Finite Group Databasebeta

← #22565189

Subgroups — #22565189

Order 8576.

576 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14524043C211yesmaximal, characteristic, complemented
2C2×#14524031C211yesmaximal, characteristic
3C2×#14524031C211yesmaximal, characteristic
4C2×C4×Dic134C211yesmaximal, characteristic, complemented
5#16863318C214yesmaximal, complemented
6#16863302C211yesmaximal, characteristic, complemented
7#16863298C211yesmaximal, characteristic
8#16863288C214yesmaximal, complemented
9C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
10C22×Dic134C2211yescharacteristic
11C22×Dic134C2211yescharacteristic
12C22×D268C2211yescharacteristic
13#14524043—24no—
14#14524031—24no—
15#14524031—24no—
16C4×Dic134—24no—
17C2×C4⋊Dic67C2214yes—
18C2×C4⋊Dic67C2211yescharacteristic
19C2×Dic67⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2214yes—
22C2×(C2×D67)⋊C4C2214yes—
23C2×(C2×D67)⋊C4C2212yes—
24C2×C4×Dic67C2212yes—
25C67⋊#95C2214yes—
26C2×C4⋊C4×C67C2212yes—
27C2×C42×C67C2211yescharacteristic
28C2×C42×C67C2211yescharacteristic
29C2×C42×C67C2211yescharacteristic
30#95×C67C2212yes—
31C23×D67C2312yes—
32C2×Dic134—22no—
33C2×Dic134—22no—
34C2×Dic134—22no—
35C2×Dic134—22no—
36C2×Dic134—22no—
37C2×Dic134D414yes—
38C2×D268—22no—
39C2×D268—22no—
40C2×D268—22no—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C22×Dic67C2312yes—
44C4⋊Dic67—22no—
45C4⋊Dic67—28no—
46Dic67⋊C4—24no—
47(C2×D67)⋊C4—24no—
48(C2×D67)⋊C4—24no—
49(C2×D67)⋊C4—28no—
50(C2×D67)⋊C4—28no—
51C4×Dic67—24no—
52C22×C4×C67C2312yes—
53C22×C4×C67C2311yescharacteristic
54C22×C4×C67C2311yescharacteristic
55C22×C4×C67C2312yes—
56C22×C4×C67C2311yescharacteristic
57C4⋊C4×C67—24no—
58C42×C67—22no—
59C42×C67—22no—
60C42×C67D414yes—
61C23×C67C2411yescharacteristic
62C22×D67—22no—
63C22×D67—22no—
64C22×D67—24no—
65C22×D67—22no—
66C22×D67—24no—
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—22no—
74C2×C4×C67C2×D412yes—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2×D412yes—
78C2×C4×C67#4114yes—
79C2×C4×C67#4112yes—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—24no—
86C2×Dic67—22no—
87C2×Dic67C2×D414yes—
88D268—44no—
89Dic134—28no—
90Dic134—44no—
91C22×C67C22×D411yescharacteristic
92C22×C67C22×D411yescharacteristic
93C22×C67C2×#4112yes—
94C22×C67C2×#4111yescharacteristic
95C22×C672−1+412yes—
96C2×D67—42no—
97C2×D67—42no—
98C2×D67—42no—
99C2×D67—44no—
100C2×D67—44no—
101C4×C67—22no—
102C4×C67—22no—
103C4×C67—24no—
104C4×C67—42no—
105C4×C67#12114yes—
106Dic67—24no—
107Dic67—42no—
108Dic67—42no—
109C2×C67C2×#12111yescharacteristic
110C2×C67#53211yescharacteristic
111C2×C67#53611yescharacteristic
112C2×C67#54812yes—
113C2×C67#55612yes—
114D67—82no—
115#2545—671nomaximal, Sylow
116C67#254511yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—671no—
118C2×#124—671no—
119C2×#124—671no—
120C2×C4×Q8—671no—
121#396—674no—
122#391—671no—
123#390—671no—
124#388—674no—
125C4×C4⋊C4D6711yescharacteristic, complemented
126C22×Q8—671no—
127C22×Q8—671no—
128C22×D4—671no—
129C4⋊Q8—1344no—
130#124—1344no—
131#124—1344no—
132C4×Q8—1344no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—671no—
135C2×C4⋊C4—674no—
136C2×C4⋊C4C2×D6712yes—
137C2×C22⋊C4—672no—
138C2×C22⋊C4—674no—
139C2×C22⋊C4—672no—
140C2×C22⋊C4—674no—
141C2×C42—672no—
142C2×C42C2×D6711yescharacteristic
143C2×C42C2×D6711yescharacteristic
144C2×C42C2×D6711yescharacteristic
145#95—674no—
146#95C2×D6712yes—
147C24—672no—
148C2×Q8—674no—
149C2×Q8—1342no—
150C2×Q8—1342no—
151C2×Q8—1342no—
152C2×Q8—1342no—
153C2×Q8—1342no—
154C2×D4—1342no—
155C2×D4—1342no—
156C2×D4—1342no—
157C22×C4—672no—
158C22×C4—672no—
159C22×C4—672no—
160C22×C4C22×D6712yes—
161C22×C4C22×D6711yescharacteristic
162C22×C4C22×D6711yescharacteristic
163C22×C4C22×D6712yes—
164C22×C4C22×D6711yescharacteristic
165C4⋊C4—24no—
166C4⋊C4—1348no—
167C4⋊C4—1344no—
168C4⋊C4—1342no—
169C22⋊C4—1348no—
170C22⋊C4—1344no—
171C22⋊C4—1344no—
172C22⋊C4—1348no—
173C42—22no—
174C42—22no—
175C42—1344no—
176C42D26814yes—
177C23—1342no—
178C23—1342no—
179C23—1344no—
180C23—1342no—
181C23—1344no—
182C23C23×D6711yescharacteristic
183Q8—2684no—
184Q8—1348no—
185D4—2684no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—674no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4—1344no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4D4×D6712yes—
202C2×C4C2×D26812yes—
203C2×C4C2×D26812yes—
204C2×C4C2×D26812yes—
205C2×C4#1176162814yes—
206C2×C4#1176162912yes—
207C22—2684no—
208C22—2684no—
209C22—2682no—
210C22—2682no—
211C22—2682no—
212C22C2×D4×D6711yescharacteristic
213C22C22×D26811yescharacteristic
214C22C2×#1176162812yes—
215C22C2×#1176162911yescharacteristic
216C22#1452409012yes—
217C4—24no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—2682no—
222C4—2682no—
223C4—1344no—
224C4#1452401814yes—
225C2—5362no—
226C2C2×#1452401811yescharacteristic
227C2#1686394611yescharacteristic
228C2#1686397211yescharacteristic
229C2#1686405512yes—
230C2#1686411912yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC4×C4⋊C4×C67C2
hypercenterC4×C4⋊C4D67
hypocenterC67#2545
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14524043C2semidirect
4. C2×C4×Dic134C2semidirect
5. #16863318C2semidirect
6. #16863302C2semidirect
8. #16863288C2semidirect
9. C4×C4⋊C4×C67C2semidirect
116. C67#2545semidirect
125. C4×C4⋊C4D67semidirect