Finite Group Databasebeta

← #13146671

Subgroups — #13146671

Order 1728.

322 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 43 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#11721309C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3C22⋊#6188C211yesmaximal, characteristic
4#11720704C212yesmaximal, complemented
5#11720702C212yesmaximal, complemented
6C22×#1084C2211yescharacteristic
7C22×(C3⋊Dic3)⋊C3—21no—
8C2×#6196C2211yescharacteristic
9C2×#6188C2212yes—
10#3757⋊C3—21no—
11#88449—21no—
12#88449C2211yescharacteristic
13#88449C2212yes—
14#88448—21no—
15C2×Dic3⋊Dic3—61no—
16#63964—61no—
17C23×H(3,𝔽3)D411yescharacteristic
18C2×#1084—22no—
19C2×#1084D411yescharacteristic
20C2×#1084C2311yescharacteristic
21C2×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊Dic3)⋊C3—21no—
23C2×(C3⋊Dic3)⋊C3—22no—
24C2×(C3⋊Dic3)⋊C3—42no—
25#6196—22no—
26#6188—22no—
27#4722—91nomaximal
28C22×C3⋊Dic3—21no—
29C22×C3×Dic3—61no—
30C22×C3×Dic3—61no—
31C22×C3×Dic3—61no—
32C2×#627—61no—
33#3757—21no—
34C3×#217—61no—
35#3724—61no—
36Dic3⋊Dic3—61no—
37Dic3⋊Dic3—62no—
38Dic3⋊Dic3—61no—
39Dic3⋊Dic3—62no—
40C22×H(3,𝔽3)—21no—
41C22×H(3,𝔽3)—21no—
42C22×H(3,𝔽3)Q1612yes—
43C22×H(3,𝔽3)C2×D411yescharacteristic
44#1084—21no—
45#1084#4112yes—
46(C3⋊Dic3)⋊C3—41no—
47(C3⋊Dic3)⋊C3—41no—
48(C3⋊Dic3)⋊C3—42no—
49C2×C4⋊Dic3—91no—
50#924—91no—
51C2×Dic3⋊C4—181no—
52#878—181no—
53C3×C22⋊C8—91no—
54#818—92no—
55#816—92no—
56C23×C32—21no—
57C23×C32—21no—
58C2×C3⋊Dic3—22no—
59C2×C3⋊Dic3—21no—
60C2×C3⋊Dic3—21no—
61C2×C3⋊Dic3—42no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—61no—
70C2×C3×Dic3—122no—
71C2×C3×Dic3—122no—
72#627—122no—
73#484—271noSylow
74C2×H(3,𝔽3)—22no—
75C2×H(3,𝔽3)#12311yescharacteristic
76C2×H(3,𝔽3)C2×Q1611yescharacteristic
77C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
78C22×C3×C4—91no—
79C22×C3×C4—181no—
80C22×Dic3—61no—
81C22×Dic3—61no—
82C22×Dic3—61no—
83C22×Dic3—181no—
84C2×Dic6—91no—
85C2×Dic6—181no—
86C2×C3×C8—92no—
87C3×C22⋊C4—181no—
88#217—61no—
89#217—181no—
90C4⋊Dic3—92no—
91C4⋊Dic3—91no—
92C4⋊Dic3—181no—
93C4⋊Dic3—181no—
94Dic3⋊C4—181no—
95Dic3⋊C4—182no—
96Dic3⋊C4—181no—
97Dic3⋊C4—181no—
98Dic3⋊C4—182no—
99C22×C32—22no—
100C22×C32—22no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—41no—
106C22×C32—41no—
107C3⋊Dic3—41no—
108C3⋊Dic3—41no—
109C3⋊Dic3—42no—
110C3×Dic3—62no—
111C3×Dic3—62no—
112C3×Dic3—62no—
113C3×Dic3—121no—
114C3×Dic3—121no—
115C3×Dic3—121no—
116C3×Dic3—122no—
117C22⋊Q8—271no—
118C2×C4⋊C4—271no—
119#107—272no—
120#103—272no—
121C22⋊C8—91no—
122H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
123C23×C3—61no—
124C23×C3—61no—
125C23×C3GO+(4,𝔽2)11yescharacteristic
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—182no—
129C2×C3×C4—182no—
130C2×C3×C4—181no—
131C2×C3×C4—181no—
132C2×C3×C4—362no—
133C2×Dic3—61no—
134C2×Dic3—61no—
135C2×Dic3—61no—
136C2×Dic3—61no—
137C2×Dic3—64no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140C2×Dic3—62no—
141C2×Dic3—122no—
142C2×Dic3—122no—
143C2×Dic3—181no—
144C2×Dic3—182no—
145C2×Dic3—181no—
146C2×Dic3—362no—
147Dic6—182no—
148Dic6—362no—
149C3×C8—182no—
150C2×C32—24no—
151C2×C32—21no—
152C2×C32—21no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—42no—
158C2×Q8—271no—
159C22×C4—91no—
160C22×C4—181no—
161C2×C8—92no—
162C4⋊C4—272no—
163C4⋊C4—271no—
164C4⋊C4—541no—
165C4⋊C4—541no—
166C22⋊C4—181no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—62no—
170C22×C3—62no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—61no—
174C22×C3—61no—
175C22×C3—121no—
176C22×C3—121no—
177C22×C3#377612yes—
178C22×C3C2×GO+(4,𝔽2)11yescharacteristic
179C3×C4—92no—
180C3×C4—181no—
181C3×C4—361no—
182C3×C4—362no—
183C3×C4—361no—
184Dic3—62no—
185Dic3—62no—
186Dic3—62no—
187Dic3—121no—
188Dic3—121no—
189Dic3—122no—
190Dic3—121no—
191Dic3—361no—
192Dic3—362no—
193Dic3—361no—
194C32—21no—
195C32—21no—
196C23#1088⋊C211yescharacteristic
197Q8—542no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—182no—
201C2×C4—181no—
202C2×C4—181no—
203C2×C4—182no—
204C2×C4—362no—
205C8—182no—
206C2×C3—22no—
207C2×C3—64no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—61no—
212C2×C3—61no—
213C2×C3—61no—
214C2×C3—122no—
215C2×C3#6423311yescharacteristic
216C2×C3#6423611yescharacteristic
217C2×C3C2×#377611yescharacteristic
218C22—21no—
219C22—21no—
220C22#8858912yes—
221C22C2×#1088⋊C211yescharacteristic
222C4—92no—
223C4—181no—
224C4—362no—
225C4—361no—
226C4—361no—
227C3—61no—
228C3—61no—
229C3#1059350011yescharacteristic
230C2—22no—
231C2#1172230911yescharacteristic
232C2C2×#8858911yescharacteristic
233C2#1172231411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11720704C2semidirect
5. #11720702C2semidirect
122. H(3,𝔽3)#484semidirect

Maximal subgroups

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

subgroupindex
#117213092
C2×#884492
C22⋊#61882
#117207042
#117207022
#47229