Finite Group Databasebeta

← #22565583

Subgroups — #22565583

Order 8576.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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×C4⋊C4×D67C211yesmaximal, characteristic
3#16863327C211yesmaximal, characteristic, complemented
4#16863317C211yesmaximal, characteristic, complemented
5#16863316C211yesmaximal, characteristic, complemented
6#16863295C212yesmaximal, complemented
7#16863293C212yesmaximal, complemented
8#16863270C212yesmaximal
9#16863267C212yesmaximal
10#16863243C211yesmaximal, characteristic
11#398×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×C4×D67C2211yescharacteristic
14C22×C4×D67C2211yescharacteristic
15C22×C4×D67C2211yescharacteristic
16#14524023—24no—
17C2×C4⋊Dic67C2212yes—
18C2×C4⋊Dic67C2211yescharacteristic
19C2×C4⋊Dic67C2212yes—
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2212yes—
24C4⋊C4×D67—24no—
25C2×(C2×D67)⋊C4C2212yes—
26C2×(C2×D67)⋊C4C2212yes—
27C2×C4×Dic67C2212yes—
28C67⋊#95C2211yescharacteristic
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2212yes—
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2211yescharacteristic
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C2×Dic134—22no—
39C2×Dic134—42no—
40C22×Dic67C2312yes—
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C2×C4×D67—22no—
46C2×C4×D67—22no—
47C2×C4×D67—22no—
48C2×C4×D67—22no—
49C2×C4×D67—42no—
50C2×C4×D67—42no—
51C2×C4×D67D414yes—
52C4⋊Dic67—22no—
53C4⋊Dic67—24no—
54C4⋊Dic67—42no—
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4—24no—
58Dic67⋊C4—42no—
59(C2×D67)⋊C4—24no—
60(C2×D67)⋊C4Q818yescomplemented
61C4×Dic67—42no—
62C22×C4×C67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2312yes—
66C22×C4×C67C2311yescharacteristic
67C4⋊C4×C67—22no—
68C4⋊C4×C67—22no—
69C4⋊C4×C67—24no—
70C23×C67C2411yescharacteristic
71C22×D67—21no—
72C22×D67—21no—
73C22×D67—22no—
74C22×D67C2×D412yes—
75C22×D67C2×Q814yes—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67C2×D412yes—
87C2×C4×C67C2×Q814yes—
88C2×C4×C67#4112yes—
89C2×Dic67—21no—
90C2×Dic67—22no—
91C2×Dic67—21no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—21no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67C2×D412yes—
100C2×Dic67C2×Q814yes—
101C2×Dic67#4112yes—
102C4×D67—28no—
103C4×D67—44no—
104C4×D67—44no—
105Dic134—44no—
106C22×C67C22×D411yescharacteristic
107C22×C67C22×Q812yes—
108C22×C67C2×#4111yescharacteristic
109C22×C67C2×#4111yescharacteristic
110C22×C67H(5,𝔽2)12yes—
111C2×D67—22no—
112C2×D67—22no—
113C2×D67—24no—
114C2×D67—24no—
115C2×D67C4⋊Q814yes—
116C4×C67—22no—
117C4×C67—22no—
118C4×C67—24no—
119C4×C67—42no—
120C4×C67—41no—
121Dic67—22no—
122Dic67—22no—
123Dic67—24no—
124Dic67—42no—
125Dic67—41no—
126C2×C67#52511yescharacteristic
127C2×C67C2×C4⋊Q811yescharacteristic
128C2×C67#53411yescharacteristic
129C2×C67#55412yes—
130C2×C67#55412yes—
131D67—24no—
132#2583—671nomaximal, Sylow
133C67#258311yescharacteristic, complemented, Sylow
134C2×C22⋊Q8—671no—
135C22×C4⋊C4—671no—
136#398—671no—
137#398D6711yescharacteristic, complemented
138#395—671no—
139#395—671no—
140#389—672no—
141#389—672no—
142#385—672no—
143#385—672no—
144C4×C22⋊C4—671no—
145C22×Q8—671no—
146C23×C4—671no—
147C23×C4—671no—
148C23×C4—671no—
149C22⋊Q8—1344no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—672no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—671no—
155C2×C4⋊C4—672no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—1344no—
158C2×C4⋊C4C2×D6711yescharacteristic
159C2×C4⋊C4C2×D6711yescharacteristic
160C2×C4⋊C4C2×D6712yes—
161C2×C22⋊C4—672no—
162C2×C22⋊C4—672no—
163C2×C42—672no—
164#95—671no—
165#95—672no—
166#95—672no—
167#95—672no—
168#95C2×D6711yescharacteristic
169#95C2×D6712yes—
170C24—671no—
171C2×Q8—2682no—
172C2×Q8—1342no—
173C22×C4—2682no—
174C22×C4—2682no—
175C22×C4—674no—
176C22×C4—672no—
177C22×C4—671no—
178C22×C4—671no—
179C22×C4—672no—
180C22×C4—671no—
181C22×C4—1342no—
182C22×C4—1342no—
183C22×C4—1342no—
184C22×C4—1342no—
185C22×C4C22×D6712yes—
186C22×C4C22×D6711yescharacteristic
187C22×C4C22×D6711yescharacteristic
188C22×C4C22×D6712yes—
189C22×C4C22×D6711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—2682no—
194C4⋊C4—2682no—
195C4⋊C4—1342no—
196C4⋊C4—1342no—
197C4⋊C4—1344no—
198C4⋊C4—1344no—
199C4⋊C4—1342no—
200C22⋊C4—678no—
201C22⋊C4—1344no—
202C42—2682no—
203C23—672no—
204C23—674no—
205C23—1341no—
206C23—1341no—
207C23—1342no—
208C23C23×D6711yescharacteristic
209Q8—2684no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—2684no—
221C2×C4—2684no—
222C2×C4—672no—
223C2×C4—674no—
224C2×C4—672no—
225C2×C4—1348no—
226C2×C4—1342no—
227C2×C4—1341no—
228C2×C4—1342no—
229C2×C4—1342no—
230C2×C4—1342no—
231C2×C4—1341no—
232C2×C4—1342no—
233C2×C4—1341no—
234C2×C4—1342no—
235C2×C4—1342no—
236C2×C4D4×D6712yes—
237C2×C4Q8×D6714yes—
238C2×C4#1176162712yes—
239C22—674no—
240C22—1344no—
241C22—1344no—
242C22—1342no—
243C22—1342no—
244C22C2×D4×D6711yescharacteristic
245C22C2×Q8×D6712yes—
246C22#41×D6711yescharacteristic
247C22C2×#1176162711yescharacteristic
248C22#1452408912yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—42no—
253C4—41no—
254C4—2681no—
255C4—2682no—
256C4—1344no—
257C4—1342no—
258C4—1342no—
259C2—1344no—
260C2#1686390611yescharacteristic
261C2C4⋊Q8×D6711yescharacteristic
262C2#1686395711yescharacteristic
263C2#1686410212yes—
264C2#1686410412yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#398×C67C2
hypercenter#398D67
hypocenterC67#2583
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #16863327C2semidirect
4. #16863317C2semidirect
5. #16863316C2semidirect
6. #16863295C2semidirect
7. #16863293C2semidirect
11. #398×C67C2semidirect
60. (C2×D67)⋊C4Q8semidirect
133. C67#2583semidirect
137. #398D67semidirect