Finite Group Databasebeta

← #13147419

Subgroups — #13147419

Order 1728.

565 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 63 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×#88748C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic, complemented
3C2×#88626C211yesmaximal, characteristic, complemented
4#11720784C214yesmaximal, complemented
5C2×Q8×H(3,𝔽3)C411yescharacteristic
6#88748—22no—
7#88748—22no—
8C2×C4⋊#272C412yes—
9C2×C4⋊#272C411yescharacteristic
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×#6188C2212yes—
13C2×#6188C2212yes—
14#88627—22no—
15#88626—22no—
16C2×C3×#239—62no—
17C22×#272C2×C412yes—
18C22×#272C2×C411yescharacteristic
19Q8×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×#1084C2311yescharacteristic
23C4⋊#272—24no—
24C4⋊#272—22no—
25C4×#272—22no—
26C4×#272—22no—
27C4×#272D412yes—
28C4×#272D412yes—
29#6188—22no—
30#6188—22no—
31C3×#408—91nomaximal
32C2×C32×Q8—22no—
33C3×#239—68no—
34C3×#239—68no—
35C2×C3×D12—64no—
36C2×C3×D12—62no—
37C2×C3×C4×D3—62no—
38C2×C3×C4×D3—64no—
39C22×H(3,𝔽3)C22×C411yescharacteristic
40C2×#272—22no—
41C2×#272—22no—
42C2×#272—24no—
43C2×#272C22⋊C412yes—
44C4×H(3,𝔽3)—22no—
45C4×H(3,𝔽3)C22⋊C412yes—
46#1084C2×D411yescharacteristic
47#1084C2×D411yescharacteristic
48C2×C3×#41—91no—
49C2×#239—62no—
50C2×C3×M16—91no—
51C22×C3×C8—91no—
52C3×C22⋊C8—94no—
53C22×C3×D3—62no—
54C22×C3×D3—64no—
55C32×Q8—28no—
56C2×C32×C4—24no—
57C2×C32×C4—22no—
58C2×C3×Dic3—62no—
59C3×D12—616no—
60C3×D12—68no—
61C3×C4×D3—64no—
62C3×C4×D3—64no—
63C3×C4×D3—68no—
64C3×C4×D3—68no—
65#408—91noSylow
66C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
67C2×H(3,𝔽3)#13112yes—
68#272—22no—
69#272—42no—
70C3×#41—182no—
71C3×#41—182no—
72C2×C3×Q8—62no—
73C2×C3×Q8#15711yescharacteristic
74C2×C3×D4—92no—
75C2×C3×D4—91no—
76C22×C3×C4—91no—
77C22×C3×C4—92no—
78#239—68no—
79#239—68no—
80C2×D12—64no—
81C2×D12—62no—
82C2×C4×D3—62no—
83C2×C4×D3—64no—
84C3×M16—182no—
85C2×C3×C8—92no—
86C2×C3×C8—92no—
87C2×C3×C8—182no—
88C22×C32—22no—
89C2×C3×D3—64no—
90C2×C3×D3—68no—
91C2×C3×D3—124no—
92C2×C3×D3—128no—
93C32×C4—28no—
94C32×C4—24no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C2×#41—91no—
98C2×M16—91no—
99C22×C8—91no—
100C22⋊C8—94no—
101H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
102C23×C3—91no—
103C23×C3—92no—
104C22×D3—62no—
105C22×D3—64no—
106C3×Q8—22no—
107C3×Q8—68no—
108C3×D4—184no—
109C3×D4—182no—
110C2×C3×C4—64no—
111C2×C3×C4—62no—
112C2×C3×C4—91no—
113C2×C3×C4—92no—
114C2×C3×C4—92no—
115C2×C3×C4—182no—
116C2×C3×C4—182no—
117C2×C3×C4C2×#15712yes—
118C2×C3×C4C2×#15711yescharacteristic
119C2×Dic3—62no—
120D12—616no—
121D12—68no—
122C4×D3—64no—
123C4×D3—64no—
124C4×D3—68no—
125C4×D3—68no—
126C3×C8—182no—
127C3×C8—182no—
128C2×C32—24no—
129C2×C32—22no—
130C3×D3—124no—
131C3×D3—128no—
132#41—182no—
133#41—182no—
134C2×Q8#108811yescharacteristic
135C2×D4—92no—
136C2×D4—91no—
137C22×C4—91no—
138C22×C4—92no—
139M16—182no—
140C2×C8—92no—
141C2×C8—92no—
142C2×C8—182no—
143C22×C3—62no—
144C22×C3—92no—
145C22×C3—182no—
146C22×C3—184no—
147C22×C3—182no—
148C22×C3C22×#15711yescharacteristic
149C2×D3—64no—
150C2×D3—68no—
151C2×D3—124no—
152C2×D3—128no—
153C3×C4—22no—
154C3×C4—68no—
155C3×C4—64no—
156C3×C4—91no—
157C3×C4—91no—
158C3×C4C22⋊#15712yes—
159Dic3—62no—
160Dic3—62no—
161C32—22no—
162C23—91no—
163C23—92no—
164Q8—22no—
165D4—184no—
166D4—182no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—182no—
171C2×C4—182no—
172C2×C4C2×#108812yes—
173C2×C4C2×#108811yescharacteristic
174C8—182no—
175C8—182no—
176C2×C3—64no—
177C2×C3—62no—
178C2×C3—182no—
179C2×C3—362no—
180C2×C3#6428912yes—
181C2×C3C2×C22⋊#15711yescharacteristic
182D3—124no—
183D3—128no—
184C22—92no—
185C22—182no—
186C22—184no—
187C22—182no—
188C22C22×#108811yescharacteristic
189C4—22no—
190C4—91no—
191C4—91no—
192C4C22⋊#108812yes—
193C3—62no—
194C3#1059383411yescharacteristic
195C2—182no—
196C2—362no—
197C2#1172245112yes—
198C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×#88748C2
cosocleC2×#1084C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88627C2semidirect
3. C2×#88626C2semidirect
4. #11720784C2semidirect
101. H(3,𝔽3)#408semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#886272
C2×#886262
#117207842
C3×#4089