Finite Group Databasebeta

← #22565413

Subgroups — #22565413

Order 8576.

500 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 171 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
2C2×#14524008C211yesmaximal, characteristic
3#16863317C211yesmaximal, characteristic
4#16863316C211yesmaximal, characteristic
5#16863315C211yesmaximal, characteristic
6#16863300C212yesmaximal
7#16863293C212yesmaximal
8#16863291C212yesmaximal
9#16863265C212yesmaximal
10#16863247C211yesmaximal, characteristic
11#395×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13C22×Dic134C2211yescharacteristic
14#14524041—24no—
15#14524008—24no—
16C2×C4⋊Dic67C2212yes—
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2212yes—
22C2×Dic67⋊C4C2212yes—
23C2×Dic67⋊C4C2212yes—
24C2×C4×Dic67C2212yes—
25C2×C4×Dic67C2211yescharacteristic
26C2×C4×Dic67C2211yescharacteristic
27C2×C4×Dic67C2211yescharacteristic
28C67⋊#95C2211yescharacteristic
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C22×Q8×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—22no—
39C2×Dic134—22no—
40C2×Dic134—42no—
41C2×Dic134Q814yescomplemented
42C22×Dic67C2312yes—
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2311yescharacteristic
45C22×Dic67C2312yes—
46C22×Dic67C2311yescharacteristic
47C22×Dic67C2311yescharacteristic
48C4⋊Dic67—42no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—24no—
53Dic67⋊C4—42no—
54Dic67⋊C4—42no—
55Dic67⋊C4Q818yescomplemented
56C4×Dic67—22no—
57C4×Dic67—22no—
58C4×Dic67—24no—
59C4×Dic67D414yes—
60C2×Q8×C67—22no—
61C2×Q8×C67—42no—
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2312yes—
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C22×C4×C67C2312yes—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—42no—
69C23×C67C2411yescharacteristic
70Q8×C67—44no—
71C2×C4×C67—21no—
72C2×C4×C67—22no—
73C2×C4×C67—21no—
74C2×C4×C67—22no—
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×C67C2×D412yes—
82C2×C4×C67C2×Q814yes—
83C2×C4×C67C2×Q812yes—
84C2×Dic67—21no—
85C2×Dic67—22no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—21no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67C2×D412yes—
94C2×Dic67C2×D412yes—
95C2×Dic67C2×Q814yes—
96C2×Dic67C2×Q812yes—
97C2×Dic67C2×Q812yes—
98C2×Dic67C2×Q814yes—
99C2×Dic67#4112yes—
100Dic134—28no—
101Dic134—44no—
102C22×C67C22×D411yescharacteristic
103C22×C67C22×Q811yescharacteristic
104C22×C67C22×Q812yes—
105C22×C67C2×#4111yescharacteristic
106C22×C67H(5,𝔽2)12yes—
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—24no—
110C4×C67—41no—
111C4×C67—42no—
112Dic67—22no—
113Dic67—24no—
114Dic67—42no—
115Dic67—41no—
116Dic67C22⋊Q814yes—
117Dic67C4⋊Q814yes—
118C2×C67C2×C22⋊Q811yescharacteristic
119C2×C67C2×C4⋊Q811yescharacteristic
120C2×C67#53511yescharacteristic
121C2×C67#55412yes—
122C2×C67Q8212yes—
123#2567—671nomaximal, Sylow
124C67#256711yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—671no—
126C2×C4×Q8—671no—
127#395—671no—
128#395—671no—
129#395—671no—
130#395D6711yescharacteristic
131#391—672no—
132#389—672no—
133#389—672no—
134#384—672no—
135C4×C4⋊C4—671no—
136C22×Q8—671no—
137C22×Q8—671no—
138C22×Q8C2×D6711yescharacteristic
139C4⋊Q8—1344no—
140C4×Q8—1344no—
141C2×C4⋊C4—672no—
142C2×C4⋊C4—672no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4—672no—
145C2×C4⋊C4—671no—
146C2×C4⋊C4—671no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4C2×D6711yescharacteristic
150C2×C4⋊C4C2×D6712yes—
151C2×C42—671no—
152C2×C42—671no—
153C2×C42—671no—
154C2×C42—672no—
155#95—671no—
156#95—672no—
157#95—672no—
158#95—672no—
159#95C2×D6711yescharacteristic
160#95C2×D6712yes—
161C2×Q8—22no—
162C2×Q8—42no—
163C2×Q8—2682no—
164C2×Q8—674no—
165C2×Q8—1342no—
166C2×Q8—1342no—
167C2×Q8—1342no—
168C22×C4—671no—
169C22×C4—671no—
170C22×C4—671no—
171C22×C4—672no—
172C22×C4—672no—
173C22×C4—671no—
174C22×C4C22×D6712yes—
175C22×C4C22×D6711yescharacteristic
176C22×C4C22×D6711yescharacteristic
177C22×C4C22×D6711yescharacteristic
178C22×C4C22×D6712yes—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—2682no—
182C4⋊C4—2682no—
183C4⋊C4—2682no—
184C4⋊C4—678no—
185C4⋊C4—1342no—
186C4⋊C4—1344no—
187C4⋊C4—1344no—
188C4⋊C4—1342no—
189C42—674no—
190C42—1342no—
191C42—1342no—
192C42—1344no—
193C23C23×D6711yescharacteristic
194Q8—44no—
195Q8—2684no—
196Q8—1348no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—672no—
208C2×C4—672no—
209C2×C4—674no—
210C2×C4—672no—
211C2×C4—672no—
212C2×C4—674no—
213C2×C4—672no—
214C2×C4—1341no—
215C2×C4—1342no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4—1342no—
219C2×C4—1341no—
220C2×C4—1342no—
221C2×C4—1341no—
222C2×C4—1342no—
223C2×C4D4×D6712yes—
224C2×C4Q8×D6714yes—
225C2×C4Q8×D6712yes—
226C22C2×D4×D6711yescharacteristic
227C22C2×Q8×D6712yes—
228C22C2×Q8×D6711yescharacteristic
229C22#41×D6711yescharacteristic
230C22#1452408812yes—
231C4—24no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—2682no—
237C4—2681no—
238C4—674no—
239C4—674no—
240C4—1344no—
241C4—1342no—
242C2C22⋊Q8×D6711yescharacteristic
243C2C4⋊Q8×D6711yescharacteristic
244C2#1686396111yescharacteristic
245C2#1686410012yes—
246C2#1686412612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2567
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
41. C2×Dic134Q8semidirect
55. Dic67⋊C4Q8semidirect
124. C67#2567semidirect