Finite Group Databasebeta

← #22571292

Subgroups — #22571292

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#16864008C211yesmaximal, characteristic, complemented
2#16863791C211yesmaximal, characteristic, complemented
3#16863775C211yesmaximal, characteristic, complemented
4#16863764C211yesmaximal, characteristic, complemented
5#16863750C211yesmaximal, characteristic, complemented
6#16863615C212yesmaximal, complemented
7#16863603C212yesmaximal, complemented
8#16863558C212yesmaximal
9#16863552C212yesmaximal, complemented
10#16863457C211yesmaximal, characteristic, complemented
11#536×C67C211yesmaximal, characteristic
12C2×#11761618C2212yes—
13C2×#11761618C2212yes—
14C2×#11761618C2212yes—
15C2×#11761616C2212yes—
16#14524043C2211yescharacteristic
17#14524036—22no—
18#14524035C2211yescharacteristic
19#14524033—22no—
20#14524022C2212yes—
21C4×Dic134C2212yes—
22C4⋊(C67⋊C8)C2212yes—
23#14523967C2212yes—
24#14523967C2212yes—
25#14523967C2212yes—
26#14523964C2212yes—
27C22⋊(C67⋊C8)C2212yes—
28#14523954C2211yescharacteristic
29C4×C67⋊C8C2211yescharacteristic
30C2×#41×C67C2211yescharacteristic
31C22×Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33C4⋊Q8×C67C2211yescharacteristic
34#124×C67C2211yescharacteristic
35#124×C67C2211yescharacteristic
36#123×C67—22no—
37C22⋊Q8×C67—22no—
38#117×C67C2211yescharacteristic
39C2×Dic134C2312yes—
40#11761618—42no—
41#11761618—42no—
42#11761618—42no—
43#11761616—42no—
44C2×C67⋊C8C2312yes—
45C2×C67⋊C8C2312yes—
46C4⋊Dic67—22no—
47C4⋊Dic67C2312yes—
48Dic67⋊C4—22no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51C22⋊Dic67—22no—
52C4×Dic67—22no—
53#41×C67—42no—
54C2×Q8×C67—21no—
55C2×Q8×C67—42no—
56C2×Q8×C67C2311yescharacteristic
57C2×Q8×C67C2311yescharacteristic
58C2×Q8×C67C2311yescharacteristic
59C2×D4×C67—21no—
60C2×D4×C67C2311yescharacteristic
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67C2311yescharacteristic
64C4⋊C4×C67—22no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67D412yes—
67C22⋊C4×C67—22no—
68C22⋊C4×C67—22no—
69C22⋊C4×C67D412yes—
70C42×C67C2311yescharacteristic
71C42×C67C2311yescharacteristic
72C23×C67—21no—
73C23×C67C2×D411yescharacteristic
74Q8×C67—22no—
75Q8×C67—22no—
76Q8×C67—22no—
77Q8×C67—42no—
78D4×C67—22no—
79D4×C67—42no—
80C2×C4×C67—22no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67C2×D412yes—
87C2×C4×C67C2×D412yes—
88C2×C4×C67C2×D411yescharacteristic
89C2×C4×C67C2411yescharacteristic
90C2×Dic67—22no—
91C2×Dic67—22no—
92Dic134—22no—
93Dic134—22no—
94C67⋊C8—22no—
95C67⋊C8—22no—
96C22×C67—21no—
97C22×C67—22no—
98C22×C67—41no—
99C22×C67—42no—
100C22×C67C22×D411yescharacteristic
101C4×C67—21no—
102C4×C67—22no—
103C4×C67—22no—
104C4×C67—41no—
105C4×C67—41no—
106C4×C67—41no—
107C4×C67H(5,𝔽2)12yes—
108Dic67—42no—
109Dic67—42no—
110C2×C67—41no—
111C2×C67—81no—
112C2×C67#53511yescharacteristic
113C2×C67#57812yes—
114#3223—671nomaximal, Sylow
115C67#322311yescharacteristic, complemented, Sylow
116#541—671no—
117#536D6711yescharacteristic
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×#41C2×D6711yescharacteristic
128C22×Q8C2×D6711yescharacteristic
129C2×Q16—672no—
130C2×Q16—672no—
131C2×Q16—672no—
132C2×SD16—672no—
133C4⋊Q8—671no—
134C4⋊Q8C2×D6711yescharacteristic
135C4⋊Q8C2×D6711yescharacteristic
136#126—1342no—
137#125—671no—
138#125—1342no—
139#124C2×D6711yescharacteristic
140#124C2×D6711yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—672no—
144C4×Q8—672no—
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—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—672no—
158C2×Q8C22×D6711yescharacteristic
159C2×Q8C22×D6711yescharacteristic
160C2×Q8C22×D6711yescharacteristic
161C2×D4—21no—
162C2×D4C22×D6711yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
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—672no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4C22⋊D6712yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—1342no—
183C22⋊C4C22⋊D6712yes—
184C42—1342no—
185C42C22×D6711yescharacteristic
186C42C22×D6711yescharacteristic
187C23—21no—
188C23C2×C22⋊D6711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—1342no—
194Q8—1342no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1342no—
204C2×C4—1342no—
205C2×C4C2×C22⋊D6712yes—
206C2×C4C2×C22⋊D6712yes—
207C2×C4C2×C22⋊D6711yescharacteristic
208C2×C4C23×D6711yescharacteristic
209C8—1342no—
210C8—1342no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D6711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—2682no—
223C4—2682no—
224C4#1452408812yes—
225C2—41no—
226C2—81no—
227C2#1686396311yescharacteristic
228C2#1686421112yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#536×C67C2
socleC22×C67C22×D4
hypercenter#536D67
hypocenterC67#3223
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #16864008C2semidirect
2. #16863791C2semidirect
3. #16863775C2semidirect
4. #16863764C2semidirect
5. #16863750C2semidirect
6. #16863615C2semidirect
7. #16863603C2semidirect
9. #16863552C2semidirect
10. #16863457C2semidirect
115. C67#3223semidirect