Finite Group Databasebeta

← #41013503

Subgroups — #41013503

Order 27008.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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#41×Dic211C211yesmaximal, characteristic
2#27213120C211yesmaximal, characteristic
3#27212907C212yesmaximal, complemented
4#27212903C212yesmaximal, complemented
5#413×C211C211yesmaximal, characteristic
6Q8×Dic211—21no—
7D4×Dic211—21no—
8D4×Dic211—22no—
9#20269936C412yescomplemented
10#20269936C2212yes—
11#20269936C2211yescharacteristic
12C2×C4×Dic211C412yescomplemented
13C2×C4×Dic211C2211yescharacteristic
14#20269892—22no—
15C2×#41×C211C2211yescharacteristic
16C2×M16×C211C2212yes—
17#101×C211—21no—
18#100×C211—21no—
19C22×Dic211—22no—
20C22×Dic211—21no—
21C4⋊Dic211—22no—
22C4⋊Dic211—21no—
23C4⋊Dic211—22no—
24C22⋊Dic211—22no—
25C22⋊Dic211—21no—
26C22⋊Dic211—22no—
27C4×Dic211—22no—
28C4×Dic211—21no—
29C4×Dic211—22no—
30C4×Dic211C2×C411yescharacteristic
31C4×Dic211C2×C412yes—
32C4×Dic211C2×C411yescharacteristic
33#41×C211—22no—
34#41×C211—22no—
35C2×Q8×C211D411yescharacteristic
36C2×D4×C211D411yescharacteristic
37C2×D4×C211D412yes—
38C22×C4×C211C2×C412yes—
39C22×C4×C211C2311yescharacteristic
40M16×C211—22no—
41M16×C211—22no—
42C2×C8×C211—22no—
43C2×Dic211—21no—
44C2×Dic211—22no—
45C2×Dic211—21no—
46C2×Dic211—42no—
47C2×Dic211—44no—
48C2×Dic211—42no—
49C23×C211C22⋊C412yes—
50C23×C211C2×D411yescharacteristic
51Q8×C211—22no—
52D4×C211—24no—
53D4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—22no—
56C2×C4×C211C22⋊C412yes—
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211#4112yes—
60C2×C4×C211#4112yes—
61C8×C211—42no—
62C22×C211—22no—
63C22×C211—41no—
64C22×C211—42no—
65C22×C211C2×C22⋊C411yescharacteristic
66C22×C211#12312yes—
67C4×C211—22no—
68C4×C211#11711yescharacteristic
69C4×C211#11711yescharacteristic
70C4×C211#12312yes—
71Dic211—22no—
72Dic211—42no—
73Dic211—41no—
74Dic211—82no—
75C2×C211—21no—
76C2×C211—22no—
77C2×C211—42no—
78C2×C211#38111yescharacteristic
79C211#182711yescharacteristic, complemented, Sylow
80#1827—2111nomaximal, Sylow
81C4×#41—2111no—
82#413D21111yescharacteristic
83#410—2111no—
84#339—2112no—
85H(3,ℤ/4)—2112no—
86C2×#41C2×D21111yescharacteristic
87C2×M16C2×D21112yes—
88C4×Q8—4221no—
89C4×D4—4221no—
90C4×D4—4222no—
91#117—2112no—
92#117—2112no—
93#117—2111no—
94C2×C42—2112no—
95C2×C42—2111no—
96#101—21no—
97#100—21no—
98#99—4222no—
99#41—22no—
100#41—22no—
101C2×Q8C22⋊D21111yescharacteristic
102C2×D4C22⋊D21111yescharacteristic
103C2×D4D84412yes—
104C22×C4—4222no—
105C22×C4—4221no—
106C22×C4C4×D21112yes—
107C22×C4C22×D21111yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—4222no—
112C4⋊C4—4222no—
113C4⋊C4—4221no—
114C22⋊C4—4222no—
115C22⋊C4—4222no—
116C22⋊C4—4221no—
117C42—4222no—
118C42—4222no—
119C42—4221no—
120C42—2111no—
121C42—2112no—
122C42—2111no—
123C23(C2×D211)⋊C412yes—
124C23C2×D84411yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—8442no—
131C2×C4—8444no—
132C2×C4—8442no—
133C2×C4—4222no—
134C2×C4—4221no—
135C2×C4—4221no—
136C2×C4(C2×D211)⋊C412yes—
137C2×C4C2×C4×D21111yescharacteristic
138C2×C4C2×C22⋊D21111yescharacteristic
139C2×C4#1680996112yes—
140C2×C4#1680996112yes—
141C8—42no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×(C2×D211)⋊C411yescharacteristic
146C22#2026996112yes—
147C4—22no—
148C4—8441no—
149C4—8442no—
150C4—16882no—
151C4—4222no—
152C4#2026993311yescharacteristic
153C4#2026993311yescharacteristic
154C4#2026995812yes—
155C2—21no—
156C2—22no—
157C2—42no—
158C2#2721300711yescharacteristic
subgroupG/H
centerC4#20269933
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#413×C211C2
socleC2×C211#381
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenter#413D211
hypocenterC211#1827
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #27212907C2semidirect
4. #27212903C2semidirect
9. #20269936C4semidirect
12. C2×C4×Dic211C4semidirect
79. C211#1827semidirect

Maximal subgroups

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

subgroupindex
#41×Dic2112
#272131202
#272129072
#272129032
#413×C2112
#1827211