Finite Group Databasebeta

← #13165616

Subgroups — #13165616

Order 1728.

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

normal subgroups only

This group has 564 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×#3774C211yesmaximal, characteristic, complemented
2C2×#88924C211yesmaximal, characteristic, complemented
3#11722964C212yesmaximal, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722773C212yesmaximal, complemented
6#10593423—31nomaximal
7C2×#6291—22no—
8C2×#6289C2211yescharacteristic
9C2×C3×#625C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3774—22no—
13#88924—22no—
14#88815C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C2×#3787—31no—
17C2×#3774D311yescharacteristic, complemented
18#63958—62no—
19#63912—32no—
20#63743—31no—
21#63739—32no—
22C22×C3⋊(C3⋊D3)D411yescharacteristic
23C22×C32×D3—22no—
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3D411yescharacteristic
27#6291—44no—
28#6289—21no—
29#6289—21no—
30C3×#625—22no—
31C3×#625—22no—
32C3⋊#622—21no—
33C3×#622D412yescomplemented
34#5157—91nomaximal
35C2×#638—62no—
36C2×C4×C3⋊D3—31no—
37C2×C3×#67—62no—
38C2×#626—62no—
39C2×#626—62no—
40C2×#626—122no—
41C2×#625C2×D312yes—
42C2×#624—62no—
43C2×#622—31no—
44C2×#622C2×D311yescharacteristic
45#3787—62no—
46#3774—22no—
47C3×#217—62no—
48C3×(C2×D3)⋊C4—62no—
49#3724—32no—
50Dic3⋊Dic3—62no—
51(C2×D3)⋊Dic3—22no—
52C22×C33C2×D411yescharacteristic
53C2×C3⋊(C3⋊D3)—21no—
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)D812yescomplemented
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C2×C32×D3—42no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—21no—
63C2×C3×D8—91no—
64#938—92no—
65C2×D3×C8—91no—
66#884—182no—
67#832—92no—
68#818—91no—
69C23×C32—62no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—122no—
73C22×C3×D3—22no—
74C22×C3×D3—22no—
75C22×C3×D3—42no—
76C2×C32×C4—31no—
77#638—124no—
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3C22×D311yescharacteristic
80C4×C3⋊D3—61no—
81C4×C3⋊D3—61no—
82C3×#67—122no—
83C3×#67—122no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—122no—
89#626—124no—
90#626—124no—
91#626—128no—
92#625—22no—
93#625—22no—
94#624—122no—
95#624—122no—
96#622—61no—
97#622#6712yescomplemented
98#466—271noSylow
99C2×C33#12111yescharacteristic
100C2×C33C2×D811yescharacteristic
101C2×C33#13511yescharacteristic
102C3⋊(C3⋊D3)—22no—
103C32×D3—82no—
104C2×C3×D4—92no—
105C2×#67—62no—
106C2×#67—182no—
107C2×#67—182no—
108C2×#67—362no—
109C2×D12—182no—
110C2×C4×D3—91no—
111C2×C4×D3—182no—
112C3×D8—182no—
113C2×C3×C8—91no—
114C3×C22⋊C4—182no—
115#217—62no—
116(C2×D3)⋊C4—62no—
117C4⋊Dic3—92no—
118Dic3⋊C4—182no—
119C2×C3⋊C8—91no—
120D3×C8—182no—
121C22×C32—22no—
122C22×C32—42no—
123C22×C32—122no—
124C22×C32—122no—
125C22×C32—122no—
126C22×C32D3×D411yescharacteristic
127C2×C3⋊D3—32no—
128C2×C3⋊D3—64no—
129C2×C3⋊D3—61no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—124no—
132C2×C3⋊D3—122no—
133C2×C3⋊D3—122no—
134C2×C3⋊D3—242no—
135C2×C3⋊D3—242no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—44no—
140C2×C3×D3—42no—
141C2×C3×D3—42no—
142C2×C3×D3—42no—
143C2×C3×D3—82no—
144C2×C3×D3—82no—
145C32×C4—61no—
146C3⋊Dic3—42no—
147C3⋊Dic3#23711yescharacteristic
148C3⋊Dic3C2×#6711yescharacteristic
149C3×Dic3—42no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—122no—
153C3×Dic3—124no—
154C3×Dic3—122no—
155C2×D8—91no—
156C22×C8—271no—
157#121—272no—
158#107—271no—
159#102—272no—
160C33#46611yescharacteristic, complemented, Sylow
161C23×C3—62no—
162C23×C3—62no—
163C23×C3—122no—
164C22×D3—22no—
165C22×D3—91no—
166C22×D3—182no—
167C22×D3—362no—
168C3×D4—182no—
169C3×D4—182no—
170C2×C3×C4—62no—
171C2×C3×C4—91no—
172C2×C3×C4—182no—
173#67—122no—
174#67—122no—
175#67—364no—
176#67—364no—
177#67—368no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—122no—
182C2×Dic3GO+(4,𝔽2)11yescharacteristic
183D12—364no—
184C4×D3—181no—
185C4×D3—181no—
186C4×D3—362no—
187C4×D3—362no—
188C3×C8—92no—
189C3⋊C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—242no—
197C2×C32D3×D811yescharacteristic
198C2×C32#91211yescharacteristic
199C2×C32#93811yescharacteristic
200C3⋊D3—62no—
201C3⋊D3—124no—
202C3⋊D3—244no—
203C3×D3—82no—
204C3×D3—84no—
205C3×D3—82no—
206C2×D4—92no—
207C2×D4—542no—
208C22×C4—271no—
209D8—182no—
210C2×C8—91no—
211C2×C8—271no—
212C2×C8—542no—
213C4⋊C4—272no—
214C22⋊C4—182no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—122no—
218C22×C3—122no—
219C22×C3—122no—
220C22×C3—124no—
221C22×C3—122no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—242no—
225C22×C3—242no—
226C22×C3C2×GO+(4,𝔽2)11yescharacteristic
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—92no—
231C2×D3—184no—
232C2×D3—181no—
233C2×D3—181no—
234C2×D3—364no—
235C2×D3—362no—
236C2×D3—362no—
237C2×D3—722no—
238C2×D3—722no—
239C3×C4—91no—
240C3×C4—91no—
241C3×C4—122no—
242C3×C4—362no—
243Dic3—21no—
244Dic3—62no—
245Dic3—62no—
246Dic3—122no—
247Dic3—124no—
248Dic3—122no—
249C32—22no—
250C32—42no—
251C32#515711yescharacteristic, complemented
252C23—62no—
253C23—271no—
254D4—182no—
255D4—182no—
256D4—1084no—
257C2×C4—31no—
258C2×C4—91no—
259C2×C4—182no—
260C2×C4—541no—
261C2×C4—541no—
262C8—92no—
263C8—541no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—242no—
271C2×C3—244no—
272C2×C3—242no—
273C2×C3#6422311yescharacteristic
274C2×C3#6422511yescharacteristic
275C2×C3#6424211yescharacteristic
276D3—82no—
277D3—182no—
278D3—364no—
279D3—724no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—272no—
284C22—541no—
285C22—541no—
286C22D3×GO+(4,𝔽2)11yescharacteristic
287C4—61no—
288C4—91no—
289C4—91no—
290C4—362no—
291C3—22no—
292C3—42no—
293C3#1059342311yescharacteristic, complemented
294C2—242no—
295C2—542no—
296C2#1172411411yescharacteristic
297C2#1172412711yescharacteristic
298C2#1172413511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#466
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×#3774C2semidirect
2. C2×#88924C2semidirect
3. #11722964C2semidirect
4. #11722797C2semidirect
5. #11722773C2semidirect
17. C2×#3774D3semidirect
33. C3×#622D4semidirect
55. C2×C3⋊(C3⋊D3)D8semidirect
97. #622#67semidirect
160. C33#466semidirect
251. C32#5157semidirect
293. C3#10593423semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C2×#889242
#117229642
#117227972
#117227732
#105934233
#51579