Finite Group Databasebeta

← #13165636

Subgroups — #13165636

Order 1728.

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

normal subgroups only

This group has 556 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×#3774C211yesmaximal, characteristic, complemented
2C2×#88926C211yesmaximal, characteristic, complemented
3#11722964C212yesmaximal, complemented
4#11722796C211yesmaximal, characteristic, complemented
5#11722773C212yesmaximal, complemented
6#10593429—31nomaximal
7C2×#6291—22no—
8C2×#6289C2211yescharacteristic
9C2×C3×#625C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3774—22no—
13#88926—22no—
14#88815C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C2×#3788—31no—
17C2×#3774D311yescharacteristic, complemented
18#63958—62no—
19#63912—32no—
20#63742—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#5159—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#3788—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)—22no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C3×C3⋊Dic3—42no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3#4111yescharacteristic
61C32×Dic3—21no—
62C2×C3×D8—91no—
63#938—92no—
64C2×ΓL(1,𝔽25)—91no—
65#884—182no—
66#832—92no—
67#817—91no—
68C23×C32—62no—
69C22×C3⋊D3—31no—
70C22×C3⋊D3—62no—
71C22×C3⋊D3—122no—
72C22×C3×D3—22no—
73C22×C3×D3—22no—
74C22×C3×D3—42no—
75C2×C32×C4—31no—
76#638—124no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3C22×D311yescharacteristic
79C4×C3⋊D3—61no—
80C4×C3⋊D3—61no—
81C3×#67—122no—
82C3×#67—122no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—122no—
88#626—124no—
89#626—124no—
90#626—128no—
91#625—22no—
92#625—22no—
93#624—122no—
94#624—122no—
95#622—61no—
96#622#6712yescomplemented
97#469—271noSylow
98C2×C33#12111yescharacteristic
99C2×C33AGL(1,ℤ/8)12yes—
100C3⋊(C3⋊D3)—41no—
101C32×D3—82no—
102C2×C3×D4—92no—
103C2×#67—62no—
104C2×#67—182no—
105C2×#67—182no—
106C2×#67—362no—
107C2×D12—182no—
108C2×C4×D3—91no—
109C2×C4×D3—182no—
110C3×D8—182no—
111C2×C3×C8—91no—
112C3×C22⋊C4—182no—
113#217—62no—
114(C2×D3)⋊C4—62no—
115C4⋊Dic3—92no—
116Dic3⋊C4—182no—
117C2×C3⋊C8—91no—
118ΓL(1,𝔽25)—182no—
119C22×C32—22no—
120C22×C32—42no—
121C22×C32—122no—
122C22×C32—122no—
123C22×C32—122no—
124C22×C32D3×D411yescharacteristic
125C2×C3⋊D3—64no—
126C2×C3⋊D3—62no—
127C2×C3⋊D3—61no—
128C2×C3⋊D3—122no—
129C2×C3⋊D3—122no—
130C2×C3⋊D3—124no—
131C2×C3⋊D3—242no—
132C2×C3⋊D3—242no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—42no—
136C2×C3×D3—44no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—42no—
140C2×C3×D3—82no—
141C2×C3×D3—82no—
142C32×C4—61no—
143C3⋊Dic3—42no—
144C3⋊Dic3#23711yescharacteristic
145C3⋊Dic3C2×#6711yescharacteristic
146C3×Dic3—42no—
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—124no—
151C3×Dic3—122no—
152C2×D8—91no—
153C2×M16—271no—
154#121—272no—
155#106—271no—
156#102—272no—
157C33#46911yescharacteristic, complemented, Sylow
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—122no—
161C22×D3—22no—
162C22×D3—91no—
163C22×D3—182no—
164C22×D3—362no—
165C3×D4—182no—
166C3×D4—182no—
167C2×C3×C4—62no—
168C2×C3×C4—91no—
169C2×C3×C4—182no—
170#67—122no—
171#67—122no—
172#67—364no—
173#67—364no—
174#67—368no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—122no—
179C2×Dic3GO+(4,𝔽2)11yescharacteristic
180D12—364no—
181C4×D3—181no—
182C4×D3—181no—
183C4×D3—362no—
184C4×D3—362no—
185C3×C8—92no—
186C3⋊C8—181no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—242no—
194C2×C32#91112yes—
195C2×C32#93811yescharacteristic
196C3⋊D3—121no—
197C3⋊D3—124no—
198C3⋊D3—244no—
199C3×D3—82no—
200C3×D3—84no—
201C3×D3—82no—
202C2×D4—92no—
203C2×D4—542no—
204C22×C4—271no—
205D8—182no—
206M16—542no—
207C2×C8—91no—
208C2×C8—271no—
209C4⋊C4—272no—
210C22⋊C4—182no—
211C22×C3—22no—
212C22×C3—42no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3—122no—
216C22×C3—124no—
217C22×C3—122no—
218C22×C3—122no—
219C22×C3—122no—
220C22×C3—242no—
221C22×C3—242no—
222C22×C3C2×GO+(4,𝔽2)11yescharacteristic
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—42no—
226C2×D3—184no—
227C2×D3—182no—
228C2×D3—181no—
229C2×D3—362no—
230C2×D3—362no—
231C2×D3—364no—
232C2×D3—722no—
233C2×D3—722no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—122no—
237C3×C4—362no—
238Dic3—21no—
239Dic3—62no—
240Dic3—62no—
241Dic3—122no—
242Dic3—124no—
243Dic3—122no—
244C32—22no—
245C32—42no—
246C32#515911yescharacteristic, complemented
247C23—62no—
248C23—271no—
249D4—182no—
250D4—182no—
251D4—1084no—
252C2×C4—31no—
253C2×C4—91no—
254C2×C4—182no—
255C2×C4—541no—
256C2×C4—541no—
257C8—92no—
258C8—541no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—42no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3—242no—
266C2×C3—244no—
267C2×C3—242no—
268C2×C3#6422412yes—
269C2×C3#6424211yescharacteristic
270D3—82no—
271D3—364no—
272D3—361no—
273D3—724no—
274C22—122no—
275C22—122no—
276C22—122no—
277C22—541no—
278C22—542no—
279C22D3×GO+(4,𝔽2)11yescharacteristic
280C4—61no—
281C4—91no—
282C4—91no—
283C4—362no—
284C3—22no—
285C3—42no—
286C3#1059342911yescharacteristic, complemented
287C2—242no—
288C2—1081no—
289C2#1172411411yescharacteristic
290C2#1172414012yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#469
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×#88926C2semidirect
3. #11722964C2semidirect
4. #11722796C2semidirect
5. #11722773C2semidirect
17. C2×#3774D3semidirect
33. C3×#622D4semidirect
96. #622#67semidirect
157. C33#469semidirect
246. C32#5159semidirect
286. C3#10593429semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C2×#889262
#117229642
#117227962
#117227732
#105934293
#51599