Finite Group Databasebeta

← #22565980

Subgroups — #22565980

Order 8576.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14524041C211yesmaximal, characteristic, complemented
2C2×#14524023C211yesmaximal, characteristic
3C2×#14524019C211yesmaximal, characteristic
4C2×#14524000C211yesmaximal, characteristic
5#16863330C214yesmaximal, complemented
6#16863325C214yesmaximal
7#16863302C211yesmaximal, characteristic, complemented
8#16863261C211yesmaximal, characteristic, complemented
9#391×C67C211yesmaximal, characteristic, complemented
10C22×Dic134C2211yescharacteristic
11C22×C4×D67C2211yescharacteristic
12#14524041—24no—
13#14524023—24no—
14#14524019—24no—
15#14524000—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2212yes—
18C2×Dic67⋊C4C2214yes—
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2212yes—
23C2×(C2×D67)⋊C4C2212yes—
24C2×C4×Dic67C2211yescharacteristic
25C67⋊#95C2214yes—
26C67⋊#95C2214yes—
27C22×Q8×C67C2211yescharacteristic
28C2×C4⋊C4×C67C2211yescharacteristic
29C2×C42×C67C2211yescharacteristic
30#95×C67C2214yes—
31C23×D67C2311yescharacteristic
32C2×Dic134—22no—
33C2×Dic134—22no—
34C2×Dic134—22no—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C22×Dic67C2311yescharacteristic
39C2×C4×D67—22no—
40C2×C4×D67—22no—
41C2×C4×D67D414yes—
42C4⋊Dic67—22no—
43C4⋊Dic67—24no—
44Dic67⋊C4—24no—
45Dic67⋊C4—24no—
46Dic67⋊C4—28no—
47(C2×D67)⋊C4—24no—
48(C2×D67)⋊C4—24no—
49(C2×D67)⋊C4—24no—
50C4×Dic67D414yes—
51C2×Q8×C67—22no—
52C2×Q8×C67—22no—
53C2×Q8×C67—22no—
54C22×C4×C67C2312yes—
55C22×C4×C67C2312yes—
56C22×C4×C67C2312yes—
57C22×C4×C67C2311yescharacteristic
58C4⋊C4×C67—22no—
59C42×C67—22no—
60C23×C67C2411yescharacteristic
61C22×D67—21no—
62C22×D67—21no—
63C22×D67—22no—
64C22×D67—22no—
65C22×D67C2×D412yes—
66Q8×C67—44no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—24no—
74C2×C4×C67—22no—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2×D412yes—
77C2×C4×C67#4112yes—
78C2×Dic67—21no—
79C2×Dic67—21no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—24no—
87C2×Dic67C2×D414yes—
88C2×Dic67C2×D412yes—
89C4×D67—28no—
90Dic134—44no—
91C22×C67C22×D411yescharacteristic
92C22×C67C22×D411yescharacteristic
93C22×C67C2×#4111yescharacteristic
94C22×C672−1+412yes—
95C22×C672−1+412yes—
96C2×D67—22no—
97C2×D67—22no—
98C2×D67—22no—
99C2×D67—42no—
100C2×D67—42no—
101C4×C67—42no—
102C4×C67—42no—
103C4×C67—42no—
104C4×C67#12114yes—
105Dic67—22no—
106Dic67—24no—
107Dic67—42no—
108Dic67—42no—
109C2×C67C2×#12111yescharacteristic
110C2×C67#53611yescharacteristic
111C2×C67#53612yes—
112C2×C67#53611yescharacteristic
113C2×C67#54112yes—
114D67—42no—
115#2621—671nomaximal, Sylow
116C67#262111yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—671no—
118C2×C22⋊Q8—671no—
119C2×C22⋊Q8—671no—
120C2×#117—671no—
121#398—674no—
122#397—674no—
123#391—671no—
124#391D6711yescharacteristic, complemented
125#382—671no—
126C22×Q8—671no—
127C22×Q8C2×D6711yescharacteristic
128C23×C4—671no—
129C4⋊Q8—1344no—
130C22⋊Q8—1344no—
131C22⋊Q8—1344no—
132#117—1344no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—671no—
136C2×C4⋊C4—672no—
137C2×C4⋊C4—674no—
138C2×C4⋊C4C2×D6711yescharacteristic
139C2×C22⋊C4—672no—
140C2×C22⋊C4—672no—
141C2×C22⋊C4—672no—
142C2×C42—671no—
143C2×C42C2×D6711yescharacteristic
144#95—674no—
145#95—674no—
146#95C2×D6714yes—
147C24—671no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1342no—
152C2×Q8—1342no—
153C2×Q8—1342no—
154C22×C4—672no—
155C22×C4—671no—
156C22×C4—674no—
157C22×C4—672no—
158C22×C4—672no—
159C22×C4—1342no—
160C22×C4—1342no—
161C22×C4C22×D6712yes—
162C22×C4C22×D6712yes—
163C22×C4C22×D6712yes—
164C22×C4C22×D6711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—1344no—
167C4⋊C4—1344no—
168C4⋊C4—1344no—
169C4⋊C4—1348no—
170C4⋊C4—1342no—
171C22⋊C4—1344no—
172C22⋊C4—1344no—
173C22⋊C4—1344no—
174C42—22no—
175C42—674no—
176C23—672no—
177C23—1341no—
178C23—1341no—
179C23—1342no—
180C23—1342no—
181C23C23×D6711yescharacteristic
182Q8—44no—
183Q8—2684no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—674no—
193C2×C4—672no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4—1342no—
198C2×C4—1344no—
199C2×C4—1341no—
200C2×C4—1341no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4—1348no—
204C2×C4D4×D6712yes—
205C2×C4D4×D6712yes—
206C2×C4#1176162812yes—
207C22—2682no—
208C22—2682no—
209C22—1342no—
210C22—1342no—
211C22—1342no—
212C22C2×D4×D6711yescharacteristic
213C22C2×D4×D6711yescharacteristic
214C22C2×#1176162811yescharacteristic
215C22#1452409012yes—
216C22#1452409112yes—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—2682no—
221C4—2682no—
222C4—1342no—
223C4—1344no—
224C4#1452401614yes—
225C2—2682no—
226C2C2×#1452401611yescharacteristic
227C2#1686396611yescharacteristic
228C2#1686396912yes—
229C2#1686397111yescharacteristic
230C2#1686400412yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#391×C67C2
hypercenter#391D67
hypocenterC67#2621
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14524041C2semidirect
5. #16863330C2semidirect
7. #16863302C2semidirect
8. #16863261C2semidirect
9. #391×C67C2semidirect
116. C67#2621semidirect
124. #391D67semidirect