Finite Group Databasebeta

← #22573118

Subgroups — #22573118

Order 8576.

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

normal subgroups only

This group has 340 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
1Q8×Dic134C211yesmaximal, characteristic, complemented
2#16864107C211yesmaximal, characteristic
3#16863804C212yesmaximal, complemented
4#16863799C211yesmaximal, characteristic, complemented
5#16863701C212yesmaximal, complemented
6#16863687C211yesmaximal, characteristic
7#16863675C212yesmaximal
8#16863523C211yesmaximal, characteristic, complemented
9#16863499C212yesmaximal
10C4×#11761615C211yesmaximal, characteristic
11#477×C67C211yesmaximal, characteristic, complemented
12C2×#11761615C2211yescharacteristic
13#14524043—21no—
14#14524043C2211yescharacteristic
15#14524042—22no—
16#14524042C2212yes—
17#14524034C2212yes—
18#14524023—21no—
19#14524019—22no—
20Q8×Dic67C2212yes—
21C4×Dic134—21no—
22C4×Dic134C2211yescharacteristic
23C4×D268C2211yescharacteristic
24#14524005C2212yes—
25C4⋊C4×D67—21no—
26#14523976C2212yes—
27#14523976C2212yes—
28#14523975C2212yes—
29#14523974C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523967C2211yescharacteristic
32#14523965C2212yes—
33#14523963C2211yescharacteristic
34#14523953C2212yes—
35C4×C67⋊C8C2211yescharacteristic
36C4⋊Q8×C67C2211yescharacteristic
37C4×Q8×C67C2211yescharacteristic
38#107×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#103×C67C2212yes—
41C2×Dic134—21no—
42C2×Dic134—22no—
43C2×Dic134D411yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C4×D67—21no—
46C2×C4×D67—22no—
47#11761615Q814yescomplemented
48C2×C67⋊C8C2311yescharacteristic
49C2×C67⋊C8C2311yescharacteristic
50C4⋊Dic67—21no—
51C4⋊Dic67—22no—
52C4⋊Dic67—22no—
53C4⋊Dic67D411yescharacteristic
54C4⋊Dic67C2312yes—
55C4⋊Dic67C2311yescharacteristic
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58Dic67⋊C4D412yes—
59(C2×D67)⋊C4—21no—
60(C2×D67)⋊C4—22no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C2×Q8×C67—21no—
64C2×Q8×C67C2311yescharacteristic
65C2×C8×C67C2312yes—
66C4⋊C4×C67—21no—
67C4⋊C4×C67—21no—
68C4⋊C4×C67C2312yes—
69C4⋊C4×C67C2311yescharacteristic
70C42×C67—21no—
71C42×C67C2311yescharacteristic
72C22×D67—21no—
73Q8×C67—21no—
74Q8×C67—81no—
75Q8×C67C2×Q812yes—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67C2×D411yescharacteristic
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2411yescharacteristic
82C2×Dic67—22no—
83C2×Dic67—21no—
84C2×Dic67C2×D412yes—
85C2×Dic67C2×D412yes—
86D268—21no—
87D268C2×Q812yes—
88C4×D67—42no—
89C4×D67—82no—
90Dic134—22no—
91Dic134—82no—
92Dic134—81no—
93C8×C67—22no—
94C67⋊C8—21no—
95C67⋊C8C2×Q812yes—
96C22×C67C22×D411yescharacteristic
97C2×D67—22no—
98C2×D67—41no—
99C2×D67—41no—
100C4×C67—22no—
101C4×C67—21no—
102C4×C67—41no—
103C4×C67—42no—
104C4×C67#13712yes—
105C4×C67C22×Q811yescharacteristic
106C4×C672−1+411yescharacteristic
107Dic67—22no—
108Dic67—22no—
109Dic67—41no—
110Dic67—42no—
111C2×C67D4×Q811yescharacteristic
112C2×C67C2×#13711yescharacteristic
113C2×C67#57611yescharacteristic
114D67—42no—
115#3385—671nomaximal, Sylow
116C67#338511yescharacteristic, complemented, Sylow
117Q82—671no—
118#554—671no—
119#501—672no—
120#500—671no—
121#479—672no—
122#477D6711yescharacteristic, complemented
123#476—671no—
124#474—672no—
125#446—671no—
126#440—672no—
127C4×SD16—671no—
128C2×SD16—671no—
129C4⋊Q8—672no—
130C4⋊Q8—671no—
131C4⋊Q8—1341no—
132C4⋊Q8—1342no—
133C4⋊Q8C2×D6711yescharacteristic
134#125—672no—
135C22⋊Q8—1341no—
136C22⋊Q8—1342no—
137C4×Q8—671no—
138C4×Q8—672no—
139C4×Q8—1341no—
140C4×Q8C2×D6711yescharacteristic
141C4×D4—672no—
142C4×D4—671no—
143C2×C4⋊C4—1341no—
144#107—672no—
145#107—672no—
146#107C2×D6712yes—
147#106—672no—
148#106—671no—
149C4⋊C8—672no—
150C4⋊C8C2×D6711yescharacteristic
151#103—671no—
152#103C2×D6712yes—
153#102—672no—
154#102—671no—
155#97—672no—
156C4×C8—671no—
157C2×Q8—21no—
158C2×Q8—671no—
159C2×Q8—1342no—
160C2×Q8—1341no—
161C2×Q8C22×D6711yescharacteristic
162C2×D4—671no—
163C22×C4—1342no—
164C22×C4—1341no—
165SD16—674no—
166C2×C8—671no—
167C2×C8—671no—
168C2×C8C22×D6712yes—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—671no—
172C4⋊C4—672no—
173C4⋊C4—672no—
174C4⋊C4—671no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1341no—
180C4⋊C4C22×D6711yescharacteristic
181C4⋊C4C22×D6712yes—
182C22⋊C4—1342no—
183C22⋊C4—1341no—
184C42—21no—
185C42—672no—
186C42—1342no—
187C42C22×D6711yescharacteristic
188C23—1341no—
189Q8—21no—
190Q8—81no—
191Q8—5361no—
192Q8—5362no—
193Q8—1342no—
194Q8Q8×D6712yes—
195D4—672no—
196D4—1341no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—2682no—
201C2×C4—5362no—
202C2×C4—672no—
203C2×C4—672no—
204C2×C4—1342no—
205C2×C4—1341no—
206C2×C4D4×D6711yescharacteristic
207C2×C4D4×D6711yescharacteristic
208C2×C4C23×D6711yescharacteristic
209C8—22no—
210C8—672no—
211C8—1341no—
212C22—2681no—
213C22—2681no—
214C22—1342no—
215C22C2×D4×D6711yescharacteristic
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—2681no—
221C4—2682no—
222C4—1342no—
223C4—1342no—
224C4#1452407512yes—
225C4C2×Q8×D6711yescharacteristic
226C4#1452409011yescharacteristic
227C2—2682no—
228C2#1686405911yescharacteristic
229C2C2×#1452407511yescharacteristic
230C2#1686420011yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#477×C67C2
socleC22×C67C22×D4
hypercenter#477D67
hypocenterC67#3385
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. Q8×Dic134C2semidirect
3. #16863804C2semidirect
4. #16863799C2semidirect
5. #16863701C2semidirect
8. #16863523C2semidirect
11. #477×C67C2semidirect
47. #11761615Q8semidirect
116. C67#3385semidirect
122. #477D67semidirect