Finite Group Databasebeta

← #22566126

Subgroups — #22566126

Order 8576.

520 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 141 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#16863326C211yesmaximal, characteristic, complemented
3#16863324C212yesmaximal, complemented
4#16863321C211yesmaximal, characteristic, complemented
5#16863316C211yesmaximal, characteristic, complemented
6#16863310C212yesmaximal
7#16863308C211yesmaximal, characteristic, complemented
8#16863280C212yesmaximal, complemented
9#16863269C212yesmaximal, complemented
10#16863259C211yesmaximal, characteristic, complemented
11C2×C4×D4×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13#14524022—24no—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C22⋊Dic67C2212yes—
21C2×C22⋊Dic67C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C22×D4×C67C2211yescharacteristic
30C23×C4×C67C2211yescharacteristic
31C23×C4×C67C2211yescharacteristic
32C4×D4×C67—24no—
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C22⋊C4×C67C2211yescharacteristic
36C2×C42×C67C2211yescharacteristic
37C2×Dic134—22no—
38C2×Dic134—42no—
39C22×Dic67C2312yes—
40C22×Dic67C2312yes—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C4⋊Dic67—22no—
44Dic67⋊C4—24no—
45Dic67⋊C4—24no—
46Dic67⋊C4—42no—
47C22⋊Dic67—24no—
48C22⋊Dic67—24no—
49C22⋊Dic67—42no—
50C22⋊Dic67—42no—
51C24×C67C2311yescharacteristic
52C24×C67C2311yescharacteristic
53C2×D4×C67—22no—
54C2×D4×C67—22no—
55C2×D4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—42no—
58C22×C4×C67—42no—
59C22×C4×C67D414yes—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2311yescharacteristic
65C4⋊C4×C67—22no—
66C22⋊C4×C67—22no—
67C22⋊C4×C67—22no—
68C42×C67—22no—
69C23×C67—21no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—21no—
73C23×C67—22no—
74C23×C67—22no—
75C23×C67—22no—
76C23×C67—22no—
77C23×C67C2×D412yes—
78C23×C67#4112yes—
79C23×C67C2411yescharacteristic
80D4×C67—28no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—44no—
90C2×C4×C67—44no—
91C2×C4×C67C2×D412yes—
92C2×C4×C67C2×D412yes—
93C2×C4×C67#4112yes—
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—
104C2×Dic67—22no—
105C2×Dic67—22no—
106Dic134—44no—
107C22×C67—22no—
108C22×C67—22no—
109C22×C67—22no—
110C22×C67—22no—
111C22×C67—24no—
112C22×C67—41no—
113C22×C67—41no—
114C22×C67—42no—
115C22×C67—42no—
116C22×C67—42no—
117C22×C67C22×D411yescharacteristic
118C22×C67C2×#4111yescharacteristic
119C22×C67C2×#4111yescharacteristic
120C22×C67H(5,𝔽2)12yes—
121C22×C672−1+412yes—
122C4×C67—22no—
123C4×C67—22no—
124C4×C67—22no—
125C4×C67—41no—
126C4×C67—41no—
127Dic67—42no—
128Dic67—42no—
129Dic67—42no—
130Dic67—42no—
131C2×C67—42no—
132C2×C67—42no—
133C2×C67#52511yescharacteristic
134C2×C67#53411yescharacteristic
135C2×C67#53611yescharacteristic
136C2×C67#53912yes—
137C2×C67#54212yes—
138#2638—671nomaximal, Sylow
139C67#263811yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—671no—
141C2×C4×D4D6711yescharacteristic
142#398—671no—
143#397—671no—
144#397—672no—
145#395—671no—
146#394—672no—
147#393—671no—
148#387—672no—
149#385—672no—
150#381—671no—
151C22×Q8—671no—
152C22×D4C2×D6711yescharacteristic
153C23×C4C2×D6711yescharacteristic
154C23×C4C2×D6711yescharacteristic
155C22⋊Q8—1344no—
156C4×D4—24no—
157C2×C4⋊C4—672no—
158C2×C4⋊C4—672no—
159C2×C4⋊C4—671no—
160C2×C4⋊C4—672no—
161C2×C4⋊C4C2×D6711yescharacteristic
162C2×C22⋊C4—672no—
163C2×C22⋊C4—672no—
164C2×C22⋊C4—672no—
165C2×C22⋊C4—672no—
166C2×C22⋊C4C2×D6711yescharacteristic
167C2×C22⋊C4C2×D6711yescharacteristic
168C2×C42C2×D6711yescharacteristic
169#95—672no—
170#95—671no—
171#95—671no—
172#95—672no—
173#95—672no—
174#95—672no—
175#95—672no—
176C24C22×D6711yescharacteristic
177C24C22×D6711yescharacteristic
178C2×Q8—2682no—
179C2×Q8—1342no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—672no—
187C22×C4—672no—
188C22×C4—672no—
189C22×C4—672no—
190C22×C4C22⋊D6714yes—
191C22×C4C22×D6711yescharacteristic
192C22×C4C22×D6711yescharacteristic
193C22×C4C22×D6711yescharacteristic
194C22×C4C22×D6711yescharacteristic
195C22×C4C22×D6711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—2682no—
198C4⋊C4—1344no—
199C4⋊C4—1342no—
200C4⋊C4—1344no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—2682no—
204C22⋊C4—2682no—
205C22⋊C4—1344no—
206C22⋊C4—1344no—
207C42—22no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23C2×C22⋊D6712yes—
217C23#1176162812yes—
218C23C23×D6711yescharacteristic
219Q8—2684no—
220D4—28no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—44no—
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×C4—1342no—
241C2×C4—1342no—
242C2×C4—1342no—
243C2×C4C2×C22⋊D6712yes—
244C2×C4C2×C22⋊D6712yes—
245C2×C4#1176162812yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—24no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22C22×C22⋊D6711yescharacteristic
257C22C2×#1176162811yescharacteristic
258C22C2×#1176162811yescharacteristic
259C22#1452408812yes—
260C22#1452409012yes—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—2682no—
267C4—2682no—
268C4—2682no—
269C4—2682no—
270C2—42no—
271C2—42no—
272C2#1686390711yescharacteristic
273C2#1686395511yescharacteristic
274C2#1686396711yescharacteristic
275C2#1686398812yes—
276C2#1686401112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C4×D4×C67C2
hypercenterC2×C4×D4D67
hypocenterC67#2638
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14524022C2semidirect
2. #16863326C2semidirect
3. #16863324C2semidirect
4. #16863321C2semidirect
5. #16863316C2semidirect
7. #16863308C2semidirect
8. #16863280C2semidirect
9. #16863269C2semidirect
10. #16863259C2semidirect
139. C67#2638semidirect