Finite Group Databasebeta

← #13165615

Subgroups — #13165615

Order 1728.

524 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 56 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×D3×#622C211yesmaximal, characteristic, complemented
3#11722953C212yesmaximal, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722773C212yesmaximal, complemented
6#10593520—31nomaximal
7C2×D3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#625—22no—
9C2×C3×#625C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3774—22no—
13D3×#622—22no—
14#88815C2212yes—
15(C2×D3)⋊(C3⋊Dic3)—22no—
16C22×#622—31no—
17C2×#3774D311yescharacteristic, complemented
18#63980—32no—
19#63970—62no—
20#63743—31no—
21#63739—32no—
22C22×C32×D3—22no—
23C22×C32×D3D411yescharacteristic
24C2×C3⋊(C3⋊Dic3)D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27D3×C3⋊Dic3—21no—
28D3×C3⋊Dic3—21no—
29C3×#625—22no—
30C3×#625—22no—
31C3×#625—44no—
32C3⋊#622—21no—
33C3×#622D412yescomplemented
34#5157—91nomaximal
35C22×C3⋊Dic3—31no—
36C2×C3×#67—62no—
37C2×C3×#67—62no—
38C2×C3×#67—62no—
39C2×#625—22no—
40C2×#625C2×D312yes—
41C2×D3×Dic3—62no—
42C2×#622—31no—
43C2×#622—62no—
44C2×#622C2×D311yescharacteristic
45#3774—22no—
46#3757—62no—
47#3724—32no—
48Dic3⋊Dic3—62no—
49(C2×D3)⋊Dic3—62no—
50(C2×D3)⋊Dic3—62no—
51(C2×D3)⋊Dic3—122no—
52C22×C33C2×D411yescharacteristic
53C2×C32×D3—21no—
54C2×C32×D3—21no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C2×C32×D3D812yescomplemented
59C3⋊(C3⋊Dic3)—21no—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3C2×D411yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C2×C3×D8—91no—
64#939—182no—
65#938—92no—
66C2×D3×C8—91no—
67#832—92no—
68#818—91no—
69C23×C32—31no—
70C23×C32—62no—
71C22×C3×D3—22no—
72C22×C3×D3—22no—
73C22×C3×D3—22no—
74C22×C3×D3—42no—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—62no—
78C2×C3⋊Dic3—61no—
79C2×C3⋊Dic3—61no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C22×D311yescharacteristic
82C3×#67—122no—
83C3×#67—122no—
84C3×#67—124no—
85C3×#67—124no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—122no—
90#625—22no—
91#625—22no—
92#625—44no—
93D3×Dic3—122no—
94D3×Dic3—122no—
95#622—61no—
96#622#6712yescomplemented
97#466—271noSylow
98C2×C33#12111yescharacteristic
99C2×C33C2×D811yescharacteristic
100C2×C33#13511yescharacteristic
101C32×D3—22no—
102C32×D3—82no—
103C2×C3×D4—92no—
104C2×C3×D4—182no—
105C2×#67—62no—
106C2×#67—62no—
107C2×#67—62no—
108C22×Dic3—182no—
109C2×C4×D3—91no—
110C3×D8—182no—
111C2×C3×C8—91no—
112#217—182no—
113#217—182no—
114#217—362no—
115(C2×D3)⋊C4—182no—
116C4⋊Dic3—92no—
117Dic3⋊C4—182no—
118C2×C3⋊C8—91no—
119D3×C8—182no—
120C22×C32—22no—
121C22×C32—32no—
122C22×C32—42no—
123C22×C32—61no—
124C22×C32—61no—
125C22×C32—122no—
126C22×C32—122no—
127C22×C32—122no—
128C22×C32D3×D411yescharacteristic
129C2×C3×D3—24no—
130C2×C3×D3—42no—
131C2×C3×D3—42no—
132C2×C3×D3—42no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—44no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—82no—
140C2×C3×D3—82no—
141C3⋊Dic3—42no—
142C3⋊Dic3—61no—
143C3⋊Dic3—122no—
144C3⋊Dic3—242no—
145C3⋊Dic3#23711yescharacteristic
146C3⋊Dic3C2×#6711yescharacteristic
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C2×D8—91no—
153C22×C8—271no—
154#121—272no—
155#107—271no—
156#102—272no—
157C33#46611yescharacteristic, complemented, Sylow
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—62no—
161C23×C3—122no—
162C22×D3—22no—
163C22×D3GO+(4,𝔽2)11yescharacteristic
164C3×D4—182no—
165C3×D4—182no—
166C3×D4—364no—
167C2×C3×C4—91no—
168C2×C3×C4—182no—
169#67—122no—
170#67—122no—
171#67—124no—
172#67—124no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—91no—
177C2×Dic3—122no—
178C2×Dic3—182no—
179C2×Dic3—362no—
180C2×Dic3—362no—
181C2×Dic3—362no—
182C4×D3—181no—
183C4×D3—181no—
184C3×C8—92no—
185C3⋊C8—181no—
186C2×C32—22no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—62no—
193C2×C32—242no—
194C2×C32D3×D811yescharacteristic
195C2×C32#91211yescharacteristic
196C2×C32#93811yescharacteristic
197C3×D3—44no—
198C3×D3—82no—
199C3×D3—84no—
200C3×D3—82no—
201C2×D4—92no—
202C2×D4—182no—
203C22×C4—271no—
204D8—182no—
205C2×C8—91no—
206C2×C8—271no—
207C2×C8—542no—
208C4⋊C4—272no—
209C22⋊C4—542no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—64no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3—122no—
216C22×C3—122no—
217C22×C3—122no—
218C22×C3—124no—
219C22×C3—122no—
220C22×C3—122no—
221C22×C3—122no—
222C22×C3—242no—
223C22×C3—242no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C2×D3—21no—
226C2×D3—21no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3#377412yescomplemented
231C3×C4—91no—
232C3×C4—91no—
233C3×C4—362no—
234Dic3—62no—
235Dic3—62no—
236Dic3—122no—
237Dic3—122no—
238Dic3—181no—
239Dic3—242no—
240Dic3—362no—
241Dic3—722no—
242C32—22no—
243C32—42no—
244C32#515711yescharacteristic, complemented
245C23—31no—
246C23—62no—
247D4—182no—
248D4—182no—
249D4—364no—
250C2×C4—91no—
251C2×C4—182no—
252C2×C4—271no—
253C2×C4—541no—
254C2×C4—541no—
255C8—92no—
256C8—541no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3—124no—
264C2×C3—242no—
265C2×C3—244no—
266C2×C3—242no—
267C2×C3C2×#377411yescharacteristic
268C2×C3#6423711yescharacteristic
269C2×C3#6424211yescharacteristic
270D3—22no—
271D3—82no—
272C22—32no—
273C22—61no—
274C22—61no—
275C22—122no—
276C22—122no—
277C22—122no—
278C22D3×GO+(4,𝔽2)11yescharacteristic
279C4—91no—
280C4—91no—
281C4—362no—
282C4—541no—
283C3—22no—
284C3—42no—
285C3#1059352011yescharacteristic, complemented
286C2—62no—
287C2—242no—
288C2#1172411411yescharacteristic
289C2D3×#377411yescharacteristic
290C2#1172413211yescharacteristic
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×D3×#622C2semidirect
3. #11722953C2semidirect
4. #11722797C2semidirect
5. #11722773C2semidirect
17. C2×#3774D3semidirect
33. C3×#622D4semidirect
58. C2×C32×D3D8semidirect
96. #622#67semidirect
157. C33#466semidirect
230. C2×D3#3774semidirect
244. C32#5157semidirect
285. C3#10593520semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C2×D3×#6222
#117229532
#117227972
#117227732
#105935203
#51579