Finite Group Databasebeta

← #22568222

Subgroups — #22568222

Order 8576.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14524022C211yesmaximal, characteristic, complemented
2#16863329C212yesmaximal, complemented
3#16863326C211yesmaximal, characteristic, complemented
4#16863324C212yesmaximal, complemented
5#16863321C211yesmaximal, characteristic, complemented
6#16863319C212yesmaximal, complemented
7#16863316C211yesmaximal, characteristic, complemented
8#16863310C212yesmaximal
9#16863308C211yesmaximal, characteristic, complemented
10#16863256C211yesmaximal, characteristic, complemented
11C2×#123×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13#14524022—24no—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×C4⋊Dic67C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C22⋊Dic67C2212yes—
21C2×C22⋊Dic67C2212yes—
22C2×C22⋊Dic67C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C22×D4×C67C2211yescharacteristic
30C23×C4×C67C2211yescharacteristic
31#123×C67—24no—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C22⋊C4×C67C2211yescharacteristic
36C2×C22⋊C4×C67C2211yescharacteristic
37C2×Dic134—22no—
38C2×Dic134—42no—
39C22×Dic67C2312yes—
40C22×Dic67C2312yes—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C4⋊Dic67—22no—
44C4⋊Dic67—42no—
45Dic67⋊C4—24no—
46Dic67⋊C4—42no—
47Dic67⋊C4—42no—
48C22⋊Dic67—24no—
49C22⋊Dic67—42no—
50C22⋊Dic67—42no—
51C22⋊Dic67—42no—
52C24×C67C2311yescharacteristic
53C24×C67C2311yescharacteristic
54C2×D4×C67—22no—
55C2×D4×C67—42no—
56C22×C4×C67—42no—
57C22×C4×C67D414yes—
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2311yescharacteristic
63C4⋊C4×C67—22no—
64C4⋊C4×C67—22no—
65C22⋊C4×C67—22no—
66C22⋊C4×C67—22no—
67C22⋊C4×C67—22no—
68C23×C67—22no—
69C23×C67—21no—
70C23×C67—22no—
71C23×C67—21no—
72C23×C67—21no—
73C23×C67—22no—
74C23×C67—21no—
75C23×C67—22no—
76C23×C67—21no—
77C23×C67C2×D412yes—
78C23×C67C2411yescharacteristic
79D4×C67—44no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—22no—
86C2×C4×C67—21no—
87C2×C4×C67—22no—
88C2×C4×C67—21no—
89C2×C4×C67—44no—
90C2×C4×C67C2×D412yes—
91C2×C4×C67C2×D412yes—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67—22no—
100C2×Dic67—22no—
101C2×Dic67—22no—
102C2×Dic67—22no—
103C2×Dic67—22no—
104Dic134—44no—
105C22×C67—22no—
106C22×C67—22no—
107C22×C67—22no—
108C22×C67—42no—
109C22×C67—42no—
110C22×C67—41no—
111C22×C67—41no—
112C22×C67—41no—
113C22×C67—42no—
114C22×C67—42no—
115C22×C67C22×D411yescharacteristic
116C22×C67H(5,𝔽2)12yes—
117C22×C67H(5,𝔽2)11yescharacteristic
118C22×C672−1+412yes—
119C22×C672−1+411yescharacteristic
120C4×C67—22no—
121C4×C67—22no—
122C4×C67—41no—
123C4×C67—41no—
124C4×C67—41no—
125Dic67—42no—
126Dic67—42no—
127Dic67—42no—
128Dic67—42no—
129C2×C67—42no—
130C2×C67—81no—
131C2×C67#53411yescharacteristic
132C2×C67#53611yescharacteristic
133C2×C67#53711yescharacteristic
134C2×C67#56212yes—
135C2×C67#56312yes—
136#2814—671nomaximal, Sylow
137C67#281411yescharacteristic, complemented, Sylow
138C2×#123D6711yescharacteristic
139C2×C22⋊Q8—671no—
140#398—671no—
141#398—672no—
142#397—671no—
143#397—672no—
144#396—672no—
145#395—671no—
146#394—672no—
147#393—671no—
148#380—671no—
149C22×Q8—671no—
150C22×D4C2×D6711yescharacteristic
151C23×C4C2×D6711yescharacteristic
152#123—24no—
153C22⋊Q8—1344no—
154C2×C4⋊C4—672no—
155C2×C4⋊C4—672no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—672no—
158C2×C4⋊C4—671no—
159C2×C4⋊C4C2×D6711yescharacteristic
160C2×C4⋊C4C2×D6711yescharacteristic
161C2×C22⋊C4—672no—
162C2×C22⋊C4—672no—
163C2×C22⋊C4—672no—
164C2×C22⋊C4—672no—
165C2×C22⋊C4C2×D6711yescharacteristic
166C2×C22⋊C4C2×D6711yescharacteristic
167C2×C22⋊C4C2×D6711yescharacteristic
168#95—671no—
169#95—671no—
170#95—672no—
171#95—672no—
172#95—672no—
173#95—672no—
174C24C22×D6711yescharacteristic
175C24C22×D6711yescharacteristic
176C2×Q8—2682no—
177C2×Q8—1342no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—42no—
181C22×C4—672no—
182C22×C4—672no—
183C22×C4—672no—
184C22×C4—672no—
185C22×C4C22⋊D6714yes—
186C22×C4C22×D6711yescharacteristic
187C22×C4C22×D6711yescharacteristic
188C22×C4C22×D6711yescharacteristic
189C22×C4C22×D6711yescharacteristic
190C22×C4C22×D6711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2682no—
194C4⋊C4—2682no—
195C4⋊C4—2682no—
196C4⋊C4—1344no—
197C4⋊C4—1342no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—2682no—
202C22⋊C4—2682no—
203C22⋊C4—2682no—
204C22⋊C4—1344no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—21no—
214C23C2×C22⋊D6712yes—
215C23C23×D6711yescharacteristic
216Q8—2684no—
217D4—44no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—1342no—
229C2×C4—1342no—
230C2×C4—1342no—
231C2×C4—1342no—
232C2×C4—1342no—
233C2×C4—1342no—
234C2×C4—1342no—
235C2×C4—1342no—
236C2×C4—1342no—
237C2×C4—1342no—
238C2×C4—1342no—
239C2×C4—1342no—
240C2×C4C2×C22⋊D6712yes—
241C2×C4C2×C22⋊D6712yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22C22×C22⋊D6711yescharacteristic
253C22#1452408812yes—
254C22#1452408911yescharacteristic
255C22#1452409012yes—
256C22#1452409011yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—2682no—
263C4—2682no—
264C4—2682no—
265C4—2682no—
266C2—42no—
267C2—81no—
268C2#1686395511yescharacteristic
269C2#1686396711yescharacteristic
270C2#1686397411yescharacteristic
271C2#1686414312yes—
272C2#1686414912yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#123×C67C2
hypercenterC2×#123D67
hypocenterC67#2814
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14524022C2semidirect
2. #16863329C2semidirect
3. #16863326C2semidirect
4. #16863324C2semidirect
5. #16863321C2semidirect
6. #16863319C2semidirect
7. #16863316C2semidirect
9. #16863308C2semidirect
10. #16863256C2semidirect
137. C67#2814semidirect