Finite Group Databasebeta

← #41012535

Subgroups — #41012535

Order 27008.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213856C211yesmaximal, characteristic
2#27213526C211yesmaximal, characteristic, complemented
3#27213441C211yesmaximal, characteristic, complemented
4#27213437C211yesmaximal, characteristic
5#27212894C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×Dic844—22no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269969—21no—
12#20269955—22no—
13#20269953—21no—
14C4×D844—21no—
15C4×D844C2211yescharacteristic
16C4⋊C4×D211—21no—
17#20269910—21no—
18#20269909—21no—
19C4⋊(C211⋊C8)C2211yescharacteristic
20#20269901—21no—
21#20269899—21no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422D412yes—
27C2×D844D411yescharacteristic
28C2×C4×D211—22no—
29C2×C4×D211—21no—
30Dic844—44no—
31C2×C211⋊C8—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—22no—
38Dic211⋊C4—22no—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—21no—
41C2×Q8×C211D411yescharacteristic
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49Dic422—42no—
50Dic422—42no—
51D844—21no—
52D844Q1612yescomplemented
53C2×Dic211—21no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C4×D211—41no—
57C4×D211—41no—
58C4×D211—82no—
59C211⋊C8—41no—
60Q8×C211—21no—
61Q8×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—22no—
68C8×C211—41no—
69C22×C211#12011yescharacteristic
70C2×D211—22no—
71C2×D211—41no—
72C2×D211—41no—
73C4×C211—21no—
74C4×C211—41no—
75C4×C211—41no—
76C4×C211C2×Q1611yescharacteristic
77C4×C211#13511yescharacteristic
78C4×C211#13711yescharacteristic
79C4×C211#13711yescharacteristic
80Dic211—41no—
81Dic211—42no—
82Dic211—81no—
83C2×C211#45111yescharacteristic
84C2×C211#45211yescharacteristic
85C2×C211#45311yescharacteristic
86D211—42no—
87C211#163211yescharacteristic, complemented, Sylow
88#1632—2111nomaximal, Sylow
89#554—2111no—
90#494—2111no—
91#477—2111no—
92#476—2111no—
93#333—2111no—
94Q8⋊C8D21111yescharacteristic, complemented
95D4⋊C8—2111no—
96C2×Q16—4222no—
97C4⋊Q8—2111no—
98C4⋊Q8—2111no—
99#125—4221no—
100C22⋊Q8—4222no—
101C22⋊Q8—4221no—
102C4×Q8C2×D21111yescharacteristic
103C4×D4—4221no—
104C4×D4—2111no—
105C2×C4⋊C4—4221no—
106#107—4221no—
107#106—4221no—
108C4⋊C8—2111no—
109C4⋊C8C2×D21111yescharacteristic
110#103—4221no—
111#102—4221no—
112C4×C8C2×D21111yescharacteristic
113C2×Q8—4221no—
114C2×Q8—2112no—
115C2×Q8D84411yescharacteristic
116C2×D4—2111no—
117C22×C4—4222no—
118C22×C4—4221no—
119Q16—8444no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—4221no—
123C4⋊C4—21no—
124C4⋊C4—4222no—
125C4⋊C4—4222no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—4221no—
129C4⋊C4—4221no—
130C4⋊C4—2111no—
131C4⋊C4D84411yescharacteristic
132C22⋊C4—4222no—
133C22⋊C4—4221no—
134C42—21no—
135C42C22×D21111yescharacteristic
136C23—4221no—
137Q8—21no—
138Q8—21no—
139Q8—8442no—
140Q8—8442no—
141D4—4221no—
142D4—2112no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—8441no—
146C2×C4—8441no—
147C2×C4—8441no—
148C2×C4—16882no—
149C2×C4—4221no—
150C2×C4—4222no—
151C2×C4D4×D21111yescharacteristic
152C2×C4D4×D21111yescharacteristic
153C2×C4C2×D84411yescharacteristic
154C8—22no—
155C8—41no—
156C8—8441no—
157C22—8441no—
158C22—8441no—
159C22—4222no—
160C22#2026994711yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—8442no—
165C4—8441no—
166C4—16881no—
167C4Q16×D21111yescharacteristic
168C4#2026999911yescharacteristic
169C4#2027000611yescharacteristic
170C4#2027001011yescharacteristic
171C2—8442no—
172C2#2721330311yescharacteristic
173C2#2721331011yescharacteristic
174C2#2721331511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenterQ8⋊C8D211
hypocenterC211#1632
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213526C2semidirect
3. #27213441C2semidirect
5. #27212894C2semidirect
7. Q8⋊C8×C211C2semidirect
52. D844Q16semidirect
87. C211#1632semidirect
94. Q8⋊C8D211semidirect

Maximal subgroups

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

subgroupindex
#272138562
#272135262
#272134412
#272134372
#272128942
#272128752
Q8⋊C8×C2112
#1632211