Finite Group Databasebeta

← #30639376

Subgroups — #30639376

Order 16256.

224 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861137C211yesmaximal, characteristic
3#22549870C211yesmaximal, characteristic
4#22549844C211yesmaximal, characteristic
5#22549832C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861144—22no—
13#16861137—22no—
14#16861132C411yescharacteristic, complemented
15#16861132C411yescharacteristic, complemented
16#16861129C411yescharacteristic, complemented
17#16861129C411yescharacteristic, complemented
18(C2×D127)⋊C8—21no—
19(C2×D127)⋊C8—21no—
20#16861120—21no—
21C4×C127⋊C8—21no—
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24C2×C42×C127C2211yescharacteristic
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523232—21no—
34#14523232D412yes—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C411yescharacteristic
41C2×C127⋊C8C2×C411yescharacteristic
42C2×C127⋊C8D411yescharacteristic
43C2×C127⋊C8D411yescharacteristic
44C22×C4×C127—21no—
45C22×C4×C127C2311yescharacteristic
46M16×C127—21no—
47M16×C127—21no—
48C2×C8×C127—21no—
49C4⋊C4×C127C2×C411yescharacteristic
50C4⋊C4×C127C2×C411yescharacteristic
51C22⋊C4×C127—21no—
52C42×C127—21no—
53C42×C127—21no—
54C42×C127—21no—
55C22×D127—21no—
56Dic254—21no—
57Dic254—21no—
58D508—21no—
59D508—21no—
60C22⋊D127—41no—
61C22⋊D127—41no—
62C2×Dic127—21no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—21no—
66C127⋊C8—21no—
67C127⋊C8—22no—
68C127⋊C8—21no—
69C127⋊C8—21no—
70C23×C127#4111yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127C22×C411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C8×C127—41no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127C4×D411yescharacteristic
86C22×C127#12411yescharacteristic
87C22×C127#12611yescharacteristic
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—41no—
91C4×C127—42no—
92C4×C127—41no—
93C4×C127—41no—
94C4×C127#11711yescharacteristic
95C4×C127C4×D411yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12311yescharacteristic
98Dic127—41no—
99C2×C127—21no—
100C2×C127—21no—
101C2×C127#38811yescharacteristic
102C2×C127#44311yescharacteristic
103C2×C127#44411yescharacteristic
104D127—81no—
105#1929—1271nomaximal, Sylow
106C127#192911yescharacteristic, complemented, Sylow
107C2×#108—1271no—
108C2×#104—1271no—
109#416—1271no—
110#408—1271no—
111#405—1271no—
112#339D12711yescharacteristic, complemented
113#336—1271no—
114C2×#41—1271no—
115C2×M16—1271no—
116C2×M16C2×D12711yescharacteristic
117C22×C8—1271no—
118C22×C8—1271no—
119#117C2×D12711yescharacteristic
120C2×C42C2×D12711yescharacteristic
121#108—2542no—
122#104—2542no—
123#103—1271no—
124#103—1271no—
125#102—1271no—
126#102—1271no—
127C22⋊C8—2541no—
128C22⋊C8—2541no—
129#97—2541no—
130C4×C8—2541no—
131#41—2541no—
132#41—2541no—
133#41—2541no—
134#41—2541no—
135C2×Q8—1271no—
136C2×D4—1271no—
137C2×D4—2541no—
138C22×C4—21no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—21no—
142M16—21no—
143M16—1272no—
144M16—2541no—
145C2×C8—21no—
146C2×C8—1271no—
147C2×C8—1271no—
148C2×C8—1271no—
149C2×C8—1271no—
150C2×C8—2541no—
151C2×C8—2541no—
152C2×C8—2541no—
153C2×C8—2541no—
154C2×C8—2541no—
155C4⋊C4C4×D12711yescharacteristic
156C4⋊C4C4×D12711yescharacteristic
157C22⋊C4—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C23—2541no—
162C23#1452324711yescharacteristic
163Q8—2541no—
164Q8—2541no—
165D4—5081no—
166D4—5081no—
167D4—2541no—
168D4—2541no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—5081no—
175C2×C4—5081no—
176C2×C4—2541no—
177C2×C4C2×C4×D12711yescharacteristic
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4#1452324811yescharacteristic
181C2×C4#1452324811yescharacteristic
182C2×C4#1452324911yescharacteristic
183C8—41no—
184C8—2541no—
185C8—2541no—
186C8—2541no—
187C8—2541no—
188C8—2542no—
189C22—21no—
190C22—21no—
191C22—5081no—
192C22—5081no—
193C22—5081no—
194C22(C22⋊D127)⋊C411yescharacteristic
195C22#1686119511yescharacteristic
196C22#1686120411yescharacteristic
197C4—42no—
198C4—41no—
199C4—41no—
200C4—5081no—
201C4#1686116611yescharacteristic
202C4#1686117111yescharacteristic
203C4#1686118211yescharacteristic
204C4#1686119411yescharacteristic
205C2—21no—
206C2—21no—
207C2—10161no—
208C2#2254976511yescharacteristic
209C2#2254999011yescharacteristic
210C2#2254999511yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C4×D4
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1929
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#16861144C2semidirect
5. #22549832C2semidirect
6. #22549631C2semidirect
7. #339×C127C2semidirect
14. #16861132C4semidirect
15. #16861132C4semidirect
16. #16861129C4semidirect
17. #16861129C4semidirect
106. C127#1929semidirect
112. #339D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
C2×#168611372
#225498702
#225498442
#225498322
#225496312
#339×C1272
#1929127