Finite Group Databasebeta

← #41014310

Subgroups — #41014310

Order 27008.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#16809952C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic
3#27213156C211yesmaximal, characteristic
4#27213155C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic
6#27213104C211yesmaximal, characteristic
7C2×#101×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809952—21no—
10C2×#16809952—21no—
11C2×#16809952—22no—
12C2×#16809952C414yescomplemented
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15#20269936C2211yescharacteristic
16#20269912—22no—
17#20269902—22no—
18#20269901—22no—
19#20269895—22no—
20#20269887—21no—
21C8×Dic211—21no—
22C22×Q8×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25#101×C211—22no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422C2×C412yes—
32C22×Dic211—21no—
33#16809952—24no—
34#16809952—24no—
35#16809946—21no—
36#16809946—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8C2×C412yes—
39C2×C211⋊C8D411yescharacteristic
40C2×C211⋊C8D411yescharacteristic
41C2×C211⋊C8D412yes—
42C4⋊Dic211D412yes—
43C22⋊Dic211—21no—
44C4×Dic211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211C2×C412yes—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—22no—
64C2×Dic211—22no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C211⋊C8—41no—
68C211⋊C8C22⋊C412yes—
69C211⋊C8C22⋊C412yes—
70C23×C211#4111yescharacteristic
71Q8×C211—22no—
72Q8×C211—22no—
73Q8×C211—41no—
74Q8×C211—41no—
75C2×C4×C211—22no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C22×C411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C8×C211—22no—
85C8×C211—41no—
86C22×C211—22no—
87C22×C211C4×D411yescharacteristic
88C22×C211#12111yescharacteristic
89C22×C211#12411yescharacteristic
90C4×C211—42no—
91C4×C211C2×C22⋊C411yescharacteristic
92C4×C211C4×D411yescharacteristic
93C4×C211#12111yescharacteristic
94C4×C211#12711yescharacteristic
95Dic211—42no—
96Dic211—42no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211#39011yescharacteristic
100C2×C211#47312yes—
101C211#197911yescharacteristic, complemented, Sylow
102#1979—2111nomaximal, Sylow
103C22×Q16—2111no—
104C2×#108—2111no—
105#419—2111no—
106#419—2111no—
107C2×#101—2111no—
108C2×#101D21111yescharacteristic
109#405—2111no—
110C22×Q8—2111no—
111C22×Q8C2×D21111yescharacteristic
112C2×Q16—4221no—
113C2×Q16—4221no—
114C2×Q16—4222no—
115C2×Q16—2114no—
116C2×M16—2111no—
117C2×M16C2×D21111yescharacteristic
118C2×M16C2×D21111yescharacteristic
119C22×C8—2111no—
120#117—2111no—
121#108—4222no—
122#103—4222no—
123#103—4222no—
124#101—22no—
125#101—4222no—
126#97—4221no—
127C4×C8—4221no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—4221no—
134C2×Q8—4221no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—2112no—
139C2×Q8C4×D21112yes—
140C22×C4—21no—
141C22×C4—4221no—
142C22×C4C22×D21111yescharacteristic
143Q16—4224no—
144Q16—4224no—
145M16—21no—
146M16—21no—
147M16—21no—
148M16—4221no—
149M16—4221no—
150M16C22⋊D21112yes—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—4221no—
154C2×C8—2112no—
155C2×C8—2111no—
156C2×C8—2111no—
157C2×C8—2112no—
158C4⋊C4—2112no—
159C22⋊C4—4221no—
160C42—4221no—
161C23#1680996211yescharacteristic
162Q8—22no—
163Q8—22no—
164Q8—41no—
165Q8—41no—
166Q8—8441no—
167Q8—8441no—
168Q8—4222no—
169Q8—4222no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—4222no—
176C2×C4—4222no—
177C2×C4C2×C4×D21111yescharacteristic
178C2×C4D4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C2×C4D4×D21111yescharacteristic
181C2×C4C2×C22⋊D21111yescharacteristic
182C2×C4#1680996111yescharacteristic
183C8—22no—
184C8—41no—
185C8—8441no—
186C8—2112no—
187C8—2112no—
188C22—22no—
189C22C4×C22⋊D21111yescharacteristic
190C22#2026995011yescharacteristic
191C22#2026996311yescharacteristic
192C4—42no—
193C4—8442no—
194C4—8442no—
195C4C22⋊C4×D21111yescharacteristic
196C4(C22⋊D211)⋊C411yescharacteristic
197C4#2026994911yescharacteristic
198C4#2026997311yescharacteristic
199C2—21no—
200C2—21no—
201C2#2721304611yescharacteristic
202C2#2721342012yes—
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#121
prime radicalC23×C211#41
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1979
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
12. C2×#16809952C4semidirect
101. C211#1979semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C2×#202699122
#272131562
#272131552
C2×#202698952
#272131042
C2×#101×C2112
#1979211