Finite Group Databasebeta

← #13147342

Subgroups — #13147342

Order 1728.

296 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 51 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#11721760C211yesmaximal, characteristic
2#11720785C211yesmaximal, characteristic, complemented
3#11720784C211yesmaximal, characteristic
4#11720784C211yesmaximal, characteristic
5#11720783C211yesmaximal, characteristic, complemented
6C4⋊#6188C211yesmaximal, characteristic, complemented
7C4×#6188C211yesmaximal, characteristic, complemented
8C2×C4×#272C2211yescharacteristic
9C2×#6188C2211yescharacteristic
10C2×#6188C2211yescharacteristic
11C2×#6188C2211yescharacteristic
12C2×#6188C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C411yescharacteristic
14#88542C411yescharacteristic
15#88542C411yescharacteristic
16C4⋊#1084C411yescharacteristic
17C4×#1084C2211yescharacteristic
18C4×#1084C2211yescharacteristic
19C3×#892—62no—
20C22×#272C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×#1084C2×C411yescharacteristic
25C2×#1084C2×C411yescharacteristic
26C2×#1084C2311yescharacteristic
27C4×#272—21no—
28C4×#272—21no—
29#6188—21no—
30#6188—21no—
31#6188—21no—
32#6188—21no—
33C3×#433—91nomaximal
34C2×C3×C4×D3—62no—
35C32×C4⋊C4—22no—
36C3×(C2×D3)⋊C4—62no—
37C3×(C2×D3)⋊C4—62no—
38C3×C4⋊Dic3—62no—
39C3×C4×Dic3—62no—
40C3×C4×Dic3—62no—
41C22×H(3,𝔽3)C22×C411yescharacteristic
42C2×#272—21no—
43C2×#272—21no—
44C2×#272—21no—
45C4×H(3,𝔽3)—21no—
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—21no—
48#1084—21no—
49#1084—21no—
50#1084#4111yescharacteristic
51#1084#4111yescharacteristic
52C3×#117—91no—
53#892—62no—
54C3×C4⋊C8—91no—
55C3×C4⋊C8—91no—
56C3×C22⋊C8—91no—
57C3×C22⋊C8—91no—
58C3×#97—91no—
59C3×C4×C8—91no—
60C22×C3×D3—62no—
61C2×C32×C4—22no—
62C2×C32×C4—22no—
63C2×C32×C4—22no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—62no—
67C3×C4×D3—64no—
68C3×C4×D3—64no—
69#433—91noSylow
70C2×H(3,𝔽3)#11711yescharacteristic
71C2×H(3,𝔽3)#13111yescharacteristic
72C2×H(3,𝔽3)#13111yescharacteristic
73#272—41no—
74C22×C3×C4—91no—
75C2×C4×D3—62no—
76C2×C3×C8—91no—
77C2×C3×C8—91no—
78C2×C3×C8—91no—
79C2×C3×C8—91no—
80C3×C4⋊C4—62no—
81C3×C4⋊C4—91no—
82C3×C4⋊C4#15711yescharacteristic
83C3×C22⋊C4—91no—
84C3×C22⋊C4—91no—
85C3×C42—91no—
86C3×C42—91no—
87(C2×D3)⋊C4—62no—
88(C2×D3)⋊C4—62no—
89C4⋊Dic3—62no—
90C4×Dic3—62no—
91C4×Dic3—62no—
92C22×C32—22no—
93C2×C3×D3—64no—
94C2×C3×D3—122no—
95C2×C3×D3—122no—
96C32×C4—24no—
97C32×C4—42no—
98C32×C4—42no—
99C3×Dic3—62no—
100C3×Dic3—62no—
101C3×Dic3—122no—
102C3×Dic3—122no—
103#117—91no—
104C4⋊C8—91no—
105C4⋊C8—91no—
106C22⋊C8—91no—
107C22⋊C8—91no—
108#97—91no—
109C4×C8—91no—
110H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
111C23×C3—91no—
112C22×D3—62no—
113C2×C3×C4—62no—
114C2×C3×C4—62no—
115C2×C3×C4—62no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4—181no—
120C2×C3×C4—181no—
121C2×C3×C4C2×#15711yescharacteristic
122C2×C3×C4C2×#15711yescharacteristic
123C2×C3×C4C2×#15711yescharacteristic
124C2×Dic3—62no—
125C2×Dic3—62no—
126C2×Dic3—62no—
127C4×D3—64no—
128C4×D3—64no—
129C3×C8—181no—
130C3×C8—181no—
131C3×C8—181no—
132C3×C8—181no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—22no—
136C3×D3—124no—
137C22×C4—91no—
138C2×C8—91no—
139C2×C8—91no—
140C2×C8—91no—
141C2×C8—91no—
142C4⋊C4—91no—
143C4⋊C4#108811yescharacteristic
144C22⋊C4—91no—
145C22⋊C4—91no—
146C42—91no—
147C42—91no—
148C22×C3—62no—
149C22×C3—181no—
150C22×C3—181no—
151C22×C3—181no—
152C22×C3C22×#15711yescharacteristic
153C2×D3—64no—
154C2×D3—122no—
155C2×D3—122no—
156C3×C4—21no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—64no—
160C3×C4—91no—
161C3×C4—91no—
162C3×C4—122no—
163C3×C4—122no—
164C3×C4—181no—
165C3×C4—181no—
166Dic3—62no—
167Dic3—62no—
168Dic3—122no—
169Dic3—122no—
170C32—22no—
171C23—91no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—181no—
176C2×C4—181no—
177C2×C4C2×#108811yescharacteristic
178C2×C4C2×#108811yescharacteristic
179C2×C4C2×#108811yescharacteristic
180C8—181no—
181C8—181no—
182C8—181no—
183C8—181no—
184C2×C3—62no—
185C2×C3—62no—
186C2×C3—62no—
187C2×C3—361no—
188C2×C3#6428611yescharacteristic
189C2×C3#6428711yescharacteristic
190C2×C3#6428911yescharacteristic
191D3—124no—
192C22—181no—
193C22—181no—
194C22—181no—
195C22C22×#108811yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—91no—
200C4—91no—
201C4—181no—
202C4—181no—
203C3—62no—
204C3#1059375311yescharacteristic
205C2—361no—
206C2#1172244811yescharacteristic
207C2#1172244911yescharacteristic
208C2#1172245111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC4⋊C4×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×#1084C23
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11720785C2semidirect
5. #11720783C2semidirect
6. C4⋊#6188C2semidirect
7. C4×#6188C2semidirect
110. H(3,𝔽3)#433semidirect

Maximal subgroups

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

subgroupindex
#117217602
#117207852
#117207842
#117207832
C4⋊#61882
C4×#61882
C3×#4339