Finite Group Databasebeta

← #22565420

Subgroups — #22565420

Order 8576.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14524042C211yesmaximal, characteristic
2C2×#14524008C211yesmaximal, characteristic
3#16863336C211yesmaximal, characteristic
4#16863335C211yesmaximal, characteristic
5#16863334C211yesmaximal, characteristic
6#16863300C212yesmaximal
7#16863300C212yesmaximal
8#16863300C212yesmaximal
9#16863260C212yesmaximal
10#16863247C211yesmaximal, characteristic
11#400×C67C211yesmaximal, characteristic
12C22×Dic134C2212yes—
13C22×Dic134C2211yescharacteristic
14#14524042—24no—
15#14524008—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×C4×Dic67C2212yes—
22C2×C4×Dic67C2211yescharacteristic
23C2×C4×Dic67C2211yescharacteristic
24C2×C4×Dic67C2211yescharacteristic
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34#95×C67C2211yescharacteristic
35#95×C67C2212yes—
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—22no—
39C2×Dic134—24no—
40C2×Dic134—28no—
41C2×Dic134Q814yescomplemented
42C22×Dic67C2312yes—
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2312yes—
45C22×Dic67C2311yescharacteristic
46C22×Dic67C2311yescharacteristic
47C22×Dic67C2311yescharacteristic
48C4⋊Dic67—22no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—24no—
53C4×Dic67—22no—
54C4×Dic67—22no—
55C4×Dic67—24no—
56C4×Dic67D414yes—
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2312yes—
59C22×C4×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C4⋊C4×C67—22no—
63C4⋊C4×C67—22no—
64C23×C67C2411yescharacteristic
65C2×C4×C67—21no—
66C2×C4×C67—21no—
67C2×C4×C67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—24no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2×Q812yes—
77C2×Dic67—21no—
78C2×Dic67—21no—
79C2×Dic67—21no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—24no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67C2×D412yes—
87C2×Dic67C2×D412yes—
88C2×Dic67C2×Q812yes—
89C2×Dic67C2×Q812yes—
90C2×Dic67#4112yes—
91C2×Dic67#4114yes—
92Dic134—28no—
93Dic134—48no—
94C22×C67C22×D411yescharacteristic
95C22×C67C22×Q811yescharacteristic
96C22×C67C2×#4111yescharacteristic
97C22×C67C2×#4112yes—
98C22×C672−1+412yes—
99C4×C67—22no—
100C4×C67—22no—
101C4×C67—42no—
102C4×C67—41no—
103C4×C67—42no—
104Dic67—22no—
105Dic67—42no—
106Dic67—41no—
107Dic67—42no—
108Dic67C22⋊Q814yes—
109Dic67#12414yes—
110C2×C67C2×C22⋊Q811yescharacteristic
111C2×C67C2×#12411yescharacteristic
112C2×C67#53611yescharacteristic
113C2×C67#55612yes—
114C2×C67#55712yes—
115#2568—671nomaximal, Sylow
116C67#256811yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—671no—
118C2×C4×Q8—671no—
119#400—671no—
120#400—671no—
121#400—671no—
122#400D6711yescharacteristic
123#391—672no—
124#391—672no—
125#391—672no—
126#382—672no—
127C4×C4⋊C4—671no—
128C22×Q8—671no—
129C22×Q8—672no—
130C4⋊Q8—1344no—
131C4×Q8—1344no—
132C2×C4⋊C4—671no—
133C2×C4⋊C4—671no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—671no—
136C2×C4⋊C4—672no—
137C2×C4⋊C4C2×D6711yescharacteristic
138C2×C4⋊C4C2×D6711yescharacteristic
139C2×C42—671no—
140C2×C42—671no—
141C2×C42—671no—
142C2×C42—672no—
143#95—671no—
144#95—672no—
145#95—672no—
146#95—672no—
147#95—672no—
148#95—672no—
149#95—672no—
150#95C2×D6711yescharacteristic
151#95C2×D6712yes—
152#95C2×D6712yes—
153C2×Q8—674no—
154C2×Q8—1344no—
155C2×Q8—1348no—
156C2×Q8—1342no—
157C2×Q8—1342no—
158C22×C4—671no—
159C22×C4—671no—
160C22×C4—671no—
161C22×C4—672no—
162C22×C4—671no—
163C22×C4—672no—
164C22×C4C22×D6712yes—
165C22×C4C22×D6711yescharacteristic
166C22×C4C22×D6712yes—
167C22×C4C22×D6711yescharacteristic
168C22×C4C22×D6711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—1342no—
172C4⋊C4—1344no—
173C4⋊C4—1344no—
174C4⋊C4—1342no—
175C4⋊C4—1342no—
176C42—674no—
177C42—1342no—
178C42—1342no—
179C42—1344no—
180C23C23×D6711yescharacteristic
181Q8—2688no—
182Q8—1348no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—672no—
194C2×C4—672no—
195C2×C4—672no—
196C2×C4—672no—
197C2×C4—674no—
198C2×C4—672no—
199C2×C4—1341no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4—1344no—
204C2×C4—1342no—
205C2×C4—1342no—
206C2×C4—1341no—
207C2×C4—1341no—
208C2×C4D4×D6712yes—
209C2×C4Q8×D6712yes—
210C22C2×D4×D6711yescharacteristic
211C22C2×Q8×D6711yescharacteristic
212C22#41×D6711yescharacteristic
213C22#41×D6712yes—
214C22#1452409012yes—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—42no—
219C4—41no—
220C4—2682no—
221C4—2682no—
222C4—2681no—
223C4—674no—
224C4—674no—
225C4—1342no—
226C2C22⋊Q8×D6711yescharacteristic
227C2#124×D6711yescharacteristic
228C2#1686396811yescharacteristic
229C2#1686411312yes—
230C2#1686412312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#400×C67C2
hypercenter#400D67
hypocenterC67#2568
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
41. C2×Dic134Q8semidirect
116. C67#2568semidirect