Finite Group Databasebeta

← #30639380

Subgroups — #30639380

Order 16256.

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

normal subgroups only

This group has 224 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×#16861145C211yesmaximal, characteristic, complemented
2C2×#16861136C211yesmaximal, characteristic
3#22549843C211yesmaximal, characteristic
4#22549832C211yesmaximal, characteristic, complemented
5#22549635C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#416×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861145—22no—
13#16861136—22no—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)—21no—
16#16861120—21no—
17C4×C127⋊C8—21no—
18C2×#41×C127C2211yescharacteristic
19C22×C8×C127C2211yescharacteristic
20#117×C127C2211yescharacteristic
21#103×C127C411yescharacteristic, complemented
22#103×C127C411yescharacteristic, complemented
23#102×C127C411yescharacteristic, complemented
24#102×C127C411yescharacteristic, complemented
25C22×Dic127—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232D412yes—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8D411yescharacteristic
35C2×C127⋊C8D411yescharacteristic
36C4×Dic127—21no—
37C4×Dic127—21no—
38#41×C127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42C2×Q8×C127C2×C411yescharacteristic
43C2×D4×C127—21no—
44C2×D4×C127C2×C411yescharacteristic
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127C2×C411yescharacteristic
50C2×C8×C127C2×C411yescharacteristic
51C4⋊C4×C127C2×C411yescharacteristic
52C4⋊C4×C127C2×C411yescharacteristic
53C22⋊C4×C127—21no—
54C42×C127—21no—
55C2×Dic127—22no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C127⋊C8—22no—
59C127⋊C8—21no—
60C127⋊C8—21no—
61C127⋊C8—41no—
62C23×C127—21no—
63C23×C127#4111yescharacteristic
64Q8×C127—21no—
65Q8×C127—21no—
66D4×C127—21no—
67D4×C127—21no—
68D4×C127—41no—
69D4×C127—41no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C8×C127—21no—
82C8×C127—21no—
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—41no—
93C4×C127—41no—
94C4×C127#11711yescharacteristic
95C4×C127C4×D411yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12311yescharacteristic
98Dic127—42no—
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#416D12711yescharacteristic
110#408—1271no—
111#405—1271no—
112#339—1271no—
113#336—1271no—
114C2×#41C2×D12711yescharacteristic
115C2×M16—1271no—
116C2×M16—1271no—
117C22×C8—1271no—
118C22×C8C2×D12711yescharacteristic
119#117C2×D12711yescharacteristic
120C2×C42—1271no—
121#108—2542no—
122#104—2542no—
123#103Dic12711yescharacteristic, complemented
124#103Dic12711yescharacteristic, complemented
125#102Dic12711yescharacteristic, complemented
126#102Dic12711yescharacteristic, complemented
127C22⋊C8—2541no—
128C22⋊C8—2541no—
129#97—2541no—
130C4×C8—2541no—
131#41—21no—
132#41—21no—
133#41—21no—
134#41—21no—
135C2×Q8C2×Dic12711yescharacteristic
136C2×D4—21no—
137C2×D4C2×Dic12711yescharacteristic
138C22×C4—21no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—1272no—
142M16—2541no—
143M16—2541no—
144M16—2541no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—1271no—
148C2×C8—1271no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8—2541no—
152C2×C8—2541no—
153C2×C8C2×Dic12711yescharacteristic
154C2×C8C2×Dic12711yescharacteristic
155C4⋊C4C2×Dic12711yescharacteristic
156C4⋊C4C2×Dic12711yescharacteristic
157C22⋊C4—21no—
158C42—21no—
159C42—2541no—
160C42—2541no—
161C23—21no—
162C23#1452324711yescharacteristic
163Q8—21no—
164Q8—21no—
165D4—21no—
166D4—21no—
167D4—41no—
168D4—41no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—5081no—
175C2×C4—5081no—
176C2×C4—2542no—
177C2×C4C22×Dic12711yescharacteristic
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4#1452324711yescharacteristic
181C2×C4#1452324811yescharacteristic
182C2×C4#1452324811yescharacteristic
183C8—21no—
184C8—21no—
185C8—5081no—
186C8—2541no—
187C8—2541no—
188C8—2542no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22D4×Dic12711yescharacteristic
195C22#1686119511yescharacteristic
196C22#1686120211yescharacteristic
197C4—41no—
198C4—41no—
199C4—41no—
200C4—5082no—
201C4#1686116811yescharacteristic
202C4D4×Dic12711yescharacteristic
203C4#1686118211yescharacteristic
204C4#1686119111yescharacteristic
205C2—21no—
206C2—21no—
207C2—81no—
208C2#2254976911yescharacteristic
209C2#2254999311yescharacteristic
210C2#2254999811yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#416×C127C2
socleC22×C127C4×D4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#416D127
hypocenterC127#1929
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861145C2semidirect
4. #22549832C2semidirect
5. #22549635C2semidirect
6. #22549630C2semidirect
21. #103×C127C4semidirect
22. #103×C127C4semidirect
23. #102×C127C4semidirect
24. #102×C127C4semidirect
106. C127#1929semidirect
123. #103Dic127semidirect
124. #103Dic127semidirect
125. #102Dic127semidirect
126. #102Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611452
C2×#168611362
#225498432
#225498322
#225496352
#225496302
#416×C1272
#1929127