Finite Group Databasebeta

← #123×D421 #41003340

Subgroups — #123×D421 #41003340

Order 26944.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×D4×D421C211yesmaximal, characteristic
2C22×C4×D421C211yesmaximal, characteristic, complemented
3#27212351C212yesmaximal, complemented
4#27212350C211yesmaximal, characteristic, complemented
5#27212349C211yesmaximal, characteristic, complemented
6#27212348C212yesmaximal, complemented
7#27212347C211yesmaximal, characteristic, complemented
8C4⋊C4×D421C212yesmaximal, complemented
9C22⋊C4×D421C212yesmaximal, complemented
10C22⋊C4×D421C211yesmaximal, characteristic, complemented
11#123×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13C23×D421C2211yescharacteristic
14C2×D1684C2211yescharacteristic
15C2×C22⋊D421C2211yescharacteristic
16C2×C22⋊D421C2211yescharacteristic
17D4×D421—24no—
18C22×Dic421C2211yescharacteristic, complemented
19C2×C4×D421—24no—
20C2×C4×D421C2212yes—
21C2×C4×D421C2212yes—
22C2×C4×D421C2211yescharacteristic
23C2×C4×D421C2212yescomplemented
24C4⋊Dic421C2212yescomplemented
25#20269750C2212yescomplemented
26#20269750C2212yescomplemented
27C22⋊Dic421C2212yescomplemented
28C22⋊Dic421C2211yescharacteristic, complemented
29#20269748C2212yescomplemented
30#20269748C2212yescomplemented
31#20269748C2212yescomplemented
32C2×D4×C421C2211yescharacteristic
33C22×C4×C421C2211yescharacteristic, complemented
34C4⋊C4×C421C2212yescomplemented
35C22⋊C4×C421C2212yescomplemented
36C22⋊C4×C421C2211yescharacteristic, complemented
37C22×D421—24no—
38C22×D421—22no—
39C22×D421—24no—
40C22×D421D414yes—
41C22×D421C2311yescharacteristic
42C22×D421C2311yescharacteristic
43C22×D421C2311yescharacteristic
44D1684—22no—
45C22⋊D421—22no—
46C22⋊D421—22no—
47C2×Dic421—22no—
48C2×Dic421C2312yes—
49C2×Dic421C2312yes—
50C2×Dic421C2311yescharacteristic
51C4×D421—24no—
52C4×D421—24no—
53C4×D421—22no—
54C4×D421—24no—
55C23×C421C2311yescharacteristic
56C23×C421C2311yescharacteristic
57D4×C421—22no—
58C2×C4×C421—22no—
59C2×C4×C421C2312yes—
60C2×C4×C421C2312yes—
61C2×C4×C421C2311yescharacteristic
62C22×C421—22no—
63C22×C421—21no—
64C22×C421—22no—
65C22×C421C2×D412yes—
66C22×C421C2411yescharacteristic
67C2×D421—22no—
68C2×D421—28no—
69C2×D421—21no—
70C2×D421—22no—
71C2×D421—44no—
72C2×D421C2×D412yes—
73C2×D421C2×D412yes—
74C2×D421#4114yes—
75C4×C421—22no—
76C4×C421—21no—
77C4×C421—22no—
78Dic421—22no—
79Dic421—21no—
80Dic421—22no—
81C2×C421—22no—
82C2×C421—41no—
83C2×C421C22×D411yescharacteristic
84C2×C421C2×#4112yes—
85D421—22no—
86D421—41no—
87D421#12314yesdirect product factor
88C421C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—4211nomaximal, Sylow
90C22×D4—4211no—
91C23×C4—4211no—
92#123—4211no—
93#123—4211no—
94#123—4212no—
95#123—4212no—
96#123—4211no—
97#123D42111yescharacteristic, direct product factor
98C2×C4⋊C4—4212no—
99C2×C22⋊C4—4211no—
100C2×C22⋊C4—4212no—
101C24—4211no—
102C24—4211no—
103C2×D4—8424no—
104C2×D4—4211no—
105C2×D4—4211no—
106C2×D4—4211no—
107C2×D4C2×D42111yescharacteristic
108C22×C4—8424no—
109C22×C4—4211no—
110C22×C4—4212no—
111C22×C4—4212no—
112C22×C4—4211no—
113C22×C4—4212no—
114C22×C4C2×D42111yescharacteristic, complemented
115C4⋊C4—4212no—
116C4⋊C4—4212no—
117C4⋊C4—4212no—
118C4⋊C4C2×D42112yescomplemented
119C22⋊C4—4211no—
120C22⋊C4—4212no—
121C22⋊C4—4212no—
122C22⋊C4—4212no—
123C22⋊C4—4212no—
124C22⋊C4C2×D42111yescharacteristic, complemented
125C22⋊C4C2×D42112yescomplemented
126C23—8424no—
127C23—8422no—
128C23—8424no—
129C23—4211no—
130C23—4214no—
131C23—4211no—
132C23—4211no—
133C23C22×D42111yescharacteristic
134C23C22×D42111yescharacteristic
135D4—22no—
136D4—8422no—
137D4—8422no—
138D4—8422no—
139C2×C4—22no—
140C2×C4—8424no—
141C2×C4—8424no—
142C2×C4—8422no—
143C2×C4—8424no—
144C2×C4—8422no—
145C2×C4—4212no—
146C2×C4—4212no—
147C2×C4—4211no—
148C2×C4C22×D42112yes—
149C2×C4C22×D42112yes—
150C2×C4C22×D42111yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—8428no—
155C22—8422no—
156C22—8421no—
157C22—8422no—
158C22—16844no—
159C22—4212no—
160C22—4212no—
161C22—4214no—
162C22D4×D42112yes—
163C22C23×D42111yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—8422no—
168C4—8421no—
169C4—8422no—
170C2—22no—
171C2—41no—
172C2—8422no—
173C2—16841no—
174C2—4214no—
175C2C2×D4×D42111yescharacteristic
176C2#41×D42112yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#123×C421C2
prime radicalC2×D4×D421C2
hypercenter#123D421
hypocenterC421C2×#123
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D421C2semidirect
3. #27212351C2semidirect
4. #27212350C2semidirect
5. #27212349C2semidirect
6. #27212348C2semidirect
7. #27212347C2semidirect
8. C4⋊C4×D421C2semidirect
9. C22⋊C4×D421C2semidirect
10. C22⋊C4×D421C2semidirect
11. #123×C421C2semidirect
18. C22×Dic421C22semidirect
23. C2×C4×D421C22semidirect
24. C4⋊Dic421C22semidirect
25. #20269750C22semidirect
26. #20269750C22semidirect
27. C22⋊Dic421C22semidirect
28. C22⋊Dic421C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. #20269748C22semidirect
33. C22×C4×C421C22semidirect
34. C4⋊C4×C421C22semidirect
35. C22⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
87. D421#123direct (N × H)
88. C421C2×#123semidirect
97. #123D421direct (N × H)
114. C22×C4C2×D421semidirect
118. C4⋊C4C2×D421semidirect
124. C22⋊C4C2×D421semidirect
125. C22⋊C4C2×D421semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D4×D4212
C22×C4×D4212
#272123512
#272123502
#272123492
#272123482
#272123472
C4⋊C4×D4212
C22⋊C4×D4212
#123×C4212
C2×#123421