Finite Group Databasebeta

← #41012644

Subgroups — #41012644

Order 27008.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213811C211yesmaximal, characteristic
2#27213423C211yesmaximal, characteristic
3#27213353C211yesmaximal, characteristic, complemented
4#27212892C211yesmaximal, characteristic, complemented
5#27212878C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic
7#500×C211C211yesmaximal, characteristic, complemented
8C2×Q8×D211—21no—
9C2×#16809952—21no—
10#20269976—21no—
11#20269955—22no—
12#20269953—21no—
13C4×Dic422C2211yescharacteristic
14C4×D844C2211yescharacteristic
15#20269939—21no—
16#20269910—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269901—21no—
20#20269897—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#107×C211—22no—
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—22no—
26C2×Dic422D411yescharacteristic
27Q8×D211—42no—
28Q8×D211—42no—
29C2×D844D411yescharacteristic
30C2×C4×D211—22no—
31C2×C4×D211—21no—
32#16809952—42no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211D411yescharacteristic
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—22no—
38Dic211⋊C4—21no—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—21no—
41C4×Dic211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211D412yes—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—21no—
52Dic422—44no—
53Dic422D812yes—
54D844—21no—
55D844Q1612yescomplemented
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C4×D211—44no—
60C4×D211—41no—
61C4×D211—41no—
62C211⋊C8—41no—
63C211⋊C8—41no—
64Q8×C211—41no—
65Q8×C211—41no—
66Q8×C211—41no—
67Q8×C211—41no—
68C2×C4×C211—22no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—22no—
74C22×C211#12011yescharacteristic
75C2×D211—22no—
76C2×D211—41no—
77C2×D211—41no—
78C4×C211—21no—
79C4×C211—42no—
80C4×C211—81no—
81C4×C211C2×D811yescharacteristic
82C4×C211C2×Q1611yescharacteristic
83C4×C211AGL(1,ℤ/8)11yescharacteristic
84C4×C211#13711yescharacteristic
85Dic211—22no—
86Dic211—41no—
87Dic211—41no—
88C2×C211#44711yescharacteristic
89C2×C211#45111yescharacteristic
90C2×C211#45611yescharacteristic
91D211—42no—
92C211#164911yescharacteristic, complemented, Sylow
93#1649—2111nomaximal, Sylow
94D4×Q8—2111no—
95#500D21111yescharacteristic, complemented
96#474—2111no—
97#462—2111no—
98#332—2111no—
99Q8⋊C8—2111no—
100D4⋊C8—2111no—
101C22×Q8—4221no—
102C2×Q16—4221no—
103C4⋊Q8—4221no—
104C4⋊Q8C2×D21111yescharacteristic
105C4⋊Q8C2×D21111yescharacteristic
106C22⋊Q8—4221no—
107C22⋊Q8—4222no—
108C4×Q8—2111no—
109C4×D4—4221no—
110C4×D4—2111no—
111#107—22no—
112#107—4221no—
113C4⋊C8—2111no—
114C4⋊C8—2111no—
115#103—4221no—
116#102—4221no—
117C4×C8C2×D21111yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—8442no—
121C2×Q8—8442no—
122C2×Q8—4222no—
123C2×Q8—2111no—
124C2×D4—2111no—
125C22×C4—4222no—
126C22×C4—4221no—
127Q16—8442no—
128C2×C8—21no—
129C2×C8—4221no—
130C2×C8—4221no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—4222no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2111no—
138C4⋊C4C22⋊D21112yes—
139C22⋊C4—4222no—
140C22⋊C4—4221no—
141C42—4221no—
142C42C22×D21111yescharacteristic
143C23—4221no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—8444no—
149Q8—4221no—
150Q8—2112no—
151D4—4221no—
152D4—2112no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—8444no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C8—22no—
165C8—8441no—
166C8—8441no—
167C22—8441no—
168C22—8441no—
169C22—4222no—
170C22#2026994511yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—8441no—
175C4—8441no—
176C4—4222no—
177C4D8×D21111yescharacteristic
178C4Q16×D21111yescharacteristic
179C4#2027000111yescharacteristic
180C4#2027000911yescharacteristic
181C2—8442no—
182C2#2721327811yescharacteristic
183C2#2721330011yescharacteristic
184C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#500×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#500D211
hypocenterC211#1649
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #27213353C2semidirect
4. #27212892C2semidirect
5. #27212878C2semidirect
7. #500×C211C2semidirect
55. D844Q16semidirect
92. C211#1649semidirect
95. #500D211semidirect

Maximal subgroups

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

subgroupindex
#272138112
#272134232
#272133532
#272128922
#272128782
#272128752
#500×C2112
#1649211