Finite Group Databasebeta

← #22566324

Subgroups — #22566324

Order 8576.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14524023C211yesmaximal, characteristic
2C2×#14524021C211yesmaximal, characteristic
3#16863329C212yesmaximal, complemented
4#16863327C211yesmaximal, characteristic, complemented
5#16863317C212yesmaximal, complemented
6#16863317C211yesmaximal, characteristic, complemented
7#16863316C211yesmaximal, characteristic, complemented
8#16863255C212yesmaximal
9#16863253C212yesmaximal
10#95×D67C211yesmaximal, characteristic
11#398×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×Dic134C2211yescharacteristic
14C22×C4×D67C2211yescharacteristic
15C22×C4×D67C2212yes—
16#14524023—24no—
17#14524021—24no—
18C2×C4⋊Dic67C2212yes—
19C2×C4⋊Dic67C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2212yes—
23C2×Dic67⋊C4C2212yes—
24C2×(C2×D67)⋊C4C2211yescharacteristic
25C2×(C2×D67)⋊C4C2212yes—
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C67⋊#95C2211yescharacteristic
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C2×Dic134—22no—
39C2×Dic134—22no—
40C2×Dic134—42no—
41C2×Dic134—42no—
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2312yes—
45C22×Dic67C2311yescharacteristic
46C22×Dic67C2312yes—
47C2×C4×D67—42no—
48C2×C4×D67—42no—
49C2×C4×D67—42no—
50C2×C4×D67D414yes—
51C2×C4×D67Q818yescomplemented
52C4⋊Dic67—24no—
53C4⋊Dic67—42no—
54Dic67⋊C4—22no—
55Dic67⋊C4—22no—
56Dic67⋊C4—42no—
57Dic67⋊C4—42no—
58(C2×D67)⋊C4—22no—
59(C2×D67)⋊C4—22no—
60(C2×D67)⋊C4—24no—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2312yes—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—42no—
69C23×C67C2411yescharacteristic
70C22×D67—21no—
71C22×D67—22no—
72C22×D67—21no—
73C22×D67C2×D412yes—
74C22×D67C2×Q814yes—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—21no—
85C2×C4×C67—22no—
86C2×C4×C67C2×D412yes—
87C2×C4×C67C2×Q814yes—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—22no—
94C2×Dic67—21no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—21no—
99C2×Dic67C2×D412yes—
100C2×Dic67C2×Q814yes—
101C4×D67—44no—
102C4×D67—48no—
103Dic134—44no—
104Dic134—44no—
105C22×C67C22×D411yescharacteristic
106C22×C67C22×Q812yes—
107C22×C67H(5,𝔽2)11yescharacteristic
108C22×C67H(5,𝔽2)12yes—
109C22×C672−1+411yescharacteristic
110C2×D67—22no—
111C2×D67—22no—
112C2×D67—24no—
113C2×D67—24no—
114C2×D67C4⋊Q814yes—
115C4×C67—22no—
116C4×C67—24no—
117C4×C67—41no—
118C4×C67—41no—
119C4×C67—42no—
120Dic67—22no—
121Dic67—24no—
122Dic67—41no—
123Dic67—42no—
124Dic67—41no—
125C2×C67C2×C4⋊Q811yescharacteristic
126C2×C67#53411yescharacteristic
127C2×C67#53711yescharacteristic
128C2×C67#54312yes—
129C2×C67#54412yes—
130D67—24no—
131#2659—671nomaximal, Sylow
132C67#265911yescharacteristic, complemented, Sylow
133C2×C22⋊Q8—671no—
134C2×C22⋊Q8—671no—
135#398—671no—
136#398—672no—
137#398D6711yescharacteristic, complemented
138#395—671no—
139#395—672no—
140#395—671no—
141#380—672no—
142#380—672no—
143C2×#95—671no—
144C22×Q8—671no—
145C22×Q8—671no—
146C23×C4—671no—
147C23×C4—672no—
148C22⋊Q8—1344no—
149C22⋊Q8—1344no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—672no—
152C2×C4⋊C4—672no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—671no—
155C2×C4⋊C4—672no—
156C2×C4⋊C4C2×D6711yescharacteristic
157C2×C4⋊C4C2×D6711yescharacteristic
158C2×C4⋊C4C2×D6712yes—
159C2×C22⋊C4—671no—
160C2×C22⋊C4—671no—
161C2×C22⋊C4—672no—
162#95—671no—
163#95—672no—
164#95—672no—
165#95—672no—
166#95—672no—
167#95C2×D6711yescharacteristic
168#95C2×D6712yes—
169C24—671no—
170C2×Q8—2682no—
171C2×Q8—2682no—
172C2×Q8—1342no—
173C2×Q8—1342no—
174C22×C4—2682no—
175C22×C4—2682no—
176C22×C4—2682no—
177C22×C4—674no—
178C22×C4—678no—
179C22×C4—672no—
180C22×C4—671no—
181C22×C4—671no—
182C22×C4—672no—
183C22×C4—671no—
184C22×C4C22×D6711yescharacteristic
185C22×C4C22×D6712yes—
186C22×C4C22×D6711yescharacteristic
187C22×C4C22×D6711yescharacteristic
188C22×C4C22×D6712yes—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—2682no—
193C4⋊C4—2682no—
194C4⋊C4—2682no—
195C4⋊C4—1342no—
196C4⋊C4—1344no—
197C4⋊C4—1342no—
198C22⋊C4—1342no—
199C22⋊C4—1344no—
200C22⋊C4—1342no—
201C23—672no—
202C23—674no—
203C23—1341no—
204C23—1342no—
205C23—1341no—
206C23C23×D6711yescharacteristic
207Q8—2684no—
208Q8—2684no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—2688no—
221C2×C4—2684no—
222C2×C4—672no—
223C2×C4—674no—
224C2×C4—1341no—
225C2×C4—1342no—
226C2×C4—1341no—
227C2×C4—1341no—
228C2×C4—1342no—
229C2×C4—1342no—
230C2×C4—1342no—
231C2×C4—1341no—
232C2×C4—1342no—
233C2×C4—1342no—
234C2×C4—1342no—
235C2×C4D4×D6712yes—
236C2×C4Q8×D6714yes—
237C22—674no—
238C22—1344no—
239C22—1344no—
240C22—1342no—
241C22—1342no—
242C22C2×D4×D6711yescharacteristic
243C22C2×Q8×D6712yes—
244C22#1452408811yescharacteristic
245C22#1452408912yes—
246C22#1452409011yescharacteristic
247C4—24no—
248C4—22no—
249C4—41no—
250C4—41no—
251C4—42no—
252C4—2682no—
253C4—2681no—
254C4—2681no—
255C4—1344no—
256C4—1342no—
257C2—1344no—
258C2C4⋊Q8×D6711yescharacteristic
259C2#1686395711yescharacteristic
260C2#1686397511yescharacteristic
261C2#1686402112yes—
262C2#1686402612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#398×C67C2
hypercenter#398D67
hypocenterC67#2659
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #16863329C2semidirect
4. #16863327C2semidirect
5. #16863317C2semidirect
6. #16863317C2semidirect
7. #16863316C2semidirect
11. #398×C67C2semidirect
51. C2×C4×D67Q8semidirect
132. C67#2659semidirect
137. #398D67semidirect