Finite Group Databasebeta

← #13165622

Subgroups — #13165622

Order 1728.

444 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 54 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×C3×#3776C211yesmaximal, characteristic, complemented
2C2×#88924C211yesmaximal, characteristic
3#11722977C212yesmaximal
4#11722797C211yesmaximal, characteristic, complemented
5#11722775C212yesmaximal, complemented
6#10593446—31nomaximal
7C2×#6289C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3776—22no—
12#88924—22no—
13#88815C2212yes—
14#88804—22no—
15C3×#3724—22no—
16C2×#3787—31no—
17C2×#3776D311yescharacteristic, complemented
18#63888—62no—
19#63865—32no—
20#63743—31no—
21#63740—32no—
22C22×C3⋊(C3⋊D3)D411yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C2×C32×Dic3—22no—
26C2×C32×Dic3D411yescharacteristic
27#6289—21no—
28#6289—21no—
29C3×#627—22no—
30C3×#627—22no—
31C3⋊#622—21no—
32C3×#622D412yes—
33#5188—91nomaximal
34C2×C4×C3⋊D3—31no—
35C2×C3×Dic6—62no—
36C2×#627C2×D312yes—
37C2×#624—62no—
38C2×#622—31no—
39C2×#622C2×D311yescharacteristic
40#3787—62no—
41#3776—22no—
42#3752—62no—
43C3×Dic3⋊C4—62no—
44C3×Dic3⋊C4—62no—
45#3724—22no—
46#3724—32no—
47Dic3⋊Dic3—62no—
48#3722—62no—
49#3722—62no—
50#3722—122no—
51C22×C33C2×D411yescharacteristic
52C2×C3⋊(C3⋊D3)—21no—
53C2×C3⋊(C3⋊D3)—21no—
54C2×C3⋊(C3⋊D3)Q1612yescomplemented
55C3×C3⋊Dic3—42no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C32×Dic3—21no—
59C32×Dic3—42no—
60C2×C3×Q16—91no—
61#946—92no—
62C2×D3×C8—91no—
63#896—182no—
64#835—92no—
65#818—91no—
66C22×C3⋊D3—31no—
67C22×C3⋊D3—62no—
68C22×C3⋊D3—122no—
69C2×C32×C4—31no—
70C2×C32×C4—62no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3C22×D311yescharacteristic
73C4×C3⋊D3—61no—
74C4×C3⋊D3—61no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—122no—
83C3×Dic6—122no—
84C3×Dic6—122no—
85#627—22no—
86#627—22no—
87#624—122no—
88#624—122no—
89#622—61no—
90#622#6712yes—
91#467—271noSylow
92C2×C33#12111yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33#13511yescharacteristic
95C3⋊(C3⋊D3)—22no—
96C2×C3×Q8—92no—
97C2×C4×D3—91no—
98C2×C4×D3—182no—
99C2×Dic6—62no—
100C3×Q16—182no—
101C2×C3×C8—91no—
102C3×C4⋊C4—182no—
103(C2×D3)⋊C4—182no—
104(C2×D3)⋊C4—182no—
105(C2×D3)⋊C4—182no—
106(C2×D3)⋊C4—362no—
107C4⋊Dic3—92no—
108Dic3⋊C4—62no—
109Dic3⋊C4—62no—
110Dic3⋊C4—182no—
111C2×C3⋊C8—91no—
112D3×C8—182no—
113C22×C32—22no—
114C22×C32—42no—
115C22×C32D3×D411yescharacteristic
116C2×C3⋊D3—32no—
117C2×C3⋊D3—64no—
118C2×C3⋊D3—61no—
119C2×C3⋊D3—61no—
120C2×C3⋊D3—124no—
121C2×C3⋊D3—122no—
122C2×C3⋊D3—122no—
123C2×C3⋊D3—242no—
124C2×C3⋊D3—242no—
125C32×C4—61no—
126C32×C4—122no—
127C3⋊Dic3—42no—
128C3⋊Dic3#23711yescharacteristic
129C3⋊Dic3C2×#6711yescharacteristic
130C3×Dic3—42no—
131C3×Dic3—42no—
132C3×Dic3—42no—
133C3×Dic3—62no—
134C3×Dic3—62no—
135C3×Dic3—82no—
136C3×Dic3—122no—
137C3×Dic3—122no—
138C3×Dic3—242no—
139C2×Q16—91no—
140C22×C8—271no—
141C22⋊Q8—272no—
142#107—271no—
143#103—272no—
144C33#46711yescharacteristic, complemented, Sylow
145C22×D3—91no—
146C22×D3—182no—
147C22×D3—362no—
148C3×Q8—182no—
149C3×Q8—182no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—91no—
154C2×C3×C4—122no—
155C2×C3×C4—182no—
156C2×Dic3—22no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—122no—
161C2×Dic3GO+(4,𝔽2)11yescharacteristic
162C4×D3—181no—
163C4×D3—181no—
164C4×D3—362no—
165C4×D3—362no—
166Dic6—122no—
167Dic6—122no—
168C3×C8—92no—
169C3⋊C8—181no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32D3×Q1611yescharacteristic
177C2×C32#91911yescharacteristic
178C2×C32#93811yescharacteristic
179C3⋊D3—62no—
180C3⋊D3—124no—
181C3⋊D3—244no—
182C2×Q8—92no—
183C22×C4—271no—
184Q16—182no—
185C2×C8—91no—
186C2×C8—271no—
187C2×C8—542no—
188C4⋊C4—182no—
189C4⋊C4—272no—
190C22⋊C4—542no—
191C22×C3—22no—
192C22×C3—42no—
193C22×C3C2×GO+(4,𝔽2)11yescharacteristic
194C2×D3—92no—
195C2×D3—184no—
196C2×D3—181no—
197C2×D3—181no—
198C2×D3—364no—
199C2×D3—362no—
200C2×D3—362no—
201C2×D3—722no—
202C2×D3—722no—
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—122no—
206C3×C4—122no—
207C3×C4—122no—
208C3×C4—242no—
209C3×C4—362no—
210Dic3—21no—
211Dic3—42no—
212Dic3—62no—
213Dic3—62no—
214Dic3—122no—
215Dic3—122no—
216Dic3—242no—
217C32—22no—
218C32—42no—
219C32#518811yescharacteristic, complemented
220C23—271no—
221Q8—182no—
222Q8—182no—
223C2×C4—31no—
224C2×C4—62no—
225C2×C4—91no—
226C2×C4—182no—
227C2×C4—541no—
228C2×C4—541no—
229C8—92no—
230C8—541no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—42no—
237C2×C3#6422311yescharacteristic
238C2×C3#6422811yescharacteristic
239C2×C3#6424211yescharacteristic
240D3—182no—
241D3—364no—
242D3—724no—
243C22—272no—
244C22—541no—
245C22—541no—
246C22D3×GO+(4,𝔽2)11yescharacteristic
247C4—61no—
248C4—91no—
249C4—91no—
250C4—122no—
251C4—362no—
252C3—22no—
253C3—42no—
254C3#1059344611yescharacteristic, complemented
255C2—542no—
256C2#1172411411yescharacteristic
257C2#1172413111yescharacteristic
258C2#1172413711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypocenterC33#467
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3776C2semidirect
4. #11722797C2semidirect
5. #11722775C2semidirect
17. C2×#3776D3semidirect
54. C2×C3⋊(C3⋊D3)Q16semidirect
144. C33#467semidirect
219. C32#5188semidirect
254. C3#10593446semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×#889242
#117229772
#117227972
#117227752
#105934463
#51889