Finite Group Databasebeta

← #22560448

Subgroups — #22560448

Order 8576.

262 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 79 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#16864162C211yesmaximal, characteristic
2#16863444C211yesmaximal, characteristic, complemented
3#16863412C211yesmaximal, characteristic
4C2×#14523971C211yesmaximal, characteristic
5#16863167C211yesmaximal, characteristic, complemented
6#16863156C211yesmaximal, characteristic, complemented
7#423×C67C211yesmaximal, characteristic, complemented
8C2×#11761628C2211yescharacteristic
9#14524051—22no—
10#14524049—22no—
11C2×#11761612C2211yescharacteristic
12C2×#11761612C2211yescharacteristic
13M16×D67—22no—
14#14524002C2211yescharacteristic
15#14523978—22no—
16#14523971C412yescomplemented
17#14523971C412yescomplemented
18#14523969—22no—
19C2×M16×C67C2211yescharacteristic
20C2×M16×C67C2211yescharacteristic
21C2×C42×C67C2211yescharacteristic
22C4⋊C8×C67—22no—
23#11761628—21no—
24#11761628—21no—
25#11761628C2×C412yes—
26#11761628D412yes—
27C2×Dic134Q811yescharacteristic
28C2×D268Q811yescharacteristic
29C2×C22⋊D67—21no—
30C2×C4×D67—21no—
31#11761612—21no—
32#11761612—21no—
33#11761612C2×C412yes—
34#11761612D412yes—
35#11761611—24no—
36C2×C67⋊C8—21no—
37C2×C67⋊C8—21no—
38C8×D67—24no—
39C4⋊Dic67—21no—
40C22⋊Dic67—21no—
41C4×Dic67—21no—
42C22×C4×C67—21no—
43C22×C4×C67C2311yescharacteristic
44M16×C67—21no—
45M16×C67—21no—
46M16×C67—21no—
47M16×C67—21no—
48C2×C8×C67—21no—
49C2×C8×C67D412yes—
50C42×C67—21no—
51C42×C67C2×C411yescharacteristic
52C42×C67C2×C411yescharacteristic
53C23×C67#4111yescharacteristic
54C22×D67—21no—
55C2×C4×C67—21no—
56C2×C4×C67—21no—
57C2×C4×C67—42no—
58C2×C4×C67C22×C411yescharacteristic
59C2×C4×C67C2×D411yescharacteristic
60C2×C4×C67C2×D411yescharacteristic
61C2×C4×C67C2×D411yescharacteristic
62C2×C4×C67C2×Q811yescharacteristic
63C2×C4×C67#4111yescharacteristic
64C22⋊D67—22no—
65C22⋊D67—22no—
66C2×Dic67—21no—
67C2×Dic67—41no—
68D268—21no—
69D268C4⋊C412yes—
70C4×D67—22no—
71C4×D67—22no—
72Dic134—21no—
73Dic134C4⋊C412yes—
74C8×C67—22no—
75C8×C67—41no—
76C67⋊C8—22no—
77C67⋊C8—22no—
78C22×C67—41no—
79C22×C67C4×D411yescharacteristic
80C22×C67C22⋊Q811yescharacteristic
81C22×C67#12311yescharacteristic
82C2×D67—22no—
83C2×D67—81no—
84C4×C67—22no—
85C4×C67—41no—
86C4×C67C2×C4⋊C411yescharacteristic
87C4×C67C4×D411yescharacteristic
88C4×C67#12011yescharacteristic
89C4×C67C22⋊Q811yescharacteristic
90Dic67—22no—
91Dic67—81no—
92C2×C67—21no—
93C2×C67—21no—
94C2×C67—21no—
95C2×C67#38511yescharacteristic
96D67—42no—
97#1861—671nomaximal, Sylow
98C67#186111yescharacteristic, complemented, Sylow
99#568—671no—
100#430—671no—
101#423D6711yescharacteristic, complemented
102#421—671no—
103C2×#104—671no—
104#344—671no—
105#339—671no—
106C2×#41—671no—
107#131—1342no—
108#131—1342no—
109C2×M16—671no—
110C2×M16—671no—
111C2×M16—1342no—
112C2×M16C2×D6711yescharacteristic
113C2×M16C2×D6711yescharacteristic
114#117—671no—
115C2×C42C2×D6711yescharacteristic
116#108—1342no—
117C4⋊C8—22no—
118#104—672no—
119#104—672no—
120#104—1342no—
121#41—672no—
122#41—672no—
123#41—1341no—
124#41—1341no—
125C2×Q8—671no—
126C2×D4—671no—
127C2×D4—1341no—
128C22×C4—21no—
129C22×C4—1341no—
130C22×C4C22×D6711yescharacteristic
131M16—21no—
132M16—21no—
133M16—21no—
134M16—21no—
135M16—672no—
136M16—672no—
137M16—1344no—
138M16—1341no—
139M16—1341no—
140C2×C8—21no—
141C2×C8—1344no—
142C2×C8—1341no—
143C2×C8—1341no—
144C2×C8C22⋊D6712yes—
145C4⋊C4—1341no—
146C22⋊C4—1341no—
147C42—21no—
148C42—1341no—
149C42C4×D6711yescharacteristic
150C42C4×D6711yescharacteristic
151C23—1341no—
152C23#1176162811yescharacteristic
153Q8—672no—
154Q8—1341no—
155D4—672no—
156D4—1342no—
157D4—1342no—
158D4—1341no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—2681no—
163C2×C4—1341no—
164C2×C4—1342no—
165C2×C4—1342no—
166C2×C4C2×C4×D6711yescharacteristic
167C2×C4D4×D6711yescharacteristic
168C2×C4D4×D6711yescharacteristic
169C2×C4C2×C22⋊D6711yescharacteristic
170C2×C4Q8×D6711yescharacteristic
171C2×C4#1176162911yescharacteristic
172C8—22no—
173C8—41no—
174C8—1342no—
175C8—1342no—
176C22—41no—
177C22—5361no—
178C22—1342no—
179C22C4×C22⋊D6711yescharacteristic
180C22#1452402111yescharacteristic
181C22#1452402811yescharacteristic
182C4—22no—
183C4—41no—
184C4—5361no—
185C4—1342no—
186C4C4⋊C4×D6711yescharacteristic
187C4#1452400511yescharacteristic
188C4#1452401111yescharacteristic
189C4#1452401911yescharacteristic
190C2—21no—
191C2—21no—
192C2—21no—
193C2—2682no—
194C2#1686326711yescharacteristic
subgroupG/H
centerC4#14524011
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
Fitting#423×C67C2
socleC2×C67#385
hypercenter#423D67
hypocenterC67#1861
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C4 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #16863444C2semidirect
5. #16863167C2semidirect
6. #16863156C2semidirect
7. #423×C67C2semidirect
16. #14523971C4semidirect
17. #14523971C4semidirect
98. C67#1861semidirect
101. #423D67semidirect