Finite Group Databasebeta

← #22565841

Subgroups — #22565841

Order 8576.

470 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14524037C211yesmaximal, characteristic
2#16863334C211yesmaximal, characteristic, complemented
3#16863331C211yesmaximal, characteristic
4#16863315C211yesmaximal, characteristic, complemented
5#16863307C211yesmaximal, characteristic
6#16863291C212yesmaximal, complemented
7#16863287C212yesmaximal
8#16863286C212yesmaximal
9#16863280C212yesmaximal, complemented
10#16863242C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C67C211yesmaximal, characteristic, complemented
12#14524037—24no—
13C2×Dic67⋊C4C2211yescharacteristic
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2211yescharacteristic
17C2×Dic67⋊C4C2211yescharacteristic
18C2×(C2×D67)⋊C4C2211yescharacteristic
19C2×(C2×D67)⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2211yescharacteristic
22C2×(C2×D67)⋊C4C2211yescharacteristic
23C2×C4×Dic67C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C22×Q8×C67C2211yescharacteristic
32C4×Q8×C67—24no—
33C2×C4⋊C4×C67C2212yes—
34C2×C4⋊C4×C67C2211yescharacteristic
35C2×C42×C67C2212yes—
36C2×C42×C67C2211yescharacteristic
37C23×D67C2311yescharacteristic
38C22×Dic67C2312yes—
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2312yes—
43Dic67⋊C4—22no—
44Dic67⋊C4—22no—
45Dic67⋊C4—22no—
46Dic67⋊C4—24no—
47Dic67⋊C4—24no—
48(C2×D67)⋊C4—22no—
49(C2×D67)⋊C4—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—24no—
52(C2×D67)⋊C4—24no—
53C4×Dic67—24no—
54C2×Q8×C67—22no—
55C2×Q8×C67—24no—
56C22×C4×C67C2312yes—
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2312yes—
61C4⋊C4×C67—24no—
62C4⋊C4×C67—22no—
63C42×C67—24no—
64C42×C67D414yes—
65C23×C67C2411yescharacteristic
66C22×D67—21no—
67C22×D67—22no—
68C22×D67—22no—
69C22×D67—21no—
70C22×D67—21no—
71Q8×C67—28no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67C2×D412yes—
82C2×C4×C67C2×D412yes—
83C2×C4×C67C2×D412yes—
84C2×C4×C67#4114yes—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—21no—
88C2×Dic67—21no—
89C2×Dic67—21no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—21no—
95C2×Dic67—21no—
96C2×Dic67—22no—
97C2×Dic67#4114yes—
98C22×C67C22×D411yescharacteristic
99C22×C67C2×#4112yes—
100C22×C67C2×#4112yes—
101C22×C672−1+411yescharacteristic
102C22×C672−1+411yescharacteristic
103C2×D67—41no—
104C2×D67—41no—
105C2×D67—41no—
106C2×D67—42no—
107C2×D67—42no—
108C4×C67—22no—
109C4×C67—24no—
110C4×C67—22no—
111C4×C67—22no—
112C4×C67—42no—
113Dic67—24no—
114Dic67—41no—
115Dic67—41no—
116Dic67—41no—
117Dic67—42no—
118C2×C67#52511yescharacteristic
119C2×C67#53211yescharacteristic
120C2×C67#53611yescharacteristic
121C2×C67#55212yes—
122C2×C67#55612yes—
123D67—81no—
124#2604—671nomaximal, Sylow
125C67#260411yescharacteristic, complemented, Sylow
126C2×#126—671no—
127C2×C4×Q8D6711yescharacteristic, complemented
128#400—671no—
129#399—671no—
130#395—671no—
131#393—671no—
132#389—672no—
133#388—672no—
134#388—672no—
135#387—672no—
136#376—671no—
137C22×Q8C2×D6711yescharacteristic
138#126—1344no—
139C4×Q8—24no—
140C2×C4⋊C4—671no—
141C2×C4⋊C4—671no—
142C2×C4⋊C4—671no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4—672no—
145C2×C4⋊C4C2×D6712yes—
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C22⋊C4—671no—
148C2×C22⋊C4—671no—
149C2×C22⋊C4—671no—
150C2×C22⋊C4—672no—
151C2×C22⋊C4—672no—
152C2×C42—672no—
153C2×C42C2×D6712yes—
154C2×C42C2×D6711yescharacteristic
155#95—672no—
156#95—671no—
157#95—671no—
158#95—672no—
159#95—672no—
160#95—672no—
161#95—672no—
162C24—671no—
163C2×Q8—22no—
164C2×Q8—24no—
165C22×C4—672no—
166C22×C4—672no—
167C22×C4—671no—
168C22×C4—671no—
169C22×C4—671no—
170C22×C4C22×D6712yes—
171C22×C4C22×D6711yescharacteristic
172C22×C4C22×D6711yescharacteristic
173C22×C4C22×D6711yescharacteristic
174C22×C4C22×D6712yes—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—1344no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1344no—
182C22⋊C4—1344no—
183C22⋊C4—1342no—
184C22⋊C4—1342no—
185C22⋊C4—1342no—
186C22⋊C4—1344no—
187C42—24no—
188C42—1344no—
189C42C22⋊D6714yes—
190C23—1341no—
191C23—1342no—
192C23—1342no—
193C23—1341no—
194C23—1341no—
195C23C23×D6711yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—674no—
207C2×C4—1341no—
208C2×C4—1341no—
209C2×C4—1341no—
210C2×C4—1342no—
211C2×C4—1342no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1342no—
215C2×C4—1341no—
216C2×C4—1341no—
217C2×C4—1342no—
218C2×C4—1342no—
219C2×C4C2×C22⋊D6712yes—
220C2×C4C2×C22⋊D6712yes—
221C2×C4C2×C22⋊D6712yes—
222C2×C4#1176162814yes—
223C22—2682no—
224C22—2682no—
225C22—2681no—
226C22—2681no—
227C22—2681no—
228C22C22×C22⋊D6711yescharacteristic
229C22#41×D6712yes—
230C22C2×#1176162812yes—
231C22#1452409111yescharacteristic
232C22#1452409111yescharacteristic
233C4—22no—
234C4—24no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—2682no—
239C4—2681no—
240C4—2681no—
241C4—2681no—
242C4—1344no—
243C2—5361no—
244C2#1686390711yescharacteristic
245C2#1686394211yescharacteristic
246C2#1686397011yescharacteristic
247C2#1686408712yes—
248C2#1686411812yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C4×Q8×C67C2
hypercenterC2×C4×Q8D67
hypocenterC67#2604
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #16863334C2semidirect
4. #16863315C2semidirect
6. #16863291C2semidirect
9. #16863280C2semidirect
10. #16863242C2semidirect
11. C2×C4×Q8×C67C2semidirect
125. C67#2604semidirect
127. C2×C4×Q8D67semidirect