Finite Group Databasebeta

← #22573068

Subgroups — #22573068

Order 8576.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic134C211yesmaximal, characteristic, complemented
2#16864061C211yesmaximal, characteristic
3#16863793C212yesmaximal, complemented
4#16863792C211yesmaximal, characteristic, complemented
5#16863687C211yesmaximal, characteristic
6#16863685C212yesmaximal
7#16863679C212yesmaximal, complemented
8#16863520C211yesmaximal, characteristic, complemented
9C4×#11761615C211yesmaximal, characteristic
10#16863481C212yesmaximal
11#475×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67—21no—
13C2×#11761615C2211yescharacteristic
14#14524043—21no—
15#14524043C2211yescharacteristic
16#14524042—22no—
17#14524042C2212yes—
18#14524042C2212yes—
19#14524021—22no—
20#14524019—21no—
21Q8×Dic67C2212yes—
22C4×Dic134—21no—
23C4×Dic134C2211yescharacteristic
24C4×D268C2211yescharacteristic
25#14524005C2212yes—
26#14523975C2212yes—
27#14523974C2212yes—
28#14523974C2212yes—
29#14523974C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523967C2211yescharacteristic
32#14523965C2212yes—
33#14523963C2211yescharacteristic
34C4×C67⋊C8C2211yescharacteristic
35C8×Dic67C2212yes—
36C4⋊Q8×C67C2211yescharacteristic
37C4×Q8×C67C2211yescharacteristic
38#106×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#103×C67C2212yes—
41C2×Dic134—22no—
42C2×Dic134—21no—
43C2×Dic134—22no—
44C2×Dic134D411yescharacteristic
45Q8×D67—44no—
46C2×D268C2311yescharacteristic
47C2×C4×D67—21no—
48C2×C4×D67—22no—
49#11761615Q814yescomplemented
50C2×C67⋊C8C2311yescharacteristic
51C2×C67⋊C8C2311yescharacteristic
52C4⋊Dic67—21no—
53C4⋊Dic67—22no—
54C4⋊Dic67D411yescharacteristic
55C4⋊Dic67C2312yes—
56C4⋊Dic67C2311yescharacteristic
57Dic67⋊C4—22no—
58Dic67⋊C4—22no—
59Dic67⋊C4D412yes—
60(C2×D67)⋊C4—21no—
61(C2×D67)⋊C4—22no—
62C4×Dic67—22no—
63C4×Dic67C2312yes—
64C2×Q8×C67—21no—
65C2×Q8×C67C2311yescharacteristic
66C2×C8×C67C2312yes—
67C4⋊C4×C67—21no—
68C4⋊C4×C67—21no—
69C4⋊C4×C67C2312yes—
70C4⋊C4×C67C2311yescharacteristic
71C42×C67—21no—
72C42×C67C2311yescharacteristic
73C22×D67—21no—
74Q8×C67—21no—
75Q8×C67—42no—
76Q8×C67C2×Q812yescomplemented
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2411yescharacteristic
83C2×Dic67—22no—
84C2×Dic67—21no—
85C2×Dic67C2×D412yes—
86C2×Dic67C2×D412yes—
87D268—21no—
88D268C2×Q812yes—
89C4×D67—42no—
90C4×D67—44no—
91Dic134—44no—
92Dic134—82no—
93Dic134—81no—
94Dic134SD1614yescomplemented
95C8×C67—22no—
96C67⋊C8—21no—
97C67⋊C8C2×Q812yescomplemented
98C22×C67C22×D411yescharacteristic
99C2×D67—22no—
100C2×D67—41no—
101C2×D67—41no—
102C4×C67—22no—
103C4×C67—21no—
104C4×C67—42no—
105C4×C67—41no—
106C4×C67C2×SD1612yes—
107C4×C67C22×Q811yescharacteristic
108C4×C672−1+411yescharacteristic
109Dic67—22no—
110Dic67—41no—
111Dic67—42no—
112Dic67C2×SD1614yes—
113C2×C67D4×Q811yescharacteristic
114C2×C67C22×SD1611yescharacteristic
115C2×C67#57711yescharacteristic
116D67—42no—
117Q8×SD16—671nomaximal, Sylow
118C67Q8×SD1611yescharacteristic, complemented, Sylow
119Q82—671no—
120D4×Q8—671no—
121#498—671no—
122#498—672no—
123#476—672no—
124#476—671no—
125#475—672no—
126#475D6711yescharacteristic, complemented
127C8×Q8—671no—
128C4×SD16—671no—
129C4×SD16—672no—
130C22×Q8—1341no—
131C2×SD16—671no—
132C4⋊Q8—672no—
133C4⋊Q8—672no—
134C4⋊Q8—671no—
135C4⋊Q8—1341no—
136C4⋊Q8—1342no—
137C4⋊Q8C2×D6711yescharacteristic
138C22⋊Q8—1342no—
139C22⋊Q8—1341no—
140C4×Q8—671no—
141C4×Q8—672no—
142C4×Q8—1341no—
143C4×Q8C2×D6711yescharacteristic
144C4×D4—672no—
145C4×D4—671no—
146#106—672no—
147#106—672no—
148#106—671no—
149#106—672no—
150#106C2×D6712yes—
151C4⋊C8—672no—
152C4⋊C8C2×D6711yescharacteristic
153#103—671no—
154#103C2×D6712yes—
155#102—672no—
156#102—671no—
157C4×C8—672no—
158C4×C8—671no—
159C2×Q8—21no—
160C2×Q8—2684no—
161C2×Q8—671no—
162C2×Q8—1342no—
163C2×Q8—1342no—
164C2×Q8—1341no—
165C2×Q8C22×D6711yescharacteristic
166C2×D4—671no—
167C22×C4—1342no—
168C22×C4—1341no—
169SD16—674no—
170C2×C8—671no—
171C2×C8—671no—
172C2×C8C22×D6712yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—672no—
176C4⋊C4—671no—
177C4⋊C4—672no—
178C4⋊C4—671no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1342no—
182C4⋊C4—1341no—
183C4⋊C4C22×D6711yescharacteristic
184C4⋊C4C22×D6712yes—
185C22⋊C4—1342no—
186C22⋊C4—1341no—
187C42—21no—
188C42—672no—
189C42—1342no—
190C42C22×D6711yescharacteristic
191C23—1341no—
192Q8—21no—
193Q8—42no—
194Q8—2684no—
195Q8—5361no—
196Q8—5362no—
197Q8—674no—
198Q8Q8×D6712yescomplemented
199D4—672no—
200D4—1341no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—2684no—
205C2×C4—2682no—
206C2×C4—672no—
207C2×C4—672no—
208C2×C4—1342no—
209C2×C4—1341no—
210C2×C4D4×D6711yescharacteristic
211C2×C4D4×D6711yescharacteristic
212C2×C4C23×D6711yescharacteristic
213C8—22no—
214C8—672no—
215C8—1341no—
216C22—2681no—
217C22—2681no—
218C22—1342no—
219C22C2×D4×D6711yescharacteristic
220C4—22no—
221C4—21no—
222C4—42no—
223C4—41no—
224C4—2681no—
225C4—2682no—
226C4—674no—
227C4—1342no—
228C4SD16×D6712yes—
229C4C2×Q8×D6711yescharacteristic
230C4#1452409011yescharacteristic
231C2—2682no—
232C2#1686405911yescharacteristic
233C2C2×SD16×D6711yescharacteristic
234C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#475×C67C2
socleC22×C67C22×D4
hypercenter#475D67
hypocenterC67Q8×SD16
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×Dic134C2semidirect
3. #16863793C2semidirect
4. #16863792C2semidirect
7. #16863679C2semidirect
8. #16863520C2semidirect
11. #475×C67C2semidirect
49. #11761615Q8semidirect
76. Q8×C67C2×Q8semidirect
94. Dic134SD16semidirect
97. C67⋊C8C2×Q8semidirect
118. C67Q8×SD16semidirect
126. #475D67semidirect
198. Q8Q8×D67semidirect