Finite Group Databasebeta

← #22560593

Subgroups — #22560593

Order 8576.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#16864163C211yesmaximal, characteristic, complemented
2#16863413C212yesmaximal, complemented
3#16863370C211yesmaximal, characteristic, complemented
4#16863156C212yesmaximal, complemented
5#532×C67C211yesmaximal, characteristic
6#14524050—24no—
7C2×#11761612—22no—
8C2×#11761612C2212yes—
9#14524002C2212yes—
10#14523970—24no—
11#14523958—22no—
12C2×#41×C67C2211yescharacteristic
13C2×#41×C67C2211yescharacteristic
14C4⋊Q8×C67C411yescharacteristic, complemented
15#127×C67C411yescharacteristic, complemented
16#124×C67C411yescharacteristic, complemented
17#124×C67C411yescharacteristic, complemented
18#121×C67—22no—
19C4×D4×C67—22no—
20C2×C42×C67C2211yescharacteristic
21#11761612—24no—
22#11761612—22no—
23#11761612D414yescomplemented
24C2×C67⋊C8—24no—
25C2×C67⋊C8—22no—
26C4⋊Dic67—22no—
27C22⋊Dic67—22no—
28C4×Dic67—22no—
29#41×C67—22no—
30#41×C67—22no—
31#41×C67—22no—
32#41×C67—22no—
33C2×Q8×C67C2×C411yescharacteristic
34C2×Q8×C67C2×C411yescharacteristic
35C2×D4×C67—22no—
36C2×D4×C67—21no—
37C2×D4×C67C2×C411yescharacteristic
38C2×D4×C67C2×C411yescharacteristic
39C2×D4×C67D412yes—
40C22×C4×C67—21no—
41C22×C4×C67—21no—
42C22×C4×C67D412yes—
43C22×C4×C67C2311yescharacteristic
44C4⋊C4×C67—22no—
45C22⋊C4×C67—22no—
46C22⋊C4×C67—22no—
47C42×C67—21no—
48C42×C67C2×C411yescharacteristic
49C42×C67C2×C411yescharacteristic
50C23×C67—21no—
51C23×C67C22⋊C412yes—
52C23×C67C2×D411yescharacteristic
53Q8×C67—22no—
54Q8×C67—22no—
55D4×C67—24no—
56D4×C67—22no—
57D4×C67—22no—
58D4×C67—44no—
59D4×C67—42no—
60C2×C4×C67—21no—
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—21no—
64C2×C4×C67—21no—
65C2×C4×C67—42no—
66C2×C4×C67—41no—
67C2×C4×C67—41no—
68C2×C4×C67C22⋊C412yes—
69C2×C4×C67C22×C411yescharacteristic
70C2×C4×C67C2×D412yes—
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67#4112yes—
73C2×Dic67—42no—
74C67⋊C8—24no—
75C22×C67—22no—
76C22×C67—41no—
77C22×C67—42no—
78C22×C67—42no—
79C22×C67—41no—
80C22×C67C2×C22⋊C411yescharacteristic
81C22×C67#12112yes—
82C4×C67—22no—
83C4×C67—22no—
84C4×C67—41no—
85C4×C67—41no—
86C4×C67C4×D412yes—
87C4×C67#12011yescharacteristic
88C4×C67#12311yescharacteristic
89Dic67—82no—
90C2×C67—22no—
91C2×C67—21no—
92C2×C67—42no—
93C2×C67—81no—
94C2×C67#38611yescharacteristic
95#1884—671nomaximal, Sylow
96C67#188411yescharacteristic, complemented, Sylow
97#568—671no—
98#532D6711yescharacteristic
99#421—672no—
100#410—671no—
101#339—672no—
102C2×#41C2×D6711yescharacteristic
103C2×#41C2×D6711yescharacteristic
104#131—1344no—
105C2×M16—672no—
106C2×M16—1342no—
107C4⋊Q8Dic6711yescharacteristic, complemented
108#127Dic6711yescharacteristic, complemented
109#124Dic6711yescharacteristic, complemented
110#124Dic6711yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—672no—
114C2×C42C2×D6711yescharacteristic
115#104—1344no—
116#99—1342no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic6711yescharacteristic
122C2×Q8C2×Dic6711yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4C2×Dic6711yescharacteristic
126C2×D4C2×Dic6711yescharacteristic
127C2×D4C22⋊D6712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D6712yes—
131C22×C4C22×D6711yescharacteristic
132M16—674no—
133M16—1344no—
134M16—1342no—
135C2×C8—1344no—
136C2×C8—1342no—
137C4⋊C4—22no—
138C4⋊C4—1342no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—1342no—
142C42—21no—
143C42—1342no—
144C42C2×Dic6711yescharacteristic
145C42C2×Dic6711yescharacteristic
146C23—21no—
147C23C22⋊Dic6712yes—
148C23C2×C22⋊D6711yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—2682no—
165C2×C4C22⋊Dic6712yes—
166C2×C4C22×Dic6711yescharacteristic
167C2×C4D4×D6712yes—
168C2×C4C2×C22⋊D6711yescharacteristic
169C2×C4#1176162712yes—
170C8—1344no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic6711yescharacteristic
177C22#1452401512yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—5362no—
183C4D4×Dic6712yes—
184C4#1452401111yescharacteristic
185C4#1452402411yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#1686327611yescharacteristic
subgroupG/H
centerC4#14524011
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
Fitting#532×C67C2
socleC2×C67#386
hypercenter#532D67
hypocenterC67#1884
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #16864163C2semidirect
2. #16863413C2semidirect
3. #16863370C2semidirect
4. #16863156C2semidirect
14. C4⋊Q8×C67C4semidirect
15. #127×C67C4semidirect
16. #124×C67C4semidirect
17. #124×C67C4semidirect
23. #11761612D4semidirect
96. C67#1884semidirect
107. C4⋊Q8Dic67semidirect
108. #127Dic67semidirect
109. #124Dic67semidirect
110. #124Dic67semidirect