Finite Group Databasebeta

← #30650641

Subgroups — #30650641

Order 16256.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#22550531C211yesmaximal, characteristic
2#22550228C211yesmaximal, characteristic, complemented
3#22550217C212yesmaximal, complemented
4#22550153C211yesmaximal, characteristic, complemented
5#22550115C212yesmaximal, complemented
6#22550086C211yesmaximal, characteristic, complemented
7#22550036C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10#22549870C212yesmaximal, complemented
11#556×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13C2×#14523238C2212yes—
14C2×#14523235C2211yescharacteristic
15C22×C127⋊C8C2212yes—
16#16861209C2211yescharacteristic
17#16861191—22no—
18#16861188C2212yes—
19#16861180—21no—
20C4×D508C2211yescharacteristic
21C4⋊C4×D127—21no—
22#16861142C2212yes—
23#16861140C2211yescharacteristic
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861133C2211yescharacteristic
26#16861132C2212yes—
27#16861132C2212yes—
28#16861132C2212yes—
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#126×C127—22no—
35#124×C127—21no—
36C22⋊Q8×C127C2212yes—
37C4×Q8×C127—21no—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40#117×C127C2212yes—
41#14523248—24no—
42#14523248—42no—
43C2×Dic254C2312yes—
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—22no—
46C2×C4×D127—22no—
47C2×C4×D127—21no—
48#14523238—42no—
49#14523235—22no—
50C2×C127⋊C8—42no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2312yes—
53C2×C127⋊C8C2311yescharacteristic
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127C2311yescharacteristic
62C2×D4×C127D411yescharacteristic
63C22×C4×C127C2312yes—
64C4⋊C4×C127—22no—
65C4⋊C4×C127—21no—
66C4⋊C4×C127—21no—
67C4⋊C4×C127D411yescharacteristic
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C22⋊C4×C127—22no—
71C22⋊C4×C127—22no—
72C22⋊C4×C127D412yes—
73C42×C127—22no—
74C42×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127—21no—
77Dic254—22no—
78Dic254—22no—
79D508—21no—
80D508#4112yes—
81C22⋊D127—44no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C4×D127—44no—
85C4×D127—42no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88C127⋊C8—22no—
89C23×C127C2×D412yes—
90Q8×C127—21no—
91Q8×C127—21no—
92Q8×C127—42no—
93D4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2411yescharacteristic
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127C22×D411yescharacteristic
107C2×D127—22no—
108C2×D127—41no—
109C2×D127—41no—
110C4×C127—22no—
111C4×C127—21no—
112C4×C127—41no—
113C4×C127—42no—
114C4×C127—41no—
115C4×C127#13512yes—
116C4×C127C2×#4111yescharacteristic
117C4×C127H(5,𝔽2)11yescharacteristic
118Dic127—42no—
119Dic127—41no—
120C2×C127—42no—
121C2×C127#54611yescharacteristic
122C2×C127C2×#13511yescharacteristic
123C2×C127#57811yescharacteristic
124D127—42no—
125#3302—1271nomaximal, Sylow
126C127#330211yescharacteristic, complemented, Sylow
127#556D12711yescharacteristic, complemented
128#547—1271no—
129#487—1271no—
130#485—1272no—
131#474—1271no—
132#467—1272no—
133#462—1271no—
134#452—1272no—
135C4×SD16—1271no—
136C8×D4—1271no—
137#416—1272no—
138C2×#41—1272no—
139C2×Q16—1272no—
140C2×SD16—1271no—
141C22×C8—1272no—
142C4⋊Q8—1271no—
143C4⋊Q8C2×D12711yescharacteristic
144#126—22no—
145#124—21no—
146#123—2542no—
147C22⋊Q8—1272no—
148C22⋊Q8C2×D12712yes—
149#121—2541no—
150C4×Q8—21no—
151C4×Q8C2×D12711yescharacteristic
152C4×D4—1271no—
153C4×D4C2×D12711yescharacteristic
154#117C2×D12712yes—
155C2×C4⋊C4—2541no—
156#107—1272no—
157#106—1271no—
158C4⋊C8—1271no—
159#103—1271no—
160#103—1272no—
161#103—1272no—
162#103—1272no—
163#102—1271no—
164#102—1272no—
165C22⋊C8—1272no—
166C4×C8—1271no—
167#41—5082no—
168#41—2544no—
169C2×Q8—21no—
170C2×Q8—1272no—
171C2×Q8C22×D12711yescharacteristic
172C2×D4—1271no—
173C2×D4—2542no—
174C2×D4C22⋊D12711yescharacteristic
175C22×C4—2541no—
176C22×C4—2542no—
177C22×C4C22×D12712yes—
178Q16—5082no—
179SD16—2542no—
180C2×C8—1271no—
181C2×C8—1271no—
182C2×C8—1272no—
183C2×C8—5082no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—1271no—
188C4⋊C4—2542no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4C22⋊D12711yescharacteristic
192C4⋊C4C22×D12711yescharacteristic
193C4⋊C4C22×D12712yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—2542no—
197C22⋊C4—2541no—
198C22⋊C4C22⋊D12712yes—
199C42—22no—
200C42—21no—
201C42C22×D12711yescharacteristic
202C23—2541no—
203C23C2×C22⋊D12712yes—
204Q8—21no—
205Q8—21no—
206Q8—42no—
207Q8—2542no—
208Q8—2542no—
209D4—22no—
210D4—1272no—
211D4—5084no—
212D4—2541no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—5084no—
219C2×C4—5082no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4C2×C22⋊D12712yes—
223C2×C4C2×C22⋊D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2541no—
227C8—2541no—
228C8—2542no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—5081no—
233C22—5081no—
234C22—2542no—
235C22C22×C22⋊D12711yescharacteristic
236C4—21no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—5081no—
242C4—5082no—
243C4#1686122912yes—
244C4#41×D12711yescharacteristic
245C4#1686125511yescharacteristic
246C2—42no—
247C2—5082no—
248C2#2255051411yescharacteristic
249C2C2×#1686122911yescharacteristic
250C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#556×C127C2
socleC22×C127C22×D4
prime radical#22550531C2
hypercenter#556D127
hypocenterC127#3302
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #22550228C2semidirect
3. #22550217C2semidirect
4. #22550153C2semidirect
5. #22550115C2semidirect
6. #22550086C2semidirect
7. #22550036C2semidirect
8. C4×#14523235C2semidirect
9. D4×C127⋊C8C2semidirect
10. #22549870C2semidirect
11. #556×C127C2semidirect
126. C127#3302semidirect
127. #556D127semidirect

Maximal subgroups

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

subgroupindex
#225505312
#225502282
#225502172
#225501532
#225501152
#225500862
#225500362
C4×#145232352
D4×C127⋊C82
#225498702
#556×C1272
#3302127