Finite Group Databasebeta

← #13138715

Subgroups — #13138715

Order 1728.

341 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 101 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11720545C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#11720247C212yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊C16C212yesmaximal, complemented
6#10590625—31nomaximal
7D3×#812—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×(C3⋊(C3⋊C8))⋊C3C411yescharacteristic
11C8×(C3⋊D3)⋊C3C411yescharacteristic
12C8×(C3⋊D3)⋊C3C411yescharacteristic
13C8×(C3⋊D3)⋊C3C2211yescharacteristic
14C8×(C3⋊D3)⋊C3C2211yescharacteristic
15#88386C2212yescomplemented
16(C3⋊(C3⋊C16))⋊C3C2212yescomplemented
17C2×C8×C3⋊D3D311yescharacteristic
18C2×C3×D3×C8—31no—
19#63639D311yescharacteristic, complemented
20C3×#812—31no—
21C3×#812—31no—
22C3×#812—31no—
23C3×#812—61no—
24#63544—32no—
25#63543—32no—
26#63542—32no—
27D3×C3⋊C16—32no—
28C22×(C3⋊D3)⋊C3C811yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×(C3⋊Dic3)⋊C3C811yescharacteristic
31C4×(C3⋊D3)⋊C3C811yescharacteristic
32C4×(C3⋊D3)⋊C3C811yescharacteristic
33C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
34C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
35C8×H(3,𝔽3)C2×C411yescharacteristic
36C8×H(3,𝔽3)C2311yescharacteristic
37(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
38(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
39C2×#812—91no—
40D3×M32—91no—
41D3×M32—91no—
42C2×C4×C3⋊D3Dic311yescharacteristic
43C2×C3×C4×D3—31no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C2×C3⋊(C3⋊C8)Dic311yescharacteristic
48C8×C3⋊D3Dic311yescharacteristic
49C8×C3⋊D3Dic311yescharacteristic
50C8×C3⋊D3C2×D311yescharacteristic
51C8×C3⋊D3C2×D311yescharacteristic
52C2×C3×C3⋊C8—31no—
53C3×D3×C8—31no—
54C3×D3×C8—31no—
55C3×D3×C8—31no—
56C3×D3×C8—31no—
57C3⋊(C3⋊C16)C2×D312yescomplemented
58C3×C3⋊C16—32no—
59C3×C3⋊C16—32no—
60C3×C3⋊C16—32no—
61C3×C3⋊C16—62no—
62C22×H(3,𝔽3)C2×C811yescharacteristic
63C2×(C3⋊D3)⋊C3—22no—
64C2×(C3⋊D3)⋊C3C2×C811yescharacteristic
65C2×(C3⋊D3)⋊C3C2×C811yescharacteristic
66C4×H(3,𝔽3)C2×C811yescharacteristic
67C4×H(3,𝔽3)C22×C411yescharacteristic
68(C3⋊Dic3)⋊C3C2×C811yescharacteristic
69(C3⋊Dic3)⋊C3C2×C811yescharacteristic
70C22×C3×C8—91no—
71C2×D3×C8—31no—
72C2×D3×C8—91no—
73C3×M32—91no—
74C3×M32—91no—
75#812—31no—
76#812—31no—
77#812—31no—
78#812—61no—
79#812—91no—
80#812—92no—
81C2×C3⋊C16—92no—
82#798—92no—
83#798—92no—
84D3×C16—92no—
85D3×C16—92no—
86C22×C3⋊D3C3⋊C811yescharacteristic
87C22×C3×D3—31no—
88C2×C32×C4—21no—
89C2×C32×C4C4×D311yescharacteristic
90C2×C32×C4C2×Dic311yescharacteristic
91C2×C3⋊Dic3C3⋊C811yescharacteristic
92C4×C3⋊D3C3⋊C811yescharacteristic
93C4×C3⋊D3C3⋊C811yescharacteristic
94C4×C3⋊D3C2×Dic311yescharacteristic
95C4×C3⋊D3C2×Dic311yescharacteristic
96C2×C3×Dic3—31no—
97C3×C4×D3—31no—
98C3×C4×D3—31no—
99C3×C4×D3—31no—
100C3×C4×D3—31no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8C4×D311yescharacteristic
104C32×C8C2×Dic311yescharacteristic
105C32×C8C22×D311yescharacteristic
106C32×C8C22×D311yescharacteristic
107C3⋊(C3⋊C8)C2×Dic311yescharacteristic
108C3⋊(C3⋊C8)C2×Dic311yescharacteristic
109C3×C3⋊C8—31no—
110C3×C3⋊C8—31no—
111C2×M32—271noSylow
112C2×H(3,𝔽3)—21no—
113C2×H(3,𝔽3)C22×C811yescharacteristic
114(C3⋊D3)⋊C3—21no—
115(C3⋊D3)⋊C3M3212yescomplemented
116C22×C3×C4—91no—
117C2×C4×D3—31no—
118C2×C4×D3—91no—
119C2×C3×C8—31no—
120C2×C3×C8—31no—
121C2×C3×C8—61no—
122C2×C3×C8—91no—
123C2×C3×C8—91no—
124C2×C3×C8—91no—
125C2×C3×C8—91no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C2×C3⋊C8—31no—
129C2×C3⋊C8—91no—
130D3×C8—31no—
131D3×C8—31no—
132D3×C8—31no—
133D3×C8—31no—
134D3×C8—91no—
135D3×C8—91no—
136D3×C8—91no—
137D3×C8—91no—
138C3×C16—92no—
139C3×C16—92no—
140C3⋊C16—32no—
141C3⋊C16—32no—
142C3⋊C16—32no—
143C3⋊C16—62no—
144C3⋊C16—92no—
145C22×C32—21no—
146C22×C32D3×C811yescharacteristic
147C22×C32C2×C3⋊C811yescharacteristic
148C2×C3⋊D3—22no—
149C2×C3⋊D3C2×C3⋊C811yescharacteristic
150C2×C3⋊D3C2×C3⋊C811yescharacteristic
151C2×C3×D3—31no—
152C2×C3×D3—31no—
153C2×C3×D3—62no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4D3×C811yescharacteristic
157C32×C4C2×C3⋊C811yescharacteristic
158C32×C4C2×C4×D311yescharacteristic
159C32×C4C22×Dic311yescharacteristic
160C3⋊Dic3C2×C3⋊C811yescharacteristic
161C3⋊Dic3C2×C3⋊C811yescharacteristic
162C3×Dic3—31no—
163C3×Dic3—31no—
164C22×C8—91no—
165M32—91no—
166M32—91no—
167M32—272no—
168C2×C16—272no—
169H(3,𝔽3)C2×M3211yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C22×D3—31no—
172C22×D3—91no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4D3×Dic311yescharacteristic
182C2×Dic3—31no—
183C2×Dic3—91no—
184C4×D3—31no—
185C4×D3—31no—
186C4×D3—31no—
187C4×D3—31no—
188C4×D3—91no—
189C4×D3—91no—
190C4×D3—91no—
191C4×D3—91no—
192C3×C8—31no—
193C3×C8—31no—
194C3×C8—31no—
195C3×C8—31no—
196C3×C8—61no—
197C3×C8—61no—
198C3×C8—91no—
199C3×C8—91no—
200C3×C8D3×Dic311yescharacteristic
201C3×C8C2×D3211yescharacteristic
202C3⋊C8—31no—
203C3⋊C8—31no—
204C3⋊C8—91no—
205C3⋊C8—91no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—41no—
210C2×C32C2×D3×C811yescharacteristic
211C2×C32C22×C3⋊C811yescharacteristic
212C3⋊D3—21no—
213C3⋊D3#81212yescomplemented
214C3×D3—32no—
215C3×D3—61no—
216C22×C4—91no—
217C2×C8—91no—
218C2×C8—91no—
219C2×C8—91no—
220C2×C8—91no—
221C2×C8—91no—
222C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C16—92no—
224C16—92no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—61no—
228C22×C3—91no—
229C22×C3—91no—
230C22×C3—182no—
231C22×C3D3×C3⋊C811yescharacteristic
232C2×D3—31no—
233C2×D3—31no—
234C2×D3—62no—
235C2×D3—91no—
236C2×D3—91no—
237C2×D3—182no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—61no—
243C3×C4—61no—
244C3×C4—91no—
245C3×C4—91no—
246C3×C4D3×C3⋊C811yescharacteristic
247C3×C4C2×D3×Dic311yescharacteristic
248Dic3—31no—
249Dic3—31no—
250Dic3—91no—
251Dic3—91no—
252C32—21no—
253C32D3×M3211yescharacteristic, complemented
254C32C2×#81211yescharacteristic, complemented
255C23—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
262C8—91no—
263C8—91no—
264C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
265C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
266C2×C3—21no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3—61no—
272C2×C3—92no—
273C2×C3—121no—
274C2×C3—181no—
275C2×C3C2×D3×C3⋊C811yescharacteristic
276D3—32no—
277D3—61no—
278D3—92no—
279D3—181no—
280C22—91no—
281C22—91no—
282C22—182no—
283C22((C3⋊D3)⋊C3)⋊C811yescharacteristic
284C4—91no—
285C4—91no—
286C4((C3⋊D3)⋊C3)⋊C811yescharacteristic
287C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
288C3—31no—
289C3—31no—
290C3—61no—
291C3D3×#81211yescharacteristic
292C2—21no—
293C2—92no—
294C2—181no—
295C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
subgroupG/H
centerC8C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×D3×C3⋊C8
prime radicalC22×(C3⋊D3)⋊C3C8
cosocleC8×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×M32
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720545C2semidirect
3. #63639⋊C3C2semidirect
4. #11720247C2semidirect
5. ((C3⋊D3)⋊C3)⋊C16C2semidirect
15. #88386C22semidirect
16. (C3⋊(C3⋊C16))⋊C3C22semidirect
19. #63639D3semidirect
57. C3⋊(C3⋊C16)C2×D3semidirect
115. (C3⋊D3)⋊C3M32semidirect
169. H(3,𝔽3)C2×M32semidirect
213. C3⋊D3#812semidirect
253. C32D3×M32semidirect
254. C32C2×#812semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
#117205452
#63639⋊C32
#117202472
((C3⋊D3)⋊C3)⋊C162
#105906253
D3×#8123