Finite Group Databasebeta

← #22561382

Subgroups — #22561382

Order 8576.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#16863953C211yesmaximal, characteristic, complemented
2#16863412C211yesmaximal, characteristic
3C2×#14523971C211yesmaximal, characteristic
4#16863369C211yesmaximal, characteristic
5#16863167C211yesmaximal, characteristic, complemented
6#16863152C211yesmaximal, characteristic, complemented
7#339×C67C211yesmaximal, characteristic, complemented
8C2×#11761628C2211yescharacteristic
9C2×#11761612C2211yescharacteristic
10#14524043—21no—
11#14524035—21no—
12#14524022—22no—
13C4×Dic134—22no—
14#14524002C2211yescharacteristic
15#14524002C2211yescharacteristic
16#14523971—21no—
17#14523971—21no—
18#14523969—22no—
19#14523957—22no—
20C2×M16×C67C2211yescharacteristic
21#117×C67C2211yescharacteristic
22C2×C42×C67C2211yescharacteristic
23#11761628—21no—
24#11761628—21no—
25#11761628—21no—
26#11761628—21no—
27C2×Dic134—22no—
28C2×Dic134Q811yescharacteristic, complemented
29C2×D268Q811yescharacteristic, complemented
30C2×C22⋊D67—21no—
31C2×C4×D67—21no—
32#11761612—21no—
33#11761612—21no—
34C2×C67⋊C8—21no—
35C4⋊Dic67—22no—
36C4⋊Dic67—21no—
37C4⋊Dic67—21no—
38Dic67⋊C4—24no—
39C22⋊Dic67—21no—
40C22⋊Dic67D412yes—
41C4×Dic67—21no—
42C4×Dic67D412yes—
43C22×C4×C67—21no—
44C22×C4×C67C2311yescharacteristic
45M16×C67—21no—
46M16×C67—21no—
47C2×C8×C67—21no—
48C4⋊C4×C67—21no—
49C22⋊C4×C67—21no—
50C42×C67—21no—
51C42×C67—21no—
52C42×C67D411yescharacteristic
53C42×C67D411yescharacteristic
54C23×C67C2×D411yescharacteristic
55C22×D67—21no—
56C2×C4×C67—21no—
57C2×C4×C67—21no—
58C2×C4×C67—42no—
59C2×C4×C67—41no—
60C2×C4×C67C2×D411yescharacteristic
61C2×C4×C67C2×D411yescharacteristic
62C2×C4×C67C2×Q811yescharacteristic
63C2×C4×C67#4111yescharacteristic
64C2×C4×C67#4111yescharacteristic
65C2×C4×C67#4111yescharacteristic
66C22⋊D67—41no—
67C22⋊D67—41no—
68C2×Dic67—22no—
69C2×Dic67—22no—
70C2×Dic67—21no—
71D268—21no—
72D268—21no—
73C4×D67—41no—
74C4×D67—41no—
75Dic134—21no—
76Dic134—21no—
77Dic134—44no—
78C8×C67—41no—
79C67⋊C8—41no—
80C22×C67—41no—
81C22×C67#12011yescharacteristic
82C22×C67C22⋊Q811yescharacteristic
83C22×C67#12411yescharacteristic
84C2×D67—41no—
85C2×D67—81no—
86C4×C67—22no—
87C4×C67—41no—
88C4×C67—81no—
89C4×C67C22⋊Q811yescharacteristic
90C4×C67C22⋊Q811yescharacteristic
91C4×C67#12411yescharacteristic
92C4×C67#12411yescharacteristic
93Dic67—42no—
94Dic67—42no—
95Dic67—41no—
96C2×C67—21no—
97C2×C67—21no—
98C2×C67—21no—
99C2×C67#39311yescharacteristic
100D67—81no—
101#2031—671nomaximal, Sylow
102C67#203111yescharacteristic, complemented, Sylow
103#533—671no—
104#421—671no—
105C2×#104—671no—
106#410—671no—
107#344—671no—
108#339D6711yescharacteristic, complemented
109H(3,ℤ/4)—671no—
110C2×#41—671no—
111C2×M16—671no—
112C2×M16C2×D6711yescharacteristic
113C4⋊Q8—1341no—
114#125—1341no—
115C22⋊Q8—1342no—
116C4×Q8—1342no—
117#117—671no—
118#117—671no—
119#117C2×D6711yescharacteristic
120C2×C42C2×D6711yescharacteristic
121#104—1342no—
122#104—1341no—
123#104—1341no—
124#99—1342no—
125#41—1341no—
126#41—1341no—
127#41—1341no—
128#41—1341no—
129C2×Q8—671no—
130C2×Q8—1342no—
131C2×D4—671no—
132C2×D4—1341no—
133C22×C4—21no—
134C22×C4—1341no—
135C22×C4C22×D6711yescharacteristic
136M16—21no—
137M16—21no—
138M16—1341no—
139M16—1341no—
140C2×C8—21no—
141C2×C8—1341no—
142C4⋊C4—21no—
143C4⋊C4—1344no—
144C4⋊C4—1342no—
145C4⋊C4—1341no—
146C4⋊C4—1341no—
147C22⋊C4—21no—
148C22⋊C4—672no—
149C22⋊C4—1341no—
150C42—21no—
151C42—21no—
152C42—672no—
153C42—1341no—
154C42D26811yescharacteristic
155C42D26811yescharacteristic
156C23—1341no—
157C23D4×D6711yescharacteristic
158Q8—2684no—
159Q8—1341no—
160Q8—1341no—
161D4—2681no—
162D4—2681no—
163D4—1341no—
164D4—1341no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—2681no—
170C2×C4—2681no—
171C2×C4—1342no—
172C2×C4—1342no—
173C2×C4—1341no—
174C2×C4D4×D6711yescharacteristic
175C2×C4C2×D26811yescharacteristic
176C2×C4Q8×D6711yescharacteristic
177C2×C4#1176162711yescharacteristic
178C2×C4#1176162811yescharacteristic
179C2×C4#1176162811yescharacteristic
180C8—41no—
181C8—2681no—
182C22—41no—
183C22—2681no—
184C22—5361no—
185C22#1452401311yescharacteristic
186C22#1452402111yescharacteristic
187C22#1452402911yescharacteristic
188C4—22no—
189C4—41no—
190C4—81no—
191C4—2682no—
192C4—2682no—
193C4—2681no—
194C4#1452402111yescharacteristic
195C4#1452402311yescharacteristic
196C4#1452402911yescharacteristic
197C4#1452403111yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—5361no—
202C2#1686330911yescharacteristic
subgroupG/H
centerC4#14524029
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#339×C67C2
socleC2×C67#393
hypercenter#339D67
hypocenterC67#2031
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #16863953C2semidirect
5. #16863167C2semidirect
6. #16863152C2semidirect
7. #339×C67C2semidirect
28. C2×Dic134Q8semidirect
29. C2×D268Q8semidirect
102. C67#2031semidirect
108. #339D67semidirect