Finite Group Databasebeta

← #22563499

Subgroups — #22563499

Order 8576.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic67C211yesmaximal, characteristic
2C2×#14524008C211yesmaximal, characteristic
3#16863329C212yesmaximal
4#16863328C211yesmaximal, characteristic
5#16863326C211yesmaximal, characteristic
6#16863317C212yesmaximal
7#16863315C211yesmaximal, characteristic
8C4⋊C4×Dic67C212yesmaximal
9#16863247C212yesmaximal
10#16863241C211yesmaximal, characteristic
11#395×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13Q8×Dic67—24no—
14#14524008—24no—
15C2×C4⋊Dic67C2212yes—
16C2×C4⋊Dic67C2212yes—
17C2×C4⋊Dic67C2211yescharacteristic
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×C4×Dic67C2211yescharacteristic
23C2×C4×Dic67C2212yes—
24C2×C4×Dic67C2211yescharacteristic
25C2×C4×Dic67C2211yescharacteristic
26C2×C4×Dic67C2212yes—
27C67⋊#95C2211yescharacteristic
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C22×Q8×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2212yes—
34C2×C4⋊C4×C67C2211yescharacteristic
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—22no—
39C2×Dic134—22no—
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2312yes—
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C22×Dic67C2311yescharacteristic
46C4⋊Dic67—22no—
47C4⋊Dic67—24no—
48C4⋊Dic67—24no—
49Dic67⋊C4—22no—
50Dic67⋊C4—24no—
51Dic67⋊C4—24no—
52Dic67⋊C4—22no—
53C4×Dic67—22no—
54C4×Dic67—22no—
55C4×Dic67—24no—
56C4×Dic67D414yes—
57C4×Dic67Q818yescomplemented
58C2×Q8×C67—22no—
59C2×Q8×C67—24no—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2312yes—
64C22×C4×C67C2311yescharacteristic
65C4⋊C4×C67—22no—
66C4⋊C4×C67—24no—
67C23×C67C2411yescharacteristic
68Q8×C67—28no—
69C2×C4×C67—21no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2×Q814yes—
80C2×C4×C67#4114yes—
81C2×C4×C67#4112yes—
82C2×Dic67—21no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—21no—
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×Q814yes—
97C2×Dic67#4112yes—
98Dic134—28no—
99C22×C67C22×D411yescharacteristic
100C22×C67C22×Q812yes—
101C22×C67C2×#4111yescharacteristic
102C22×C67C2×#4112yes—
103C22×C67C2×#4111yescharacteristic
104C4×C67—22no—
105C4×C67—22no—
106C4×C67—24no—
107C4×C67—24no—
108C4×C67—41no—
109Dic67—22no—
110Dic67—22no—
111Dic67—24no—
112Dic67—41no—
113Dic67—42no—
114Dic67C4⋊Q814yes—
115C2×C67#52511yescharacteristic
116C2×C67#52511yescharacteristic
117C2×C67C2×C4⋊Q811yescharacteristic
118C2×C67#53312yes—
119C2×C67#53312yes—
120#2399—671nomaximal, Sylow
121C67#239911yescharacteristic, complemented, Sylow
122C2×C4×Q8—671no—
123C2×C4×Q8—671no—
124#398—671no—
125#398—672no—
126#398—671no—
127#395—671no—
128#395—672no—
129#395D6711yescharacteristic
130C4×C4⋊C4—672no—
131C4×C4⋊C4—672no—
132#376—671no—
133C22×Q8—671no—
134C22×Q8C2×D6711yescharacteristic
135C4×Q8—1344no—
136C4×Q8—1344no—
137C2×C4⋊C4—671no—
138C2×C4⋊C4—671no—
139C2×C4⋊C4—672no—
140C2×C4⋊C4—671no—
141C2×C4⋊C4—672no—
142C2×C4⋊C4—672no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4C2×D6711yescharacteristic
145C2×C4⋊C4C2×D6712yes—
146C2×C42—671no—
147C2×C42—672no—
148C2×C42—671no—
149C2×C42—672no—
150C2×C42—671no—
151#95—671no—
152#95—672no—
153#95—672no—
154#95—672no—
155#95—672no—
156#95C2×D6711yescharacteristic
157#95C2×D6712yes—
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—1342no—
161C2×Q8—1342no—
162C2×Q8—1342no—
163C22×C4—671no—
164C22×C4—672no—
165C22×C4—671no—
166C22×C4—671no—
167C22×C4—672no—
168C22×C4—671no—
169C22×C4C22×D6711yescharacteristic
170C22×C4C22×D6711yescharacteristic
171C22×C4C22×D6712yes—
172C22×C4C22×D6712yes—
173C22×C4C22×D6711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—1342no—
177C4⋊C4—1344no—
178C4⋊C4—1344no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1344no—
182C4⋊C4—1344no—
183C42—674no—
184C42—678no—
185C42—1342no—
186C42—1344no—
187C42—1342no—
188C23C23×D6711yescharacteristic
189Q8—28no—
190Q8—1348no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—672no—
201C2×C4—674no—
202C2×C4—672no—
203C2×C4—674no—
204C2×C4—672no—
205C2×C4—1341no—
206C2×C4—1342no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4—1341no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1341no—
215C2×C4—1341no—
216C2×C4D4×D6712yes—
217C2×C4Q8×D6714yes—
218C2×C4#1176162714yes—
219C2×C4#1176162912yes—
220C22C2×D4×D6711yescharacteristic
221C22C2×Q8×D6712yes—
222C22#41×D6711yescharacteristic
223C22C2×#1176162712yes—
224C22C2×#1176162911yescharacteristic
225C4—22no—
226C4—24no—
227C4—22no—
228C4—24no—
229C4—41no—
230C4—2681no—
231C4—2682no—
232C4—674no—
233C4—1342no—
234C4—1342no—
235C4—1344no—
236C2#1686390211yescharacteristic
237C2#1686390911yescharacteristic
238C2C4⋊Q8×D6711yescharacteristic
239C2#1686394812yes—
240C2#1686395112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2399
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
57. C4×Dic67Q8semidirect
121. C67#2399semidirect