Finite Group Databasebeta

← #13146613

Subgroups — #13146613

Order 1728.

439 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 65 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×C4×#1088C211yesmaximal, characteristic
2C2×#88590C211yesmaximal, characteristic
3C2×#88590C211yesmaximal, characteristic
4#11721147C212yesmaximal
5#11720707C212yesmaximal
6C22×#1088C2211yescharacteristic
7C22×#1088C2211yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C2×#6199C2212yes—
10C2×#6199C2212yes—
11C4×#1088—22no—
12#88590—22no—
13#88590C414yescomplemented
14#3750⋊C3—22no—
15#88448—22no—
16D3×Dic3⋊C4—62no—
17C22×#272C2311yescharacteristic
18C2×#1088—21no—
19C2×#1088—21no—
20C2×#1088C2×C412yes—
21C2×#1088D412yes—
22C2×#1088Q812yes—
23C2×#1088Q812yes—
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×#1084D411yescharacteristic
26C4×#272—21no—
27C4×#272—21no—
28C2×(C3⋊Dic3)⋊C3—22no—
29C2×(C3⋊Dic3)⋊C3—22no—
30#6199—22no—
31#6199—22no—
32#6199—22no—
33#6199C2×C414yes—
34#4928—91nomaximal
35C2×C3×C4×D3—62no—
36C2×D3×Dic3—62no—
37C2×D3×Dic3—62no—
38#3750—22no—
39C3×Dic3⋊C4—62no—
40#3724—62no—
41Dic3⋊Dic3—62no—
42C22×H(3,𝔽3)C2×D411yescharacteristic
43C2×#272C22×C411yescharacteristic
44C2×#272C2×D411yescharacteristic
45C2×#272C2×Q811yescharacteristic
46C2×#272C2×Q811yescharacteristic
47C2×#272#4111yescharacteristic
48C2×#272#4111yescharacteristic
49#1088—22no—
50#1088C4⋊C414yescomplemented
51C4×H(3,𝔽3)—21no—
52#1084—21no—
53(C3⋊Dic3)⋊C3—24no—
54(C3⋊Dic3)⋊C3—42no—
55C2×C3×C42—91no—
56C2×C4⋊Dic3—91no—
57C2×C4⋊Dic3—91no—
58C2×Dic3⋊C4—92no—
59D3×C4⋊C4—182no—
60C3⋊#95—92no—
61C22×C3×D3—62no—
62C2×C32×C4—22no—
63C2×C3⋊Dic3—22no—
64C2×C3⋊Dic3—22no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—62no—
68C3×C4×D3—122no—
69C3×C4×D3—122no—
70D3×Dic3—64no—
71D3×Dic3—64no—
72D3×Dic3—122no—
73D3×Dic3—122no—
74#389—271noSylow
75C2×H(3,𝔽3)C4×D411yescharacteristic
76C2×H(3,𝔽3)#12111yescharacteristic
77C2×H(3,𝔽3)C4⋊Q811yescharacteristic
78#272C2×C4⋊C411yescharacteristic
79#272C4×Q811yescharacteristic
80#272C22⋊Q811yescharacteristic
81#272#12511yescharacteristic
82C22×C3×C4—91no—
83C22×C3×C4—91no—
84C22×C3×C4—91no—
85C22×Dic3—92no—
86C22×Dic3—92no—
87C2×C4×D3—62no—
88C2×C4×D3—182no—
89C2×C4×D3—182no—
90C3×C4⋊C4—182no—
91C3×C42—182no—
92C4⋊Dic3—94no—
93C4⋊Dic3—182no—
94C4⋊Dic3—182no—
95Dic3⋊C4—62no—
96Dic3⋊C4—182no—
97Dic3⋊C4—182no—
98Dic3⋊C4—182no—
99C22×C32—22no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—62no—
104C2×C3×D3—62no—
105C2×C3×D3—62no—
106C32×C4—42no—
107C3⋊Dic3—24no—
108C3⋊Dic3—42no—
109C3×Dic3—64no—
110C3×Dic3—122no—
111C3×Dic3—122no—
112C2×C4⋊C4—272no—
113C2×C4⋊C4—271no—
114C2×C4⋊C4—271no—
115C2×C42—91no—
116#95—272no—
117H(3,𝔽3)#38911yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C22×D3—62no—
120C2×C3×C4—62no—
121C2×C3×C4—92no—
122C2×C3×C4—92no—
123C2×C3×C4—92no—
124C2×C3×C4—92no—
125C2×C3×C4—91no—
126C2×C3×C4—181no—
127C2×C3×C4—181no—
128C2×C3×C4—182no—
129C2×C3×C4—181no—
130C2×C3×C4—181no—
131C2×C3×C4—182no—
132C2×C3×C4GO+(4,𝔽2)11yescharacteristic
133C2×Dic3—62no—
134C2×Dic3—62no—
135C2×Dic3—62no—
136C2×Dic3—94no—
137C2×Dic3—182no—
138C2×Dic3—182no—
139C2×Dic3—182no—
140C2×Dic3—182no—
141C2×Dic3—182no—
142C4×D3—122no—
143C4×D3—122no—
144C4×D3—184no—
145C4×D3—184no—
146C4×D3—362no—
147C4×D3—362no—
148C2×C32—22no—
149C2×C32—22no—
150C2×C32—22no—
151C3×D3—62no—
152C3×D3—62no—
153C3×D3—62no—
154C3×D3—62no—
155C22×C4—91no—
156C22×C4—91no—
157C22×C4—91no—
158C22×C4—272no—
159C22×C4—272no—
160C4⋊C4—182no—
161C4⋊C4—274no—
162C4⋊C4—542no—
163C4⋊C4—542no—
164C42—182no—
165C22×C3—62no—
166C22×C3—91no—
167C22×C3—91no—
168C22×C3—91no—
169C22×C3—91no—
170C22×C3—91no—
171C22×C3—91no—
172C22×C3C2×GO+(4,𝔽2)11yescharacteristic
173C2×D3—62no—
174C2×D3—62no—
175C2×D3—62no—
176C2×D3—62no—
177C2×D3—62no—
178C2×D3—62no—
179C3×C4—21no—
180C3×C4—94no—
181C3×C4—122no—
182C3×C4—184no—
183C3×C4—182no—
184C3×C4—181no—
185C3×C4—362no—
186Dic3—64no—
187Dic3—122no—
188Dic3—122no—
189Dic3—184no—
190Dic3—362no—
191C32—22no—
192C23—91no—
193C2×C4—92no—
194C2×C4—92no—
195C2×C4—92no—
196C2×C4—92no—
197C2×C4—91no—
198C2×C4—181no—
199C2×C4—181no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—182no—
203C2×C4—182no—
204C2×C4—274no—
205C2×C4—542no—
206C2×C4—542no—
207C2×C4—542no—
208C2×C4#1088⋊C211yescharacteristic
209C2×C3—62no—
210C2×C3—62no—
211C2×C3—62no—
212C2×C3—91no—
213C2×C3—91no—
214C2×C3—91no—
215C2×C3—91no—
216C2×C3#6422211yescharacteristic
217C2×C3C4×GO+(4,𝔽2)11yescharacteristic
218C2×C3#6424211yescharacteristic
219D3—62no—
220D3—62no—
221D3—62no—
222D3—62no—
223C22—91no—
224C22—91no—
225C22—91no—
226C22—91no—
227C22—91no—
228C22—91no—
229C22C2×#1088⋊C211yescharacteristic
230C4—21no—
231C4—94no—
232C4—182no—
233C4—184no—
234C4—181no—
235C4—362no—
236C3—62no—
237C3#1059341011yescharacteristic
238C2—91no—
239C2—91no—
240C2—91no—
241C2—91no—
242C2#1172230011yescharacteristic
243C2C4×#1088⋊C211yescharacteristic
244C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutatorC2×#272C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical, cosocleC22×#272C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#389
LCSG ▷ C2×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. #88590C4semidirect
50. #1088C4⋊C4semidirect
117. H(3,𝔽3)#389semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
C2×#885902
#117211472
#117207072
#49289