Finite Group Databasebeta

← #13165738

Subgroups — #13165738

Order 1728.

336 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 54 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#11723015C212yesmaximal
2#11722797C211yesmaximal, characteristic
3#11722778C212yesmaximal
4C3×#63743C211yesmaximal, characteristic
5Dic3×#622C211yesmaximal, characteristic
6#10593390—31nomaximal
7C2×C3⋊#622C2211yescharacteristic
8C2×C3×#622C2211yescharacteristic
9#88815C2212yes—
10#88807—22no—
11Dic3×C3⋊Dic3C2211yescharacteristic
12C3×#3724C2212yes—
13C3×Dic3⋊Dic3—22no—
14#63870—32no—
15#63846—62no—
16C4×#622—31no—
17#63743—31no—
18#63743D311yescharacteristic
19#63742—32no—
20C2×C3⋊(C3⋊Dic3)D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3C2311yescharacteristic
23C2×C32×Dic3—22no—
24C2×C32×Dic3D411yescharacteristic
25C3⋊#622Q812yes—
26C3×#622Q812yes—
27#4878—91nomaximal
28C2×#622—31no—
29C2×#622C2×D311yescharacteristic
30#3750—62no—
31C4×C3⋊Dic3—31no—
32C3×C4⋊Dic3—62no—
33C3×Dic3⋊C4—62no—
34C3×Dic3⋊C4—62no—
35#3724—32no—
36#3724C2×D312yes—
37Dic3⋊Dic3—22no—
38Dic3⋊Dic3—62no—
39Dic3⋊Dic3—62no—
40Dic3⋊Dic3—62no—
41Dic3⋊Dic3—122no—
42Dic32—62no—
43C22×C33C2×D411yescharacteristic
44C3⋊(C3⋊Dic3)—21no—
45C3×C3⋊Dic3—42no—
46C3×C3⋊Dic3C2×Q811yescharacteristic
47C3×C3⋊Dic3C2×Q811yescharacteristic
48C32×Dic3—21no—
49C32×Dic3—42no—
50#890—92no—
51#889—182no—
52C3×#107—91no—
53#818—91no—
54C8×Dic3—91no—
55#808—92no—
56C2×C32×C4—31no—
57C2×C32×C4—62no—
58C2×C3⋊Dic3—22no—
59C2×C3⋊Dic3—31no—
60C2×C3⋊Dic3—62no—
61C2×C3⋊Dic3—122no—
62C2×C3⋊Dic3C22×D311yescharacteristic
63C2×C3×Dic3—22no—
64C2×C3×Dic3—22no—
65C2×C3×Dic3—22no—
66C2×C3×Dic3—42no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—122no—
71#622—32no—
72#622Dic612yes—
73#499—271noSylow
74C2×C33C4⋊Q811yescharacteristic
75C2×C33#13511yescharacteristic
76C2×C33#13511yescharacteristic
77C2×C3×C8—91no—
78C3×C4⋊C4—92no—
79C3×C4⋊C4—182no—
80C4⋊Dic3—62no—
81C4⋊Dic3—92no—
82C4⋊Dic3—182no—
83Dic3⋊C4—62no—
84Dic3⋊C4—62no—
85Dic3⋊C4—182no—
86Dic3⋊C4—182no—
87Dic3⋊C4—182no—
88Dic3⋊C4—362no—
89C4×Dic3—91no—
90C4×Dic3—182no—
91C2×C3⋊C8—91no—
92C22×C32—22no—
93C22×C32—42no—
94C22×C32D3×D411yescharacteristic
95C32×C4—61no—
96C32×C4—122no—
97C3⋊Dic3—42no—
98C3⋊Dic3—61no—
99C3⋊Dic3—122no—
100C3⋊Dic3—242no—
101C3⋊Dic3C2×Dic611yescharacteristic
102C3⋊Dic3D3×Q811yescharacteristic
103C3×Dic3—42no—
104C3×Dic3—42no—
105C3×Dic3—42no—
106C3×Dic3—62no—
107C3×Dic3—62no—
108C3×Dic3—82no—
109C3×Dic3—122no—
110C3×Dic3—124no—
111C3×Dic3—122no—
112#125—272no—
113#107—91no—
114#107—271no—
115#106—272no—
116C4×C8—271no—
117C33#49911yescharacteristic, complemented, Sylow
118C2×C3×C4—62no—
119C2×C3×C4—62no—
120C2×C3×C4—62no—
121C2×C3×C4—91no—
122C2×C3×C4—122no—
123C2×C3×C4—182no—
124C2×Dic3—22no—
125C2×Dic3—62no—
126C2×Dic3—62no—
127C2×Dic3—62no—
128C2×Dic3—91no—
129C2×Dic3—122no—
130C2×Dic3—182no—
131C2×Dic3—362no—
132C2×Dic3GO+(4,𝔽2)11yescharacteristic
133C3×C8—92no—
134C3⋊C8—92no—
135C2×C32—22no—
136C2×C32—22no—
137C2×C32—22no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32#88811yescharacteristic
142C2×C32#91211yescharacteristic
143C2×C32#91911yescharacteristic
144C2×C8—91no—
145C2×C8—271no—
146C4⋊C4—92no—
147C4⋊C4—182no—
148C4⋊C4—272no—
149C4⋊C4—542no—
150C42—271no—
151C22×C3—22no—
152C22×C3—42no—
153C22×C3C2×GO+(4,𝔽2)11yescharacteristic
154C3×C4—91no—
155C3×C4—91no—
156C3×C4—122no—
157C3×C4—122no—
158C3×C4—122no—
159C3×C4—242no—
160C3×C4—362no—
161Dic3—21no—
162Dic3—42no—
163Dic3—62no—
164Dic3—62no—
165Dic3—122no—
166Dic3—124no—
167Dic3—122no—
168Dic3—181no—
169Dic3—362no—
170Dic3—722no—
171C32—22no—
172C32—42no—
173C32#487811yescharacteristic, complemented
174C2×C4—31no—
175C2×C4—62no—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4—271no—
179C8—92no—
180C8—272no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3#6422211yescharacteristic
188C2×C3#6422311yescharacteristic
189C2×C3#6423711yescharacteristic
190C22D3×GO+(4,𝔽2)11yescharacteristic
191C4—61no—
192C4—91no—
193C4—91no—
194C4—122no—
195C4—362no—
196C4—541no—
197C3—22no—
198C3—42no—
199C3#1059339011yescharacteristic, complemented
200C2#1172412511yescharacteristic
201C2#1172413411yescharacteristic
202C2#1172413511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#499
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
117. C33#499semidirect
173. C32#4878semidirect
199. C3#10593390semidirect

Maximal subgroups

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

subgroupindex
#117230152
#117227972
#117227782
C3×#637432
Dic3×#6222
#105933903
#48789