Finite Group Databasebeta

← #22563372

Subgroups — #22563372

Order 8576.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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
2#16863300C212yesmaximal, complemented
3#16863291C212yesmaximal, complemented
4#16863290C212yesmaximal
5#16863284C212yesmaximal
6#16863279C212yesmaximal, complemented
7#16863277C212yesmaximal, complemented
8#16863259C211yesmaximal, characteristic
9#380×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11#14524022C418yescomplemented
12C2×C4⋊Dic67C2211yescharacteristic
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×C22⋊Dic67C2212yes—
17C2×C22⋊Dic67C2212yes—
18C2×C4×Dic67C2212yes—
19C2×C4×Dic67C2212yes—
20C67⋊#95C2211yescharacteristic
21C67⋊#95C2212yes—
22C67⋊#95C2211yescharacteristic
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C23×C4×C67C2211yescharacteristic
28C2×C4⋊C4×C67C2212yes—
29C2×C22⋊C4×C67C2211yescharacteristic
30C2×C22⋊C4×C67C2211yescharacteristic
31#95×C67C2212yes—
32C2×Dic134—42no—
33C2×Dic134C2×C414yes—
34C22×Dic67C2312yes—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C4⋊Dic67C2×C414yes—
39Dic67⋊C4—42no—
40Dic67⋊C4—42no—
41Dic67⋊C4C2×C418yes—
42C22⋊Dic67—42no—
43C22⋊Dic67C2×C418yes—
44C4×Dic67—42no—
45C4×Dic67—42no—
46C24×C67C2311yescharacteristic
47C22×C4×C67—42no—
48C22×C4×C67C2×C414yes—
49C22×C4×C67C2312yes—
50C22×C4×C67C2312yes—
51C22×C4×C67C2311yescharacteristic
52C22×C4×C67C2311yescharacteristic
53C4⋊C4×C67—42no—
54C22⋊C4×C67—41no—
55C22⋊C4×C67—41no—
56C23×C67—21no—
57C23×C67—22no—
58C23×C67—22no—
59C23×C67—21no—
60C23×C67C22×C412yes—
61C23×C67C2411yescharacteristic
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—44no—
71C2×C4×C67C22×C412yes—
72C2×C4×C67C22×C412yes—
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67C22×C414yes—
84C2×Dic67C22×C414yes—
85Dic134—44no—
86C22×C67—22no—
87C22×C67—22no—
88C22×C67—22no—
89C22×C67—42no—
90C22×C67—42no—
91C22×C67C23×C411yescharacteristic
92C22×C67H(5,𝔽2)12yes—
93C22×C67H(5,𝔽2)11yescharacteristic
94C22×C672−1+412yes—
95C22×C672−1+411yescharacteristic
96C4×C67—22no—
97C4×C67—22no—
98C4×C67—42no—
99C4×C67—42no—
100Dic67—24no—
101Dic67—24no—
102Dic67—42no—
103Dic67—42no—
104C2×C67—42no—
105C2×C67#51811yescharacteristic
106C2×C67#51911yescharacteristic
107C2×C67#52011yescharacteristic
108C2×C67#56212yes—
109C2×C67#56312yes—
110#2378—671nomaximal, Sylow
111C67#237811yescharacteristic, complemented, Sylow
112C2×C22⋊Q8—671no—
113#391—672no—
114#389—672no—
115#388—672no—
116#388—672no—
117#387—672no—
118#387—672no—
119#381—671no—
120#380D6711yescharacteristic
121C22×Q8—671no—
122C23×C4C2×D6711yescharacteristic
123C22⋊Q8—678no—
124C2×C4⋊C4—672no—
125C2×C4⋊C4—672no—
126C2×C4⋊C4—672no—
127C2×C4⋊C4—671no—
128C2×C4⋊C4C2×D6712yes—
129C2×C22⋊C4—672no—
130C2×C22⋊C4—672no—
131C2×C22⋊C4C2×D6711yescharacteristic
132C2×C22⋊C4C2×D6711yescharacteristic
133C2×C42—672no—
134C2×C42—672no—
135#95—671no—
136#95—672no—
137#95—671no—
138#95—672no—
139#95—672no—
140#95—672no—
141#95—672no—
142#95C2×D6712yes—
143C24C22×D6711yescharacteristic
144C2×Q8—2682no—
145C2×Q8—674no—
146C22×C4—42no—
147C22×C4—672no—
148C22×C4—672no—
149C22×C4—672no—
150C22×C4—672no—
151C22×C4C4×D6714yes—
152C22×C4C22×D6712yes—
153C22×C4C22×D6712yes—
154C22×C4C22×D6711yescharacteristic
155C22×C4C22×D6711yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—2682no—
158C4⋊C4—2682no—
159C4⋊C4—678no—
160C4⋊C4—674no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C22⋊C4—2682no—
164C22⋊C4—678no—
165C42—2682no—
166C42—2682no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23C2×C4×D6712yes—
172C23C23×D6711yescharacteristic
173Q8—2684no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—674no—
184C2×C4—674no—
185C2×C4—1342no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1342no—
189C2×C4—1342no—
190C2×C4—1342no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4C2×C4×D6712yes—
196C2×C4C2×C4×D6712yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D6711yescharacteristic
203C22#1452408812yes—
204C22#1452408811yescharacteristic
205C22#1452409012yes—
206C22#1452409111yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—2682no—
212C4—2682no—
213C4—1344no—
214C4—1344no—
215C2—42no—
216C2#1686386911yescharacteristic
217C2#1686387311yescharacteristic
218C2#1686387711yescharacteristic
219C2#1686414112yes—
220C2#1686414712yes—
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#380×C67C2
socleC23×C67C24
hypercenter#380D67
hypocenterC67#2378
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #16863300C2semidirect
3. #16863291C2semidirect
6. #16863279C2semidirect
7. #16863277C2semidirect
11. #14524022C4semidirect
111. C67#2378semidirect