Finite Group Databasebeta

← #22566028

Subgroups — #22566028

Order 8576.

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

normal subgroups only

This group has 508 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×#14524020C211yesmaximal, characteristic
2C22×Dic67⋊C4C211yesmaximal, characteristic
3#16863337C212yesmaximal, complemented
4#16863330C212yesmaximal, complemented
5#16863327C212yesmaximal, complemented
6#16863316C212yesmaximal, complemented
7#16863269C211yesmaximal, characteristic
8#16863257C211yesmaximal, characteristic
9#16863257C211yesmaximal, characteristic
10#16863254C211yesmaximal, characteristic
11#385×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13C23×Dic67C2211yescharacteristic
14C23×Dic67C2211yescharacteristic
15#14524020—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4—24no—
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2212yes—
23C2×Dic67⋊C4C2212yes—
24C2×C22⋊Dic67C2211yescharacteristic
25C2×C22⋊Dic67C2211yescharacteristic
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C23×C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2212yes—
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C22⋊C4×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—42no—
39C22×Dic67—22no—
40C22×Dic67—22no—
41C22×Dic67—42no—
42C22×Dic67D414yes—
43C22×Dic67Q814yescomplemented
44C22×Dic67C2311yescharacteristic
45C22×Dic67C2312yes—
46C22×Dic67C2311yescharacteristic
47C22×Dic67C2311yescharacteristic
48C22×Dic67C2311yescharacteristic
49C22×Dic67C2312yes—
50C4⋊Dic67—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—24no—
53Dic67⋊C4—22no—
54Dic67⋊C4—22no—
55Dic67⋊C4—42no—
56Dic67⋊C4—42no—
57C22⋊Dic67—22no—
58C22⋊Dic67—22no—
59C24×C67C2311yescharacteristic
60C22×C4×C67—22no—
61C22×C4×C67—42no—
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2312yes—
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2312yes—
66C4⋊C4×C67—42no—
67C22⋊C4×C67—22no—
68C22⋊C4×C67—22no—
69C23×C67—22no—
70C23×C67—22no—
71C23×C67C2×D412yes—
72C23×C67C2×Q812yes—
73C23×C67#4112yes—
74C23×C67C2411yescharacteristic
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—44no—
86C2×Dic67—28no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—44no—
99C2×Dic67C2×D412yes—
100C2×Dic67C2×D412yes—
101C2×Dic67C2×Q814yes—
102Dic134—44no—
103C22×C67—22no—
104C22×C67—22no—
105C22×C67—24no—
106C22×C67—24no—
107C22×C67C22⋊Q814yes—
108C22×C67C22×D411yescharacteristic
109C22×C67C22×Q811yescharacteristic
110C22×C67C2×#4111yescharacteristic
111C22×C67H(5,𝔽2)12yes—
112C22×C672−1+412yes—
113C4×C67—41no—
114C4×C67—42no—
115C4×C67—41no—
116C4×C67—42no—
117Dic67—24no—
118Dic67—22no—
119Dic67—22no—
120Dic67—41no—
121Dic67—41no—
122Dic67—42no—
123C2×C67—24no—
124C2×C67C2×C22⋊Q811yescharacteristic
125C2×C67#53411yescharacteristic
126C2×C67#53611yescharacteristic
127C2×C67#53912yes—
128C2×C67#54412yes—
129#2627—671nomaximal, Sylow
130C67#262711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—671no—
132C22×C4⋊C4—671no—
133#400—672no—
134#398—672no—
135#398—672no—
136#395—672no—
137#385—671no—
138#385D6711yescharacteristic
139#381—671no—
140#381—671no—
141#380—671no—
142C22×Q8—671no—
143C23×C4—671no—
144C23×C4—671no—
145C23×C4C2×D6711yescharacteristic
146C22⋊Q8—1344no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—672no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—671no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—1344no—
155C2×C4⋊C4C2×D6712yes—
156C2×C22⋊C4—671no—
157C2×C22⋊C4—671no—
158C2×C22⋊C4C2×D6711yescharacteristic
159C2×C22⋊C4C2×D6711yescharacteristic
160#95—672no—
161#95—672no—
162#95—672no—
163#95—672no—
164#95—672no—
165#95—672no—
166#95C2×D6712yes—
167C24C22×D6711yescharacteristic
168C2×Q8—2682no—
169C2×Q8—1342no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—2682no—
173C22×C4—674no—
174C22×C4—674no—
175C22×C4—672no—
176C22×C4—671no—
177C22×C4—671no—
178C22×C4—671no—
179C22×C4—672no—
180C22×C4—671no—
181C22×C4—1342no—
182C22×C4—1342no—
183C22×C4C22×D6712yes—
184C22×C4C22×D6711yescharacteristic
185C22×C4C22×D6711yescharacteristic
186C22×C4C22×D6712yes—
187C4⋊C4—42no—
188C4⋊C4—2682no—
189C4⋊C4—2682no—
190C4⋊C4—1344no—
191C4⋊C4—1344no—
192C4⋊C4—1342no—
193C4⋊C4—1342no—
194C4⋊C4—1342no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—1342no—
198C22⋊C4—1342no—
199C23—22no—
200C23—22no—
201C23D4×D6712yes—
202C23Q8×D6712yes—
203C23#1176162812yes—
204C23C23×D6711yescharacteristic
205Q8—2684no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—2684no—
218C2×C4—672no—
219C2×C4—672no—
220C2×C4—674no—
221C2×C4—1341no—
222C2×C4—1341no—
223C2×C4—1342no—
224C2×C4—1342no—
225C2×C4—1342no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—1348no—
229C2×C4—1342no—
230C2×C4—1342no—
231C2×C4—1342no—
232C2×C4—1342no—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22#1452402114yes—
238C22C2×D4×D6711yescharacteristic
239C22C2×Q8×D6711yescharacteristic
240C22C2×#1176162811yescharacteristic
241C22#1452408812yes—
242C22#1452409012yes—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—2681no—
248C4—2681no—
249C4—2682no—
250C4—1344no—
251C4—1342no—
252C4—1342no—
253C2—24no—
254C2C2×#1452402111yescharacteristic
255C2#1686395611yescharacteristic
256C2#1686396811yescharacteristic
257C2#1686398812yes—
258C2#1686402512yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#385×C67C2
hypercenter#385D67
hypocenterC67#2627
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #16863337C2semidirect
4. #16863330C2semidirect
5. #16863327C2semidirect
6. #16863316C2semidirect
43. C22×Dic67Q8semidirect
130. C67#2627semidirect