Finite Group Databasebeta

← #41014389

Subgroups — #41014389

Order 27008.

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

normal subgroups only

This group has 228 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
1C2×#20269968C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic
3M16×Dic211C211yesmaximal, characteristic
4#27212915C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic
7#343×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269968—21no—
11#20269968—21no—
12#20269968—21no—
13#20269968—21no—
14C2×C4⋊Dic211C412yes—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4C412yes—
17C2×C4×Dic211C2211yescharacteristic
18Dic211⋊C8—21no—
19Dic211⋊C8—21no—
20#20269887—21no—
21C8×Dic211—21no—
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C22×Dic211C2×C412yes—
26C22×Dic211C2×C411yescharacteristic
27C22×Dic211C2×C411yescharacteristic
28#16809946—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946Q812yes—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—22no—
35C4⋊Dic211—22no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38Dic211⋊C4—22no—
39Dic211⋊C4—22no—
40C4×Dic211D411yescharacteristic
41C4×Dic211D411yescharacteristic
42C4×Dic211D411yescharacteristic
43C4×Dic211D411yescharacteristic
44C22×C4×C211C2×C412yes—
45C22×C4×C211C2311yescharacteristic
46M16×C211—21no—
47M16×C211—21no—
48M16×C211—21no—
49M16×C211Q812yes—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—42no—
58C2×Dic211C22⋊C411yescharacteristic
59C2×Dic211C22⋊C411yescharacteristic
60C2×Dic211C22⋊C411yescharacteristic
61C2×Dic211C22⋊C411yescharacteristic
62C211⋊C8—22no—
63C211⋊C8—41no—
64C23×C211C22×C411yescharacteristic
65C2×C4×C211—22no—
66C2×C4×C211—42no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211C2×Q811yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C8×C211—22no—
74C8×C211—41no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211C2×C22⋊C411yescharacteristic
78C22×C211C4×Q811yescharacteristic
79C22×C211C4×Q811yescharacteristic
80C4×C211—42no—
81C4×C211C22⋊Q811yescharacteristic
82C4×C211C22⋊Q811yescharacteristic
83C4×C211#12411yescharacteristic
84C4×C211C4⋊Q811yescharacteristic
85Dic211—21no—
86Dic211—21no—
87Dic211—21no—
88Dic211—21no—
89Dic211—42no—
90C2×C211—21no—
91C2×C211—21no—
92C2×C211#39111yescharacteristic
93C2×C211#41311yescharacteristic
94C2×C211#41311yescharacteristic
95C211#199811yescharacteristic, complemented, Sylow
96#1998—2111nomaximal, Sylow
97C2×#125—2111no—
98#423—2111no—
99C4×M16—2111no—
100#343—2111no—
101#343—2111no—
102#343—2111no—
103#343D21111yescharacteristic
104C2×M16—2111no—
105C2×M16—2111no—
106C2×M16C2×D21111yescharacteristic
107C2×M16C2×D21111yescharacteristic
108#125—4221no—
109#125—4221no—
110#125—4221no—
111#125—4221no—
112C2×C4⋊C4—2112no—
113C2×C4⋊C4—2112no—
114C2×C4⋊C4—2111no—
115C2×C4⋊C4C2×D21111yescharacteristic
116C2×C42—2111no—
117C4⋊C8—4221no—
118C4⋊C8—4221no—
119#97—4221no—
120C4×C8—4221no—
121C22×C4—2112no—
122C22×C4—2111no—
123C22×C4—2111no—
124C22×C4C4×D21112yes—
125C22×C4C22×D21111yescharacteristic
126M16—21no—
127M16—21no—
128M16—21no—
129M16—4221no—
130M16—4221no—
131M16—4221no—
132M16—2112no—
133M16Dic42212yes—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4221no—
137C2×C8—4221no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—4222no—
141C4⋊C4—4222no—
142C4⋊C4—4222no—
143C4⋊C4—4222no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C42—2111no—
147C42—2111no—
148C42—2111no—
149C42—2111no—
150C23C2×C4×D21111yescharacteristic
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—8442no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—4222no—
157C2×C4—2111no—
158C2×C4—2111no—
159C2×C4—2111no—
160C2×C4—2111no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4Q8×D21111yescharacteristic
164C2×C4C2×Dic42211yescharacteristic
165C2×C4#1680996111yescharacteristic
166C2×C4#1680996211yescharacteristic
167C8—22no—
168C8—41no—
169C8—8441no—
170C8—4222no—
171C22—21no—
172C22—21no—
173C22C22⋊C4×D21111yescharacteristic
174C22#2026994211yescharacteristic
175C22C4×Dic42211yescharacteristic
176C4—42no—
177C4—8442no—
178C4—4221no—
179C4—4221no—
180C4—4221no—
181C4—4221no—
182C4#2026995411yescharacteristic
183C4#2026995511yescharacteristic
184C4#2026996311yescharacteristic
185C4#2026997611yescharacteristic
186C2—21no—
187C2—21no—
188C2#2721305011yescharacteristic
189C2#2721312711yescharacteristic
190C2#2721312911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutator, prime radicalC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1998
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
95. C211#1998semidirect

Maximal subgroups

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

subgroupindex
C2×#202699682
#272131692
M16×Dic2112
#272129152
#272129142
#343×C2112
#1998211