Finite Group Databasebeta

← #22565593

Subgroups — #22565593

Order 8576.

468 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14524033C211yesmaximal, characteristic
2C2×#14524008C211yesmaximal, characteristic
3#16863327C211yesmaximal, characteristic
4#16863317C211yesmaximal, characteristic
5#16863316C211yesmaximal, characteristic
6#16863293C212yesmaximal
7#16863293C212yesmaximal
8#16863292C212yesmaximal
9#16863291C212yesmaximal
10#16863247C211yesmaximal, characteristic
11#398×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13#14524033—24no—
14#14524008—24no—
15C2×C4⋊Dic67C2212yes—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2212yes—
24C2×Dic67⋊C4C2212yes—
25C2×C4×Dic67C2212yes—
26C2×C4×Dic67C2212yes—
27C2×C4×Dic67C2211yescharacteristic
28C67⋊#95C2211yescharacteristic
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—24no—
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2312yes—
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C4⋊Dic67—22no—
46C4⋊Dic67—24no—
47Dic67⋊C4—22no—
48Dic67⋊C4—24no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52Dic67⋊C4—24no—
53Dic67⋊C4—42no—
54Dic67⋊C4Q818yescomplemented
55C4×Dic67—24no—
56C4×Dic67—42no—
57C4×Dic67D414yes—
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C4⋊C4×C67—22no—
64C4⋊C4×C67—22no—
65C4⋊C4×C67—42no—
66C23×C67C2411yescharacteristic
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2×Q814yes—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—21no—
84C2×Dic67—22no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—21no—
88C2×Dic67—22no—
89C2×Dic67—21no—
90C2×Dic67—21no—
91C2×Dic67C2×D412yes—
92C2×Dic67C2×D412yes—
93C2×Dic67C2×Q814yes—
94C2×Dic67C2×Q814yes—
95C2×Dic67#4114yes—
96Dic134—28no—
97C22×C67C22×D411yescharacteristic
98C22×C67C22×Q812yes—
99C22×C67C2×#4112yes—
100C22×C67H(5,𝔽2)11yescharacteristic
101C22×C672−1+411yescharacteristic
102C4×C67—22no—
103C4×C67—24no—
104C4×C67—41no—
105C4×C67—42no—
106C4×C67—41no—
107Dic67—24no—
108Dic67—22no—
109Dic67—24no—
110Dic67—41no—
111Dic67—41no—
112Dic67C4⋊Q814yes—
113C2×C67#52511yescharacteristic
114C2×C67C2×C4⋊Q811yescharacteristic
115C2×C67#53711yescharacteristic
116C2×C67#55412yes—
117C2×C67#55712yes—
118#2584—671nomaximal, Sylow
119C67#258411yescharacteristic, complemented, Sylow
120C2×#125—671no—
121C2×C4×Q8—671no—
122#398—671no—
123#398D6711yescharacteristic
124#395—671no—
125#395—671no—
126#389—672no—
127#389—672no—
128#389—672no—
129#389—672no—
130C4×C4⋊C4—671no—
131C22×Q8—671no—
132#125—1344no—
133C4×Q8—1344no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—672no—
136C2×C4⋊C4—672no—
137C2×C4⋊C4—671no—
138C2×C4⋊C4—671no—
139C2×C4⋊C4—671no—
140C2×C4⋊C4—671no—
141C2×C4⋊C4—671no—
142C2×C4⋊C4—672no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4C2×D6711yescharacteristic
145C2×C4⋊C4C2×D6711yescharacteristic
146C2×C4⋊C4C2×D6712yes—
147C2×C42—672no—
148C2×C42—671no—
149C2×C42—672no—
150#95—671no—
151#95—672no—
152#95—672no—
153#95—672no—
154#95C2×D6711yescharacteristic
155#95C2×D6712yes—
156C2×Q8—1344no—
157C2×Q8—1342no—
158C22×C4—671no—
159C22×C4—672no—
160C22×C4—672no—
161C22×C4—671no—
162C22×C4—671no—
163C22×C4—671no—
164C22×C4C22×D6711yescharacteristic
165C22×C4C22×D6712yes—
166C22×C4C22×D6711yescharacteristic
167C22×C4C22×D6711yescharacteristic
168C22×C4C22×D6712yes—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—2682no—
173C4⋊C4—678no—
174C4⋊C4—1344no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1344no—
180C4⋊C4—1344no—
181C4⋊C4—1342no—
182C42—2682no—
183C42—674no—
184C42—1344no—
185C23C23×D6711yescharacteristic
186Q8—1348no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—672no—
199C2×C4—674no—
200C2×C4—672no—
201C2×C4—674no—
202C2×C4—674no—
203C2×C4—1341no—
204C2×C4—1341no—
205C2×C4—1342no—
206C2×C4—1341no—
207C2×C4—1341no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4—1342no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4—1341no—
214C2×C4D4×D6712yes—
215C2×C4Q8×D6714yes—
216C22C2×D4×D6711yescharacteristic
217C22C2×Q8×D6712yes—
218C22#41×D6712yes—
219C22#1452408811yescharacteristic
220C22#1452409011yescharacteristic
221C4—24no—
222C4—22no—
223C4—41no—
224C4—42no—
225C4—41no—
226C4—2681no—
227C4—2681no—
228C4—674no—
229C4—1344no—
230C4—1344no—
231C4—1342no—
232C2#1686390411yescharacteristic
233C2C4⋊Q8×D6711yescharacteristic
234C2#1686397511yescharacteristic
235C2#1686410012yes—
236C2#1686412312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#398×C67C2
hypercenter#398D67
hypocenterC67#2584
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
54. Dic67⋊C4Q8semidirect
119. C67#2584semidirect