Finite Group Databasebeta

← #22573129

Subgroups — #22573129

Order 8576.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#16864125C211yesmaximal, characteristic, complemented
2#16864061C211yesmaximal, characteristic
3#16863803C212yesmaximal, complemented
4#16863799C211yesmaximal, characteristic, complemented
5#16863695C212yesmaximal
6#16863689C212yesmaximal, complemented
7#16863677C211yesmaximal, characteristic
8#16863523C211yesmaximal, characteristic, complemented
9#16863499C212yesmaximal
10C4×#11761615C211yesmaximal, characteristic
11#475×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67—21no—
13C2×#11761615C2211yescharacteristic
14#14524043—21no—
15#14524043C2211yescharacteristic
16#14524042C2212yes—
17#14524034—22no—
18#14524034C2212yes—
19#14524021—22no—
20#14524019—21no—
21Q8×Dic67C2212yes—
22C4×Dic134—21no—
23C4×Dic134C2211yescharacteristic
24C4×D268C2211yescharacteristic
25#14524005C2212yes—
26#14523977C2212yes—
27#14523976C2212yes—
28#14523976C2212yes—
29#14523974C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523967C2211yescharacteristic
32#14523965C2212yes—
33#14523963C2211yescharacteristic
34#14523953C2212yes—
35C4×C67⋊C8C2211yescharacteristic
36C4⋊Q8×C67C2211yescharacteristic
37C4×Q8×C67C2211yescharacteristic
38#106×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#103×C67C2212yes—
41C2×Dic134—21no—
42C2×Dic134—22no—
43C2×Dic134D411yescharacteristic
44Q8×D67—44no—
45C2×D268C2311yescharacteristic
46C2×C4×D67—21no—
47C2×C4×D67—22no—
48#11761615Q814yescomplemented
49C2×C67⋊C8C2311yescharacteristic
50C2×C67⋊C8C2311yescharacteristic
51C4⋊Dic67—21no—
52C4⋊Dic67—22no—
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—44no—
90C4×D67—42no—
91Dic134—22no—
92Dic134—44no—
93Dic134—42no—
94C8×C67—22no—
95C67⋊C8—21no—
96C67⋊C8C2×Q812yescomplemented
97C22×C67C22×D411yescharacteristic
98C2×D67—22no—
99C2×D67—41no—
100C2×D67—41no—
101C4×C67—22no—
102C4×C67—21no—
103C4×C67—42no—
104C4×C67—41no—
105C4×C67AGL(1,ℤ/8)12yes—
106C4×C67C22×Q811yescharacteristic
107C4×C672−1+411yescharacteristic
108Dic67—22no—
109Dic67—22no—
110Dic67—41no—
111Dic67—42no—
112C2×C67D4×Q811yescharacteristic
113C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C67#57811yescharacteristic
115D67—42no—
116#3386—671nomaximal, Sylow
117C67#338611yescharacteristic, complemented, Sylow
118#557—671no—
119D4×Q8—671no—
120#501—672no—
121#500—671no—
122#478—672no—
123#477—672no—
124#475D6711yescharacteristic, complemented
125#474—671no—
126#446—671no—
127#440—672no—
128C4×SD16—671no—
129C22×Q8—1341no—
130C2×SD16—671no—
131C4⋊Q8—672no—
132C4⋊Q8—671no—
133C4⋊Q8—1341no—
134C4⋊Q8C2×D6711yescharacteristic
135#125—672no—
136#125—1342no—
137C22⋊Q8—1342no—
138C22⋊Q8—1341no—
139C4×Q8—671no—
140C4×Q8—672no—
141C4×Q8—1341no—
142C4×Q8C2×D6711yescharacteristic
143C4×D4—672no—
144C4×D4—671no—
145#107—672no—
146#107—672no—
147#107—672no—
148#106—671no—
149#106C2×D6712yes—
150C4⋊C8—672no—
151C4⋊C8C2×D6711yescharacteristic
152#103—671no—
153#103C2×D6712yes—
154#102—672no—
155#102—671no—
156#97—672no—
157C4×C8—671no—
158C2×Q8—21no—
159C2×Q8—2684no—
160C2×Q8—671no—
161C2×Q8—1342no—
162C2×Q8—1341no—
163C2×Q8C22×D6711yescharacteristic
164C2×D4—671no—
165C22×C4—1342no—
166C22×C4—1341no—
167SD16—674no—
168C2×C8—671no—
169C2×C8—671no—
170C2×C8C22×D6712yes—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—671no—
174C4⋊C4—672no—
175C4⋊C4—672no—
176C4⋊C4—671no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1341no—
182C4⋊C4C22×D6711yescharacteristic
183C4⋊C4C22×D6712yes—
184C22⋊C4—1342no—
185C22⋊C4—1341no—
186C42—21no—
187C42—672no—
188C42—1342no—
189C42C22×D6711yescharacteristic
190C23—1341no—
191Q8—21no—
192Q8—42no—
193Q8—2684no—
194Q8—2682no—
195Q8—1342no—
196Q8Q8×D6712yescomplemented
197D4—672no—
198D4—1341no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—2684no—
203C2×C4—2682no—
204C2×C4—672no—
205C2×C4—672no—
206C2×C4—1342no—
207C2×C4—1341no—
208C2×C4D4×D6711yescharacteristic
209C2×C4D4×D6711yescharacteristic
210C2×C4C23×D6711yescharacteristic
211C8—22no—
212C8—672no—
213C8—1341no—
214C22—2681no—
215C22—2681no—
216C22—1342no—
217C22C2×D4×D6711yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—2681no—
223C4—2682no—
224C4—1342no—
225C4—1342no—
226C4#1452407012yes—
227C4C2×Q8×D6711yescharacteristic
228C4#1452409011yescharacteristic
229C2—2682no—
230C2#1686405911yescharacteristic
231C2C2×#1452407011yescharacteristic
232C2#1686421211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#475×C67C2
socleC22×C67C22×D4
hypercenter#475D67
hypocenterC67#3386
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. #16864125C2semidirect
3. #16863803C2semidirect
4. #16863799C2semidirect
6. #16863689C2semidirect
8. #16863523C2semidirect
11. #475×C67C2semidirect
48. #11761615Q8semidirect
76. Q8×C67C2×Q8semidirect
96. C67⋊C8C2×Q8semidirect
117. C67#3386semidirect
124. #475D67semidirect
196. Q8Q8×D67semidirect