Finite Group Databasebeta

← #30639003

Subgroups — #30639003

Order 16256.

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

normal subgroups only

This group has 270 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#22549904C211yesmaximal, characteristic, complemented
4#22549872C211yesmaximal, characteristic
5#22549636C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216—22no—
10C2×#14523232—21no—
11C2×#14523232C2211yescharacteristic
12C22×C127⋊C8—21no—
13C22×C127⋊C8C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861144—22no—
17Dic127⋊C8—21no—
18Dic127⋊C8—21no—
19#16861133C411yescharacteristic, complemented
20#16861133C411yescharacteristic, complemented
21#16861130C411yescharacteristic, complemented
22#16861130C411yescharacteristic, complemented
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#104×C127—22no—
27#14523232—24no—
28#14523232—21no—
29#14523232D412yes—
30C2×C127⋊C8—24no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8C2×C411yescharacteristic
37C2×C127⋊C8C2×C411yescharacteristic
38C2×C127⋊C8D411yescharacteristic
39C2×C127⋊C8D411yescharacteristic
40C4⋊Dic127C2×C411yescharacteristic
41C4⋊Dic127C2×C411yescharacteristic
42C22⋊Dic127—21no—
43C4×Dic127—21no—
44#41×C127—21no—
45#41×C127—21no—
46#41×C127D412yes—
47#41×C127Q812yes—
48C2×Q8×C127C2×C411yescharacteristic
49C2×D4×C127—21no—
50C2×D4×C127C2×C411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C42×C127—21no—
58C42×C127—21no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C127⋊C8—21no—
62C127⋊C8—21no—
63C127⋊C8—21no—
64C127⋊C8—22no—
65C127⋊C8—21no—
66C23×C127—21no—
67C23×C127#4111yescharacteristic
68Q8×C127—21no—
69Q8×C127C4⋊C412yes—
70D4×C127—22no—
71D4×C127—22no—
72D4×C127—21no—
73D4×C127C4⋊C412yes—
74C2×C4×C127—21no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
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—41no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127C4×D411yescharacteristic
93C22×C127C22⋊Q811yescharacteristic
94C22×C127#12311yescharacteristic
95C4×C127—22no—
96C4×C127—42no—
97C4×C127C2×C4⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12011yescharacteristic
100C4×C127C22⋊Q811yescharacteristic
101Dic127—41no—
102Dic127—41no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#38511yescharacteristic
107C2×C127#44311yescharacteristic
108C2×C127#44411yescharacteristic
109#1866—1271nomaximal, Sylow
110C127#186611yescharacteristic, complemented, Sylow
111C2×#131—1271no—
112C2×#108—1271no—
113#424—1271no—
114C2×#104D12711yescharacteristic
115#416—1271no—
116#339—1271no—
117#336—1271no—
118C2×#41C2×D12711yescharacteristic
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#117—1271no—
128C2×C42C2×D12711yescharacteristic
129#108—2542no—
130C4⋊C8—2541no—
131C4⋊C8—2541no—
132#104—22no—
133#103—1271no—
134#103—1271no—
135#102—1271no—
136#102—1271no—
137#41—21no—
138#41—21no—
139#41C22⋊D12712yes—
140#41Dic25412yes—
141C2×Q8C4×D12711yescharacteristic
142C2×D4—21no—
143C2×D4C4×D12711yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C22×D12711yescharacteristic
147M16—21no—
148M16—21no—
149M16—1272no—
150M16—2544no—
151M16—2541no—
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⋊C4—1271no—
164C4⋊C4—1271no—
165C22⋊C4—2541no—
166C42—21no—
167C42—21no—
168C42—2541no—
169C23—21no—
170C23#1452324711yescharacteristic
171Q8—21no—
172Q8Dic127⋊C412yes—
173D4—22no—
174D4—22no—
175D4—21no—
176D4Dic127⋊C412yes—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4C2×C4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C2×Dic25411yescharacteristic
190C2×C4#1452324711yescharacteristic
191C8—41no—
192C8—2541no—
193C8—2541no—
194C8—2541no—
195C8—2541no—
196C8—2542no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—41no—
201C22—41no—
202C22(C22⋊D127)⋊C411yescharacteristic
203C22#1686118611yescharacteristic
204C22#1686119011yescharacteristic
205C4—22no—
206C4—42no—
207C4—5081no—
208C4—5081no—
209C4C2×Dic127⋊C411yescharacteristic
210C4(C22⋊D127)⋊C411yescharacteristic
211C4#1686117711yescharacteristic
212C4#1686118611yescharacteristic
213C2—21no—
214C2—21no—
215C2—42no—
216C2#2254974811yescharacteristic
217C2#2254999011yescharacteristic
218C2#2254999511yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C4×D4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1866
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549904C2semidirect
5. #22549636C2semidirect
6. #22549631C2semidirect
19. #16861133C4semidirect
20. #16861133C4semidirect
21. #16861130C4semidirect
22. #16861130C4semidirect
110. C127#1866semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
C2×#168611442
#225499042
#225498722
#225496362
#225496312
C2×#104×C1272
#1866127