Finite Group Databasebeta

← #30639006

Subgroups — #30639006

Order 16256.

270 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 83 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#22549903C211yesmaximal, characteristic, complemented
4C2×#16861137C211yesmaximal, characteristic
5#22549870C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861217—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523231—21no—
13C22×C127⋊C8C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C2×C8×D127—21no—
16#16861144—22no—
17C4⋊(C127⋊C8)—21no—
18C4⋊(C127⋊C8)—21no—
19#16861137—22no—
20#16861132C411yescharacteristic, complemented
21#16861132C411yescharacteristic, complemented
22#16861129C411yescharacteristic, complemented
23#16861129C411yescharacteristic, complemented
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248D412yes—
30#14523248Q812yes—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523232—21no—
36#14523232—21no—
37#14523231—24no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8C2×C411yescharacteristic
44C2×C127⋊C8C2×C411yescharacteristic
45C2×C127⋊C8D411yescharacteristic
46C2×C127⋊C8D411yescharacteristic
47C8×D127—24no—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50M16×C127—21no—
51M16×C127D412yes—
52C2×C8×C127—21no—
53C4⋊C4×C127C2×C411yescharacteristic
54C4⋊C4×C127C2×C411yescharacteristic
55C22⋊C4×C127—21no—
56C42×C127—21no—
57C42×C127—21no—
58C42×C127—21no—
59C22×D127—21no—
60Dic254—21no—
61Dic254C4⋊C412yes—
62D508—21no—
63D508C4⋊C412yes—
64C22⋊D127—22no—
65C22⋊D127—22no—
66C2×Dic127—21no—
67C4×D127—22no—
68C4×D127—22no—
69C127⋊C8—21no—
70C127⋊C8—21no—
71C127⋊C8—21no—
72C127⋊C8—21no—
73C127⋊C8—41no—
74C23×C127#4111yescharacteristic
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
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—22no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127C4×D411yescharacteristic
90C22×C127C22⋊Q811yescharacteristic
91C22×C127#12311yescharacteristic
92C2×D127—22no—
93C2×D127—41no—
94C2×D127—41no—
95C4×C127—42no—
96C4×C127—41no—
97C4×C127—41no—
98C4×C127C2×C4⋊C411yescharacteristic
99C4×C127C4×D411yescharacteristic
100C4×C127#12011yescharacteristic
101C4×C127C22⋊Q811yescharacteristic
102Dic127—22no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#38511yescharacteristic
106C2×C127#44311yescharacteristic
107C2×C127#44411yescharacteristic
108D127—42no—
109#1866—1271nomaximal, Sylow
110C127#186611yescharacteristic, complemented, Sylow
111C2×#131—1271no—
112C2×#108—1271no—
113#424—1271no—
114C2×#104—1271no—
115#416—1271no—
116#339D12711yescharacteristic, complemented
117#336—1271no—
118C2×#41—1271no—
119#131—2542no—
120#131—2542no—
121C2×M16—1271no—
122C2×M16—2541no—
123C2×M16C2×D12711yescharacteristic
124C22×C8—1271no—
125C22×C8—1271no—
126C22×C8—2541no—
127#117C2×D12711yescharacteristic
128C2×C42C2×D12711yescharacteristic
129#108—2542no—
130C4⋊C8—2541no—
131C4⋊C8—2541no—
132#104—2542no—
133#103—1271no—
134#103—1271no—
135#102—1271no—
136#102—1271no—
137#41—1272no—
138#41—1272no—
139#41—2541no—
140#41—2541no—
141C2×Q8—1271no—
142C2×D4—1271no—
143C2×D4—2541no—
144C22×C4—21no—
145C22×C4—2541no—
146C22×C4C22×D12711yescharacteristic
147M16—21no—
148M16—2544no—
149M16—2541no—
150M16—2541no—
151M16D50812yes—
152C2×C8—21no—
153C2×C8—1271no—
154C2×C8—1271no—
155C2×C8—1271no—
156C2×C8—1271no—
157C2×C8—2544no—
158C2×C8—2541no—
159C2×C8—2541no—
160C2×C8—2541no—
161C2×C8—2541no—
162C2×C8—2541no—
163C4⋊C4C4×D12711yescharacteristic
164C4⋊C4C4×D12711yescharacteristic
165C22⋊C4—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C23—2541no—
170C23#1452324711yescharacteristic
171Q8—1272no—
172Q8—2541no—
173D4—1272no—
174D4—2542no—
175D4—2542no—
176D4—2541no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—2541no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4C2×C4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4C2×D50811yescharacteristic
189C2×C4Q8×D12711yescharacteristic
190C2×C4#1452324811yescharacteristic
191C8—22no—
192C8—5081no—
193C8—2541no—
194C8—2541no—
195C8—2541no—
196C8—2541no—
197C22—21no—
198C22—21no—
199C22—5081no—
200C22—5081no—
201C22—2542no—
202C22(C22⋊D127)⋊C411yescharacteristic
203C22#1686118911yescharacteristic
204C22#1686119111yescharacteristic
205C4—42no—
206C4—41no—
207C4—41no—
208C4—2542no—
209C4C4⋊C4×D12711yescharacteristic
210C4C4×D50811yescharacteristic
211C4#1686117911yescharacteristic
212C4#1686118711yescharacteristic
213C2—21no—
214C2—21no—
215C2—5082no—
216C2#2254975011yescharacteristic
217C2#2254999011yescharacteristic
218C2#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#1866
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
2. C2×#16861144C2semidirect
3. #22549903C2semidirect
6. #22549631C2semidirect
7. #339×C127C2semidirect
20. #16861132C4semidirect
21. #16861132C4semidirect
22. #16861129C4semidirect
23. #16861129C4semidirect
110. C127#1866semidirect
116. #339D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612172
C2×#168611442
#225499032
C2×#168611372
#225498702
#225496312
#339×C1272
#1866127