Finite Group Databasebeta

← #13147409

Subgroups — #13147409

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×#88744C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic, complemented
3C2×#88626C211yesmaximal, characteristic, complemented
4C22⋊#6188C214yesmaximal, complemented
5C2×D4×H(3,𝔽3)C411yescharacteristic
6C2×#6235C412yes—
7C22×#1084C2212yes—
8#88744—22no—
9#88744—22no—
10C2×C4×#272C2211yescharacteristic
11C2×#6229C411yescharacteristic
12C2×#6188C2212yes—
13C2×#6188C2212yes—
14#88627—22no—
15#88626—22no—
16C2×C3×#237—62no—
17C23×H(3,𝔽3)C2×C412yes—
18C22×#272C2×C411yescharacteristic
19D4×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21#6235—24no—
22C2×#1084—22no—
23C2×#1084—22no—
24C2×#1084C2×C412yes—
25C2×#1084C2311yescharacteristic
26C4×#272D412yes—
27C4×#272D412yes—
28#6229—22no—
29#6188—22no—
30#6188—22no—
31C3×#408—91nomaximal
32C2×C32×D4—22no—
33C2×C3×#67—64no—
34C22×C3×Dic3—64no—
35C3×#237—68no—
36C3×#237—68no—
37C2×C3×C4×D3—62no—
38C2×C3×Dic6—62no—
39C22×H(3,𝔽3)—22no—
40C22×H(3,𝔽3)—24no—
41C22×H(3,𝔽3)C22×C411yescharacteristic
42C2×#272—22no—
43C2×#272C22⋊C412yes—
44C4×H(3,𝔽3)C22⋊C412yes—
45#1084—22no—
46#1084C2×D411yescharacteristic
47#1084C2×D411yescharacteristic
48C2×C3×#41—91no—
49C2×#237—62no—
50C2×C3×M16—91no—
51C22×C3×C8—91no—
52C3×C22⋊C8—94no—
53C23×C32—24no—
54C22×C3×D3—62no—
55C32×D4—28no—
56C2×C32×C4—22no—
57C3×#67—616no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—62no—
60C2×C3×Dic3—68no—
61C2×C3×Dic3—68no—
62C3×C4×D3—64no—
63C3×C4×D3—64no—
64C3×Dic6—68no—
65#408—91noSylow
66C2×H(3,𝔽3)—42no—
67C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
68C2×H(3,𝔽3)#13112yes—
69#272—22no—
70C3×#41—182no—
71C3×#41—182no—
72C2×C3×Q8—91no—
73C2×C3×D4—62no—
74C2×C3×D4—92no—
75C2×C3×D4#15711yescharacteristic
76C22×C3×C4—91no—
77C22×C3×C4—92no—
78C2×#67—64no—
79C22×Dic3—64no—
80#237—68no—
81#237—68no—
82C2×C4×D3—62no—
83C2×Dic6—62no—
84C3×M16—182no—
85C2×C3×C8—92no—
86C2×C3×C8—92no—
87C2×C3×C8—182no—
88C22×C32—28no—
89C22×C32—22no—
90C22×C32—48no—
91C2×C3×D3—64no—
92C2×C3×D3—124no—
93C32×C4—24no—
94C3×Dic3—62no—
95C3×Dic3—62no—
96C3×Dic3—68no—
97C2×#41—91no—
98C2×M16—91no—
99C22×C8—91no—
100C22⋊C8—94no—
101H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
102C23×C3—64no—
103C23×C3—91no—
104C23×C3C2×#15712yes—
105C22×D3—62no—
106C3×Q8—182no—
107C3×D4—22no—
108C3×D4—68no—
109C3×D4—184no—
110C2×C3×C4—62no—
111C2×C3×C4—91no—
112C2×C3×C4—92no—
113C2×C3×C4—92no—
114C2×C3×C4—92no—
115C2×C3×C4—182no—
116C2×C3×C4—182no—
117C2×C3×C4C2×#15711yescharacteristic
118#67—616no—
119C2×Dic3—64no—
120C2×Dic3—62no—
121C2×Dic3—68no—
122C2×Dic3—68no—
123C4×D3—64no—
124C4×D3—64no—
125Dic6—68no—
126C3×C8—182no—
127C3×C8—182no—
128C2×C32—24no—
129C2×C32—22no—
130C2×C32—48no—
131C3×D3—124no—
132#41—182no—
133#41—182no—
134C2×Q8—91no—
135C2×D4—92no—
136C2×D4#108811yescharacteristic
137C22×C4—91no—
138C22×C4—92no—
139M16—182no—
140C2×C8—92no—
141C2×C8—92no—
142C2×C8—182no—
143C22×C3—24no—
144C22×C3—22no—
145C22×C3—68no—
146C22×C3—62no—
147C22×C3—92no—
148C22×C3—128no—
149C22×C3—182no—
150C22×C3C22×#15711yescharacteristic
151C2×D3—64no—
152C2×D3—124no—
153C3×C4—64no—
154C3×C4—91no—
155C3×C4—91no—
156C3×C4—182no—
157C3×C4C22⋊#15712yes—
158Dic3—62no—
159Dic3—62no—
160Dic3—68no—
161C32—22no—
162C23—91no—
163C23C2×#108812yes—
164Q8—182no—
165D4—22no—
166D4—184no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—92no—
171C2×C4—182no—
172C2×C4—182no—
173C2×C4C2×#108811yescharacteristic
174C8—182no—
175C8—182no—
176C2×C3—42no—
177C2×C3—64no—
178C2×C3—62no—
179C2×C3—128no—
180C2×C3—182no—
181C2×C3#6428712yes—
182C2×C3C2×C22⋊#15711yescharacteristic
183D3—124no—
184C22—24no—
185C22—22no—
186C22—92no—
187C22—182no—
188C22C22×#108811yescharacteristic
189C4—91no—
190C4—91no—
191C4—182no—
192C4C22⋊#108812yes—
193C3—62no—
194C3#1059382311yescharacteristic
195C2—42no—
196C2—182no—
197C2#1172244912yes—
198C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×#88744C2
cosocleC2×#1084C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88627C2semidirect
3. C2×#88626C2semidirect
4. C22⋊#6188C2semidirect
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×#887442
C2×#886272
C2×#886262
C22⋊#61882
C3×#4089