Finite Group Databasebeta

← #13165642

Subgroups — #13165642

Order 1728.

436 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 52 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×#88926C211yesmaximal, characteristic
3#11722977C212yesmaximal
4#11722796C211yesmaximal, characteristic, complemented
5#11722775C212yesmaximal, complemented
6#10593449—31nomaximal
7C2×#6289C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3776—22no—
12#88926—22no—
13#88815C2212yes—
14#88804—22no—
15C3×#3724—22no—
16C2×#3788—31no—
17C2×#3776D311yescharacteristic, complemented
18#63888—62no—
19#63865—32no—
20#63742—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#5189—91nomaximal
34C2×C4×C3⋊D3—31no—
35C2×C3×Dic6—62no—
36C2×#627C2×D312yes—
37C2×#624—62no—
38C2×#622—31no—
39C2×#622C2×D311yescharacteristic
40#3788—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)—22no—
54C3×C3⋊Dic3—42no—
55C3×C3⋊Dic3C2×D411yescharacteristic
56C3×C3⋊Dic3#4111yescharacteristic
57C32×Dic3—21no—
58C32×Dic3—42no—
59C2×C3×Q16—91no—
60#946—92no—
61C2×ΓL(1,𝔽25)—91no—
62#896—182no—
63#835—92no—
64#817—91no—
65C22×C3⋊D3—31no—
66C22×C3⋊D3—62no—
67C22×C3⋊D3—122no—
68C2×C32×C4—31no—
69C2×C32×C4—62no—
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3C22×D311yescharacteristic
72C4×C3⋊D3—61no—
73C4×C3⋊D3—61no—
74C2×C3×Dic3—22no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—122no—
82C3×Dic6—122no—
83C3×Dic6—122no—
84#627—22no—
85#627—22no—
86#624—122no—
87#624—122no—
88#622—61no—
89#622#6712yes—
90#470—271noSylow
91C2×C33#12111yescharacteristic
92C2×C33#13712yes—
93C3⋊(C3⋊D3)—41no—
94C2×C3×Q8—92no—
95C2×C4×D3—91no—
96C2×C4×D3—182no—
97C2×Dic6—62no—
98C3×Q16—182no—
99C2×C3×C8—91no—
100C3×C4⋊C4—182no—
101(C2×D3)⋊C4—182no—
102(C2×D3)⋊C4—182no—
103(C2×D3)⋊C4—182no—
104(C2×D3)⋊C4—362no—
105C4⋊Dic3—92no—
106Dic3⋊C4—62no—
107Dic3⋊C4—62no—
108Dic3⋊C4—182no—
109C2×C3⋊C8—91no—
110ΓL(1,𝔽25)—182no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32D3×D411yescharacteristic
114C2×C3⋊D3—64no—
115C2×C3⋊D3—62no—
116C2×C3⋊D3—61no—
117C2×C3⋊D3—122no—
118C2×C3⋊D3—122no—
119C2×C3⋊D3—124no—
120C2×C3⋊D3—242no—
121C2×C3⋊D3—242no—
122C32×C4—61no—
123C32×C4—122no—
124C3⋊Dic3—42no—
125C3⋊Dic3#23711yescharacteristic
126C3⋊Dic3C2×#6711yescharacteristic
127C3×Dic3—42no—
128C3×Dic3—42no—
129C3×Dic3—42no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—82no—
133C3×Dic3—122no—
134C3×Dic3—122no—
135C3×Dic3—242no—
136C2×Q16—91no—
137C2×M16—271no—
138C22⋊Q8—272no—
139#106—271no—
140#103—272no—
141C33#47011yescharacteristic, complemented, Sylow
142C22×D3—91no—
143C22×D3—182no—
144C22×D3—362no—
145C3×Q8—182no—
146C3×Q8—182no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—62no—
150C2×C3×C4—91no—
151C2×C3×C4—122no—
152C2×C3×C4—182no—
153C2×Dic3—22no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—122no—
158C2×Dic3GO+(4,𝔽2)11yescharacteristic
159C4×D3—181no—
160C4×D3—181no—
161C4×D3—362no—
162C4×D3—362no—
163Dic6—122no—
164Dic6—122no—
165C3×C8—92no—
166C3⋊C8—181no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32#91812yes—
174C2×C32#93811yescharacteristic
175C3⋊D3—121no—
176C3⋊D3—124no—
177C3⋊D3—244no—
178C2×Q8—92no—
179C22×C4—271no—
180Q16—182no—
181M16—542no—
182C2×C8—91no—
183C2×C8—271no—
184C4⋊C4—182no—
185C4⋊C4—272no—
186C22⋊C4—542no—
187C22×C3—22no—
188C22×C3—42no—
189C22×C3C2×GO+(4,𝔽2)11yescharacteristic
190C2×D3—184no—
191C2×D3—182no—
192C2×D3—181no—
193C2×D3—362no—
194C2×D3—362no—
195C2×D3—364no—
196C2×D3—722no—
197C2×D3—722no—
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—122no—
201C3×C4—122no—
202C3×C4—122no—
203C3×C4—242no—
204C3×C4—362no—
205Dic3—21no—
206Dic3—42no—
207Dic3—62no—
208Dic3—62no—
209Dic3—122no—
210Dic3—122no—
211Dic3—242no—
212C32—22no—
213C32—42no—
214C32#518911yescharacteristic, complemented
215C23—271no—
216Q8—182no—
217Q8—182no—
218C2×C4—31no—
219C2×C4—62no—
220C2×C4—91no—
221C2×C4—182no—
222C2×C4—541no—
223C2×C4—541no—
224C8—92no—
225C8—541no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—42no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3#6422612yes—
233C2×C3#6424211yescharacteristic
234D3—364no—
235D3—361no—
236D3—724no—
237C22—541no—
238C22—542no—
239C22D3×GO+(4,𝔽2)11yescharacteristic
240C4—61no—
241C4—91no—
242C4—91no—
243C4—122no—
244C4—362no—
245C3—22no—
246C3—42no—
247C3#1059344911yescharacteristic, complemented
248C2—1081no—
249C2#1172411411yescharacteristic
250C2#1172414512yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypocenterC33#470
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. #11722796C2semidirect
5. #11722775C2semidirect
17. C2×#3776D3semidirect
141. C33#470semidirect
214. C32#5189semidirect
247. C3#10593449semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×#889262
#117229772
#117227962
#117227752
#105934493
#51899