Finite Group Databasebeta

← #22567910

Subgroups — #22567910

Order 8576.

472 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#16863335C211yesmaximal, characteristic, complemented
2#16863319C212yesmaximal, complemented
3#16863316C211yesmaximal, characteristic, complemented
4#16863308C212yesmaximal, complemented
5#16863280C212yesmaximal, complemented
6#16863276C212yesmaximal
7#16863268C212yesmaximal, complemented
8#16863254C212yesmaximal, complemented
9#390×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11C23×Dic67C2212yes—
12C2×C4⋊Dic67C2211yescharacteristic
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×C22⋊Dic67C2212yes—
17C2×C22⋊Dic67C2212yes—
18C2×C22⋊Dic67C2212yes—
19C67⋊#95C2211yescharacteristic
20C67⋊#95C2212yes—
21C67⋊#95C2211yescharacteristic
22C67⋊#95C2212yes—
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C22×D4×C67C2211yescharacteristic
27C2×C4⋊C4×C67C2211yescharacteristic
28C2×C22⋊C4×C67C2212yes—
29C2×C22⋊C4×C67C2212yes—
30C2×C42×C67C2211yescharacteristic
31C2×Dic134—41no—
32C2×Dic134—42no—
33C22×Dic67—24no—
34C22×Dic67—42no—
35C22×Dic67—42no—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C22×Dic67C2312yes—
39C22×Dic67C2312yes—
40C4⋊Dic67—41no—
41Dic67⋊C4—24no—
42Dic67⋊C4—42no—
43Dic67⋊C4—42no—
44C22⋊Dic67—24no—
45C22⋊Dic67—42no—
46C22⋊Dic67—42no—
47C24×C67C2312yes—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C2×D4×C67Q814yescomplemented
51C22×C4×C67C2312yes—
52C22×C4×C67C2311yescharacteristic
53C22×C4×C67C2311yescharacteristic
54C22×C4×C67C2311yescharacteristic
55C4⋊C4×C67—41no—
56C22⋊C4×C67—42no—
57C22⋊C4×C67—42no—
58C42×C67—41no—
59C23×C67—22no—
60C23×C67—22no—
61C23×C67—22no—
62C23×C67—22no—
63C23×C67—22no—
64C23×C67—22no—
65C23×C67C2×Q814yes—
66C23×C67C2411yescharacteristic
67D4×C67—44no—
68C2×C4×C67—21no—
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×C67C2×Q812yes—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—48no—
90C2×Dic67#4114yes—
91Dic134—44no—
92C22×C67—24no—
93C22×C67—24no—
94C22×C67—24no—
95C22×C67—42no—
96C22×C67—42no—
97C22×C67—42no—
98C22×C67—42no—
99C22×C67C22×Q811yescharacteristic
100C22×C67C2×#4112yes—
101C22×C67H(5,𝔽2)11yescharacteristic
102C22×C67H(5,𝔽2)12yes—
103C22×C67H(5,𝔽2)11yescharacteristic
104C4×C67—22no—
105C4×C67—42no—
106C4×C67—41no—
107C4×C67—41no—
108Dic67—24no—
109Dic67—42no—
110Dic67—42no—
111Dic67—42no—
112C2×C67—44no—
113C2×C67#53812yes—
114C2×C67#54311yescharacteristic
115C2×C67#55412yes—
116C2×C67#55511yescharacteristic
117C2×C67#56011yescharacteristic
118#2784—671nomaximal, Sylow
119C67#278411yescharacteristic, complemented, Sylow
120#400—671no—
121#396—672no—
122#395—671no—
123#393—672no—
124#390D6711yescharacteristic
125#387—672no—
126#386—672no—
127#385—672no—
128#380—672no—
129C22×Q8—671no—
130C22×D4C2×D6711yescharacteristic
131C23×C4—672no—
132C2×C4⋊C4—672no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—671no—
135C2×C4⋊C4—672no—
136C2×C4⋊C4C2×D6711yescharacteristic
137C2×C22⋊C4—672no—
138C2×C22⋊C4—672no—
139C2×C22⋊C4—672no—
140C2×C22⋊C4C2×D6712yes—
141C2×C22⋊C4C2×D6712yes—
142C2×C42C2×D6711yescharacteristic
143#95—671no—
144#95—672no—
145#95—671no—
146#95—672no—
147#95—672no—
148#95—672no—
149#95—672no—
150C24C22×D6712yes—
151C2×Q8—2682no—
152C2×Q8—2681no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic13414yescomplemented
156C22×C4—2682no—
157C22×C4—2682no—
158C22×C4—672no—
159C22×C4—672no—
160C22×C4—672no—
161C22×C4—672no—
162C22×C4—1344no—
163C22×C4C22×D6711yescharacteristic
164C22×C4C22×D6712yes—
165C22×C4C22×D6711yescharacteristic
166C22×C4C22×D6711yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—2682no—
169C4⋊C4—2682no—
170C4⋊C4—2681no—
171C4⋊C4—1344no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—2682no—
175C22⋊C4—2682no—
176C22⋊C4—1344no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic13414yes—
185C23C23×D6711yescharacteristic
186Q8—2684no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—2688no—
198C2×C4—674no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4—1342no—
204C2×C4—1342no—
205C2×C4—1342no—
206C2×C4—1342no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4C2×Dic13412yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic13411yescharacteristic
219C22#41×D6712yes—
220C22#1452408811yescharacteristic
221C22#1452408811yescharacteristic
222C22#1452408812yes—
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—2682no—
228C4—2682no—
229C4—2682no—
230C4—1344no—
231C2—44no—
232C2#1686397812yes—
233C2#1686402211yescharacteristic
234C2#1686410512yes—
235C2#1686410811yescharacteristic
236C2#1686413311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#390×C67C2
hypercenter#390D67
hypocenterC67#2784
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16863335C2semidirect
2. #16863319C2semidirect
3. #16863316C2semidirect
4. #16863308C2semidirect
5. #16863280C2semidirect
7. #16863268C2semidirect
8. #16863254C2semidirect
50. C2×D4×C67Q8semidirect
119. C67#2784semidirect
155. C2×D4Dic134semidirect