Finite Group Databasebeta

← #22566029

Subgroups — #22566029

Order 8576.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14524041C211yesmaximal, characteristic, complemented
2C2×#14523999C211yesmaximal, characteristic
3#16863337C212yesmaximal, complemented
4#16863332C212yesmaximal
5#16863317C211yesmaximal, characteristic, complemented
6#16863315C211yesmaximal, characteristic, complemented
7#16863308C211yesmaximal, characteristic
8#16863293C212yesmaximal, complemented
9#16863260C211yesmaximal, characteristic, complemented
10#16863254C212yesmaximal
11#393×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13C23×Dic67C2211yescharacteristic
14#14524041—24no—
15#14523999—24no—
16C2×C4⋊Dic67C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2212yes—
23C2×C22⋊Dic67C2211yescharacteristic
24C2×C22⋊Dic67C2212yes—
25C2×C4×Dic67C2211yescharacteristic
26C2×C4×Dic67C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2211yescharacteristic
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C22×Q8×C67C2211yescharacteristic
33C2×C22⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—42no—
39C22×Dic67—22no—
40C22×Dic67—22no—
41C22×Dic67Q814yescomplemented
42C22×Dic67C2312yes—
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C22×Dic67C2311yescharacteristic
46C22×Dic67C2311yescharacteristic
47C22×Dic67C2311yescharacteristic
48C4⋊Dic67—42no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—24no—
53Dic67⋊C4—24no—
54Dic67⋊C4—42no—
55C22⋊Dic67—22no—
56C22⋊Dic67—42no—
57C4×Dic67—22no—
58C4×Dic67D414yes—
59C24×C67C2311yescharacteristic
60C2×Q8×C67—22no—
61C2×Q8×C67—42no—
62C22×C4×C67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2312yes—
65C22×C4×C67C2311yescharacteristic
66C22⋊C4×C67—22no—
67C22⋊C4×C67—42no—
68C23×C67—21no—
69C23×C67—21no—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67C2×Q812yes—
73C23×C67C2411yescharacteristic
74Q8×C67—44no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67C2×D412yes—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—28no—
89C2×Dic67—21no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67C2×D412yes—
98C2×Dic67C2×D412yes—
99C2×Dic67C2×Q812yes—
100C2×Dic67C2×Q812yes—
101C2×Dic67#4112yes—
102Dic134—44no—
103C22×C67—22no—
104C22×C67—24no—
105C22×C67—41no—
106C22×C67—41no—
107C22×C67—42no—
108C22×C67C22×D411yescharacteristic
109C22×C67C22×Q811yescharacteristic
110C22×C67C2×#4111yescharacteristic
111C22×C67H(5,𝔽2)12yes—
112C22×C672−1+412yes—
113C4×C67—22no—
114C4×C67—42no—
115C4×C67—41no—
116C4×C67—42no—
117Dic67—22no—
118Dic67—22no—
119Dic67—42no—
120Dic67—41no—
121Dic67—42no—
122Dic67C22⋊Q814yes—
123C2×C67—42no—
124C2×C67C2×C22⋊Q811yescharacteristic
125C2×C67#53511yescharacteristic
126C2×C67#53611yescharacteristic
127C2×C67#54212yes—
128C2×C67#54412yes—
129#2628—671nomaximal, Sylow
130C67#262811yescharacteristic, complemented, Sylow
131C2×C4⋊Q8—671no—
132C2×#117—671no—
133#400—672no—
134#399—672no—
135#395—671no—
136#395—671no—
137#393—671no—
138#393D6711yescharacteristic
139#389—672no—
140#382—671no—
141#380—672no—
142C22×Q8—671no—
143C22×Q8C2×D6711yescharacteristic
144C23×C4—671no—
145C4⋊Q8—1344no—
146#117—1344no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—671no—
150C2×C4⋊C4—671no—
151C2×C4⋊C4—672no—
152C2×C4⋊C4—672no—
153C2×C4⋊C4—672no—
154C2×C22⋊C4—672no—
155C2×C22⋊C4—671no—
156C2×C22⋊C4C2×D6711yescharacteristic
157C2×C22⋊C4C2×D6712yes—
158C2×C42—671no—
159C2×C42—671no—
160#95—672no—
161#95—672no—
162#95—671no—
163#95—672no—
164#95—672no—
165#95C2×D6711yescharacteristic
166#95C2×D6712yes—
167C24C22×D6711yescharacteristic
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—2682no—
171C2×Q8—1342no—
172C22×C4—671no—
173C22×C4—674no—
174C22×C4—671no—
175C22×C4—671no—
176C22×C4—672no—
177C22×C4—672no—
178C22×C4—671no—
179C22×C4—1342no—
180C22×C4—1342no—
181C22×C4C22×D6711yescharacteristic
182C22×C4C22×D6712yes—
183C22×C4C22×D6712yes—
184C22×C4C22×D6711yescharacteristic
185C4⋊C4—2682no—
186C4⋊C4—2682no—
187C4⋊C4—1342no—
188C4⋊C4—1344no—
189C4⋊C4—1344no—
190C4⋊C4—1344no—
191C4⋊C4—1342no—
192C22⋊C4—22no—
193C22⋊C4—42no—
194C22⋊C4—2682no—
195C22⋊C4—1342no—
196C42—674no—
197C42—1342no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23Q8×D6712yes—
203C23C23×D6711yescharacteristic
204Q8—44no—
205Q8—2684no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—672no—
216C2×C4—672no—
217C2×C4—672no—
218C2×C4—672no—
219C2×C4—672no—
220C2×C4—1341no—
221C2×C4—1342no—
222C2×C4—1342no—
223C2×C4—1342no—
224C2×C4—1342no—
225C2×C4—1342no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—1348no—
229C2×C4—1341no—
230C2×C4—1341no—
231C2×C4—1342no—
232C2×C4D4×D6712yes—
233C22—24no—
234C22—22no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22C2×D4×D6711yescharacteristic
239C22C2×Q8×D6711yescharacteristic
240C22#41×D6711yescharacteristic
241C22#1452408812yes—
242C22#1452409012yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—41no—
247C4—2682no—
248C4—2682no—
249C4—2681no—
250C4—674no—
251C4—1342no—
252C4—1342no—
253C2—42no—
254C2C22⋊Q8×D6711yescharacteristic
255C2#1686396111yescharacteristic
256C2#1686396611yescharacteristic
257C2#1686401012yes—
258C2#1686402512yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#393×C67C2
hypercenter#393D67
hypocenterC67#2628
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14524041C2semidirect
3. #16863337C2semidirect
5. #16863317C2semidirect
6. #16863315C2semidirect
8. #16863293C2semidirect
9. #16863260C2semidirect
41. C22×Dic67Q8semidirect
130. C67#2628semidirect