Finite Group Databasebeta

← #22571286

Subgroups — #22571286

Order 8576.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#16863972C211yesmaximal, characteristic
2#16863788C211yesmaximal, characteristic, complemented
3#16863775C211yesmaximal, characteristic, complemented
4#16863766C211yesmaximal, characteristic, complemented
5#16863753C211yesmaximal, characteristic, complemented
6#16863618C212yesmaximal, complemented
7#16863606C212yesmaximal, complemented
8#16863556C212yesmaximal
9#16863551C212yesmaximal, complemented
10#16863457C211yesmaximal, characteristic, complemented
11#541×C67C211yesmaximal, characteristic, complemented
12C2×#11761628C2211yescharacteristic
13C22×Dic134C2211yescharacteristic
14C2×#11761618C2212yes—
15C2×#11761618C2212yes—
16C2×#11761618C2212yes—
17C2×#11761615C2212yes—
18#14524043C2211yescharacteristic
19#14524043C2211yescharacteristic
20#14524031C2211yescharacteristic
21#14524031C2211yescharacteristic
22#14524027—22no—
23#14524023—22no—
24C4⋊(C67⋊C8)C2212yes—
25#14523966C2212yes—
26#14523966C2212yes—
27#14523966C2212yes—
28#14523963C2212yes—
29C22⋊(C67⋊C8)C2212yes—
30#14523954C2211yescharacteristic
31C4×C67⋊C8C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#126×C67—22no—
34#125×C67—22no—
35#125×C67C2211yescharacteristic
36C22⋊Q8×C67C2212yes—
37C4×Q8×C67C2212yes—
38#117×C67C2211yescharacteristic
39#11761628—42no—
40C2×Dic134—21no—
41C2×Dic134—42no—
42C2×Dic134C2311yescharacteristic
43C2×Dic134C2311yescharacteristic
44C2×Dic134C2311yescharacteristic
45C2×D268C2311yescharacteristic
46C2×C22⋊D67—21no—
47C22×Dic67—21no—
48C2×C4×D67—21no—
49#11761618—42no—
50#11761618—42no—
51#11761618—42no—
52#11761615—42no—
53C2×C67⋊C8C2312yes—
54C2×C67⋊C8C2312yes—
55C4⋊Dic67—22no—
56C4⋊Dic67—22no—
57(C2×D67)⋊C4—22no—
58(C2×D67)⋊C4—22no—
59C2×Q8×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C4⋊C4×C67—22no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—22no—
64C4⋊C4×C67—22no—
65C4⋊C4×C67D412yes—
66C4⋊C4×C67C2312yes—
67C22⋊C4×C67—22no—
68C22⋊C4×C67D412yes—
69C42×C67—22no—
70C42×C67C2311yescharacteristic
71C42×C67C2311yescharacteristic
72C23×C67C2×D411yescharacteristic
73C22×D67—21no—
74Q8×C67—22no—
75Q8×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67C2×D412yes—
80C2×C4×C67C2×D412yes—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2411yescharacteristic
83C22⋊D67—42no—
84C2×Dic67—21no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—42no—
88D268—22no—
89C4×D67—42no—
90Dic134—22no—
91Dic134—22no—
92Dic134—22no—
93Dic134—42no—
94C67⋊C8—22no—
95C67⋊C8—22no—
96C22×C67—21no—
97C22×C67—22no—
98C22×C67C22×D411yescharacteristic
99C2×D67—41no—
100C2×D67—42no—
101C4×C67—21no—
102C4×C67—22no—
103C4×C67—22no—
104C4×C67—42no—
105C4×C67—42no—
106C4×C67H(5,𝔽2)12yes—
107Dic67—41no—
108Dic67—41no—
109Dic67—41no—
110C2×C67—41no—
111C2×C67#53511yescharacteristic
112C2×C67#57812yes—
113D67—81no—
114#3223—671nomaximal, Sylow
115C67#322311yescharacteristic, complemented, Sylow
116#541D6711yescharacteristic, complemented
117#536—671no—
118#497—671no—
119#494—671no—
120#491—671no—
121#488—671no—
122#464—672no—
123#462—672no—
124#452—672no—
125#451—672no—
126#433—671no—
127C2×#41—671no—
128C22×Q8—671no—
129C2×Q16—672no—
130C2×Q16—672no—
131C2×Q16—672no—
132C2×SD16—672no—
133C4⋊Q8—671no—
134C4⋊Q8—671no—
135C4⋊Q8C2×D6711yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D6711yescharacteristic
139#124—671no—
140#124—671no—
141#123—1342no—
142C22⋊Q8—1342no—
143C22⋊Q8C2×D6712yes—
144C4×Q8C2×D6712yes—
145#117C2×D6711yescharacteristic
146C4⋊C8—672no—
147#103—672no—
148#103—672no—
149#103—672no—
150#102—672no—
151C22⋊C8—672no—
152#97—671no—
153C4×C8—671no—
154#41—2682no—
155C2×Q8—2682no—
156C2×Q8—671no—
157C2×Q8—671no—
158C2×Q8—671no—
159C2×Q8—1341no—
160C2×Q8C22×D6712yes—
161C2×D4—671no—
162C2×D4—1341no—
163C22×C4—1341no—
164C22×C4—1341no—
165C22×C4C22×D6711yescharacteristic
166Q16—2682no—
167Q16—2682no—
168Q16—2682no—
169SD16—2682no—
170C2×C8—672no—
171C2×C8—672no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4C22⋊D6712yes—
179C4⋊C4C22×D6712yes—
180C22⋊C4—22no—
181C22⋊C4—1342no—
182C22⋊C4—1342no—
183C22⋊C4C22⋊D6712yes—
184C42—22no—
185C42C22×D6711yescharacteristic
186C42C22×D6711yescharacteristic
187C23—1341no—
188C23C2×C22⋊D6711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—2682no—
192Q8—1342no—
193Q8—1342no—
194Q8—1342no—
195D4—2682no—
196D4—1342no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2682no—
201C2×C4—2682no—
202C2×C4—1341no—
203C2×C4—1341no—
204C2×C4—1341no—
205C2×C4C2×C22⋊D6712yes—
206C2×C4C2×C22⋊D6712yes—
207C2×C4C2×C22⋊D6711yescharacteristic
208C2×C4C23×D6711yescharacteristic
209C8—1342no—
210C8—1342no—
211C22—22no—
212C22—21no—
213C22—2682no—
214C22—2681no—
215C22C22×C22⋊D6711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—2681no—
222C4—2681no—
223C4—2681no—
224C4#1452408912yes—
225C2—41no—
226C2—5361no—
227C2#1686396011yescharacteristic
228C2#1686421112yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#541×C67C2
socleC22×C67C22×D4
hypercenter#541D67
hypocenterC67#3223
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #16863788C2semidirect
3. #16863775C2semidirect
4. #16863766C2semidirect
5. #16863753C2semidirect
6. #16863618C2semidirect
7. #16863606C2semidirect
9. #16863551C2semidirect
10. #16863457C2semidirect
11. #541×C67C2semidirect
115. C67#3223semidirect
116. #541D67semidirect