2×D4—21no—
137C2×D4C4×D12711yescharacteristic
138C22×C4—21no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—21no—
142M16—21no—
143M16—1272no—
144M16—2541no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—1271no—
149C2×C8—1271no—
150C2×C8—2541no—
151C2×C8—2541no—
152C2×C8—2541no—
153C2×C8C22⋊D12711yescharacteristic
154C2×C8C22⋊D12711yescharacteristic
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C22⋊C4—2541no—
158C42—2541no—
159C42—2541no—
160C42—2541no—
161C23—21no—
162C23#1452324811yescharacteristic
163Q8—21no—
164Q8—21no—
165D4—21no—
166D4—21no—
167D4—41no—
168D4—41no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4C2×C4×D12711yescharacteristic
178C2×C4D4×D12711yescharacteristic
179C2×C4C2×C22⋊D12711yescharacteristic
180C2×C4#1452324711yescharacteristic
181C2×C4#1452324711yescharacteristic
182C2×C4#1452324711yescharacteristic
183C8—21no—
184C8—21no—
185C8—41no—
186C8—2541no—
187C8—2541no—
188C8—2542no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22C4×C22⋊D12711yescharacteristic
195C22#1686119511yescharacteristic
196C22#1686120211yescharacteristic
197C4—41no—
198C4—5082no—
199C4—5081no—
200C4—5081no—
201C4#1686116511yescharacteristic
202C4(C22⋊D127)⋊C411yescharacteristic
203C4#1686118011yescharacteristic
204C4#1686119011yescharacteristic
205C2—21no—
206C2—21no—
207C2—81no—
208C2#2254976411yescharacteristic
209C2#2254999111yescharacteristic
210C2#2254999611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127C4×D4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#408D127
hypocenterC127#1929
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16861144C2semidirect
4. #22549831C2semidirect
5. #22549637C2semidirect
6. #22549630C2semidirect
14. #16861133C4semidirect
15. #16861133C4semidirect
16. #16861130C4semidirect
17. #16861130C4semidirect
106. C127#1929semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
C2×#168611362
#225498722
#225498312
#225496372
#225496302
#408×C1272
#1929127