Finite Group Databasebeta

← #22572953

Subgroups — #22572953

Order 8576.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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#16864120C211yesmaximal, characteristic
3#16863757C212yesmaximal
4#16863746C211yesmaximal, characteristic, complemented
5#16863687C211yesmaximal, characteristic
6#16863679C212yesmaximal, complemented
7#16863620C212yesmaximal
8#16863523C211yesmaximal, characteristic, complemented
9#16863503C212yesmaximal, complemented
10C4×#11761615C211yesmaximal, characteristic
11#462×C67C211yesmaximal, characteristic, complemented
12C2×#11761615C2211yescharacteristic
13#14524043—21no—
14#14524043C2211yescharacteristic
15#14524042—22no—
16#14524042C2212yes—
17#14524038—22no—
18#14524032C2212yes—
19#14524023—21no—
20Q8×Dic67C2212yes—
21Q8×Dic67C2212yes—
22C4×Dic134—21no—
23C4×Dic134C2211yescharacteristic
24C4×D268C2211yescharacteristic
25#14524001—21no—
26#14523975C2212yes—
27#14523974C2211yescharacteristic
28Dic67⋊C8C2212yes—
29#14523967C2212yes—
30#14523967C2212yes—
31#14523967C2211yescharacteristic
32#14523965C2212yes—
33#14523963C2211yescharacteristic
34#14523953C2212yes—
35C4×C67⋊C8C2211yescharacteristic
36C2×Q16×C67C2212yes—
37C4⋊Q8×C67C2211yescharacteristic
38C4×Q8×C67C2211yescharacteristic
39C4⋊C8×C67C2211yescharacteristic
40#103×C67C2212yes—
41C2×Dic134—21no—
42C2×Dic134—22no—
43C2×Dic134D411yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C4×D67—21no—
46#11761615—22no—
47C2×C67⋊C8C2311yescharacteristic
48C2×C67⋊C8C2311yescharacteristic
49C4⋊Dic67—21no—
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52C4⋊Dic67D411yescharacteristic
53C4⋊Dic67C2312yes—
54C4⋊Dic67C2311yescharacteristic
55Dic67⋊C4—22no—
56Dic67⋊C4D412yes—
57(C2×D67)⋊C4—21no—
58(C2×D67)⋊C4—22no—
59(C2×D67)⋊C4—22no—
60C4×Dic67—22no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C2×Q8×C67C2312yes—
64C2×Q8×C67C2311yescharacteristic
65Q16×C67—24no—
66C2×C8×C67C2312yes—
67C4⋊C4×C67—21no—
68C4⋊C4×C67—21no—
69C4⋊C4×C67—21no—
70C4⋊C4×C67C2311yescharacteristic
71C42×C67—21no—
72C42×C67C2311yescharacteristic
73C22×D67—21no—
74Q8×C67—21no—
75Q8×C67—22no—
76Q8×C67—22no—
77Q8×C67#4112yes—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67C2411yescharacteristic
84C2×Dic67—21no—
85C2×Dic67—22no—
86C2×Dic67C2×D412yes—
87C2×Dic67C2×D412yes—
88D268—21no—
89D268—21no—
90C4×D67—42no—
91Dic134—22no—
92Dic134—82no—
93Dic134—81no—
94C8×C67—22no—
95C67⋊C8—21no—
96C67⋊C8—21no—
97C22×C67C22×D411yescharacteristic
98C2×D67—41no—
99C2×D67—41no—
100C2×D67—41no—
101C4×C67—22no—
102C4×C67—21no—
103C4×C67—42no—
104C4×C67—41no—
105C4×C67#13712yes—
106C4×C67C2×#4111yescharacteristic
107C4×C672−1+411yescharacteristic
108Dic67—22no—
109Dic67—22no—
110Dic67—41no—
111Dic67—42no—
112C2×C67#54811yescharacteristic
113C2×C67C2×#13711yescharacteristic
114C2×C67#57711yescharacteristic
115D67—81no—
116#3369—671nomaximal, Sylow
117C67#336911yescharacteristic, complemented, Sylow
118Q82—671no—
119#556—671no—
120#489—672no—
121#487—671no—
122#476—671no—
123#475—672no—
124#464—672no—
125#462D6711yescharacteristic, complemented
126#446—671no—
127#441—672no—
128C4×SD16—671no—
129C2×Q16C2×D6712yes—
130C2×SD16—671no—
131C4⋊Q8—672no—
132C4⋊Q8—671no—
133C4⋊Q8—1341no—
134C4⋊Q8—1342no—
135C4⋊Q8C2×D6711yescharacteristic
136#126—1342no—
137#124—672no—
138C22⋊Q8—1341no—
139C4×Q8—671no—
140C4×Q8—672no—
141C4×Q8—672no—
142C4×Q8—1341no—
143C4×Q8C2×D6711yescharacteristic
144C4×D4—671no—
145#117—1341no—
146#106—672no—
147#106—671no—
148C4⋊C8—672no—
149C4⋊C8C2×D6711yescharacteristic
150#103—671no—
151#103—672no—
152#103—672no—
153#103C2×D6712yes—
154#102—672no—
155#102—671no—
156#97—672no—
157C4×C8—671no—
158C2×Q8—671no—
159C2×Q8—1342no—
160C2×Q8—1341no—
161C2×Q8C22×D6712yes—
162C2×Q8C22×D6711yescharacteristic
163C2×D4—671no—
164C22×C4—1341no—
165Q16—24no—
166SD16—1342no—
167C2×C8—671no—
168C2×C8—671no—
169C2×C8C22×D6712yes—
170C4⋊C4—21no—
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—1341no—
181C4⋊C4C22×D6711yescharacteristic
182C22⋊C4—1342no—
183C22⋊C4—1342no—
184C22⋊C4—1341no—
185C42—21no—
186C42—672no—
187C42—1342no—
188C42—1342no—
189C42C22×D6711yescharacteristic
190C23—1341no—
191Q8—21no—
192Q8—22no—
193Q8—22no—
194Q8—5361no—
195Q8—5362no—
196Q8—1342no—
197Q8#1176162712yes—
198D4—1341no—
199D4—1341no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—2682no—
204C2×C4—672no—
205C2×C4—672no—
206C2×C4—1341no—
207C2×C4—1342no—
208C2×C4D4×D6711yescharacteristic
209C2×C4D4×D6711yescharacteristic
210C2×C4C23×D6711yescharacteristic
211C8—22no—
212C8—1341no—
213C8—1341no—
214C22—2681no—
215C22—2681no—
216C22—2681no—
217C22C2×D4×D6711yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—2681no—
223C4—2682no—
224C4—1342no—
225C4—1342no—
226C4#1452407512yes—
227C4C2×#1176162711yescharacteristic
228C4#1452409011yescharacteristic
229C2—5361no—
230C2#1686405211yescharacteristic
231C2C2×#1452407511yescharacteristic
232C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#462×C67C2
socleC22×C67C22×D4
hypercenter#462D67
hypocenterC67#3369
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. Q8×Dic134C2semidirect
4. #16863746C2semidirect
6. #16863679C2semidirect
8. #16863523C2semidirect
9. #16863503C2semidirect
11. #462×C67C2semidirect
117. C67#3369semidirect
125. #462D67semidirect