Finite Group Databasebeta

← #10596188⋊C3 #13153276

Subgroups — #10596188⋊C3 #13153276

Order 1728.

1606 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 254 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×D4×(C3⋊D3)⋊C3C214yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#64147⋊C3C214yesmaximal, complemented
4#64146⋊C3C214yesmaximal, complemented
5C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
6#10596188C311yesmaximal, characteristic, complemented
7C3×#5717—31nomaximal
8C23×(C3⋊D3)⋊C3C2214yescomplemented
9C2×D4×H(3,𝔽3)C2214yescomplemented
10C2×D4×H(3,𝔽3)C2212yescomplemented
11C2×#638⋊C3C2218yescomplemented
12D4×(C3⋊D3)⋊C3—216no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
15C42×H(3,𝔽3)C2211yescharacteristic, complemented
16#3757⋊C3C2214yescomplemented
17#3752⋊C3C2214yescomplemented
18C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
19C2×D4×C3⋊D3C2×C314yescomplemented
20C2×C3×D3×D4—34no—
21C32×#127—21no—
22C32×#127D311yescharacteristic, complemented
23C32×#127C2×C311yescharacteristic, complemented
24#64147C2×C314yescomplemented
25#64146C2×C314yescomplemented
26C4×C4⋊(C3⋊D3)C2×C312yescomplemented
27C3×#938—34no—
28C3×#937—34no—
29C3×C4×D12—32no—
30C23×H(3,𝔽3)C2314yes—
31C22×(C3⋊D3)⋊C3—216no—
32C22×(C3⋊D3)⋊C3—44no—
33C22×(C3⋊D3)⋊C3C2314yes—
34D4×H(3,𝔽3)—28no—
35D4×H(3,𝔽3)—42no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)C2312yes—
38#638⋊C3—216no—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C4⋊((C3⋊D3)⋊C3)D418yescomplemented
41C4×(C3⋊D3)⋊C3—28no—
42C3×D42—91no—
43#5717—31no—
44#5717—91no—
45C23×C3⋊D3C22×C314yescomplemented
46C23×C3×D3—34no—
47C2×C32×D4—24no—
48C2×C32×D4—22no—
49C2×C32×D4C2×D314yescomplemented
50C2×C32×D4C2×D312yescomplemented
51C2×C32×D4C22×C314yescomplemented
52C2×C32×D4C22×C312yescomplemented
53C2×#638C22×C318yescomplemented
54D4×C3⋊D3—216no—
55C2×C4⋊(C3⋊D3)C22×C312yes—
56C2×C4×C3⋊D3C22×C314yescomplemented
57C2×C3×#67—38no—
58C3×D3×D4—616no—
59C2×C3×D12—32no—
60C2×C3×C4×D3—34no—
61C32×C42—21no—
62C32×C42C2×D311yescharacteristic, complemented
63C32×C42C22×C311yescharacteristic, complemented
64#3757C22×C314yescomplemented
65#3752C22×C314yescomplemented
66C4⋊(C3⋊Dic3)C22×C312yescomplemented
67C3×#217—34no—
68C3×(C2×D3)⋊C4—34no—
69C3×C4⋊Dic3—32no—
70C22×H(3,𝔽3)—28no—
71C22×H(3,𝔽3)—24no—
72C22×H(3,𝔽3)C2411yescharacteristic
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—416no—
76C2×(C3⋊D3)⋊C3C2×D418yescomplemented
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)C2×D414yescomplemented
79(C3⋊Dic3)⋊C3—24no—
80C22×C3×D4—94no—
81C2×D3×D4—34no—
82C2×D3×D4—94no—
83C3×#127—31no—
84C3×#127—31no—
85C3×#127—61no—
86C3×#127C3×D311yescharacteristic
87C3×#121—94no—
88C3×#120—94no—
89C3×C4×D4—92no—
90#938—34no—
91#938—94no—
92#937—34no—
93#937—94no—
94C4×D12—32no—
95C4×D12—92no—
96C23×C32—24no—
97C23×C32C22×D314yes—
98C23×C32C23×C314yes—
99C22×C3⋊D3—216no—
100C22×C3⋊D3—44no—
101C22×C3⋊D3C23×C314yes—
102C22×C3×D3—34no—
103C22×C3×D3—616no—
104C22×C3×D3—124no—
105C32×D4—28no—
106C32×D4—28no—
107C32×D4—48no—
108C32×D4—42no—
109C32×D4—42no—
110C32×D4—44no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D312yes—
115C2×C32×C4C23×C311yescharacteristic
116C2×C32×C4C23×C312yes—
117#638—216no—
118C2×C3⋊Dic3C23×C314yes—
119C4⋊(C3⋊D3)C3×D418yescomplemented
120C4×C3⋊D3—28no—
121C3×#67—616no—
122C2×C3×Dic3—34no—
123C3×D12—38no—
124C3×C4×D3—68no—
125D42—91noSylow
126C2×H(3,𝔽3)—44no—
127C2×H(3,𝔽3)C22×D412yes—
128C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
129(C3⋊D3)⋊C3—28no—
130C24×C3—94no—
131C23×D3—34no—
132C23×D3—94no—
133C2×C3×D4—34no—
134C2×C3×D4—34no—
135C2×C3×D4—32no—
136C2×C3×D4—32no—
137C2×C3×D4—64no—
138C2×C3×D4—62no—
139C2×C3×D4—92no—
140C2×C3×D4—98no—
141C2×C3×D4—1816no—
142C2×C3×D4C2×C3×D314yes—
143C2×C3×D4C2×C3×D312yes—
144C22×C3×C4—94no—
145C2×#67—38no—
146C2×#67—98no—
147D3×D4—616no—
148D3×D4—1816no—
149C2×D12—32no—
150C2×D12—92no—
151C2×C4×D3—34no—
152C2×C4×D3—94no—
153C3×C4⋊C4—92no—
154C3×C22⋊C4—94no—
155C3×C22⋊C4—94no—
156C3×C42—31no—
157C3×C42—31no—
158C3×C42—61no—
159C3×C42C2×C3×D311yescharacteristic
160#217—34no—
161#217—94no—
162(C2×D3)⋊C4—34no—
163(C2×D3)⋊C4—94no—
164C4⋊Dic3—32no—
165C4⋊Dic3—92no—
166C22×C32—28no—
167C22×C32—24no—
168C22×C32—28no—
169C22×C32—24no—
170C22×C32—21no—
171C22×C32—48no—
172C22×C32—44no—
173C22×C32C23×D311yescharacteristic
174C22×C32C24×C311yescharacteristic
175C2×C3⋊D3—24no—
176C2×C3⋊D3—24no—
177C2×C3⋊D3—416no—
178C2×C3⋊D3C2×C3×D418yescomplemented
179C2×C3×D3—38no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C2×C3×D3—1216no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—24no—
187C32×C4D3×D414yescomplemented
188C32×C4C2×C3×D414yescomplemented
189C3⋊Dic3—24no—
190C3×Dic3—64no—
191C22×D4—94no—
192#127(C3⋊D3)⋊C311yescharacteristic, complemented
193#121—94no—
194#120—94no—
195C4×D4—92no—
196H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
197C23×C3—34no—
198C23×C3—34no—
199C23×C3—64no—
200C23×C3—94no—
201C23×C3—1816no—
202C23×C3—364no—
203C23×C3C22×C3×D314yes—
204C22×D3—34no—
205C22×D3—616no—
206C22×D3—94no—
207C22×D3—124no—
208C22×D3—1816no—
209C22×D3—364no—
210C3×D4—28no—
211C3×D4—42no—
212C3×D4—68no—
213C3×D4—68no—
214C3×D4—98no—
215C3×D4—128no—
216C3×D4—122no—
217C3×D4—122no—
218C3×D4—124no—
219C3×D4—1816no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—94no—
227C2×C3×C4—188no—
228C2×C3×C4C22×C3×D311yescharacteristic
229C2×C3×C4C22×C3×D312yes—
230#67—616no—
231#67—1816no—
232C2×Dic3—34no—
233C2×Dic3—94no—
234D12—38no—
235D12—98no—
236C4×D3—68no—
237C4×D3—188no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—44no—
241C2×C32—44no—
242C2×C32—84no—
243C2×C32C2×D3×D412yes—
244C2×C32#100411yescharacteristic
245C2×C32C22×C3×D412yes—
246C2×C32C3×H(5,𝔽2)11yescharacteristic
247C3⋊D3—28no—
248C3×D3—68no—
249C24—94no—
250C2×D4—98no—
251C2×D4—92no—
252C2×D4—1816no—
253C2×D4C2×(C3⋊D3)⋊C314yescomplemented
254C2×D4C2×(C3⋊D3)⋊C312yescomplemented
255C22×C4—94no—
256C4⋊C4—92no—
257C22⋊C4—94no—
258C22⋊C4—94no—
259C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
260C22×C3—28no—
261C22×C3—24no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—68no—
265C22×C3—68no—
266C22×C3—64no—
267C22×C3—64no—
268C22×C3—61no—
269C22×C3—98no—
270C22×C3—128no—
271C22×C3—124no—
272C22×C3—184no—
273C22×C3—184no—
274C22×C3—3616no—
275C22×C3C23×C3×D311yescharacteristic
276C2×D3—38no—
277C2×D3—64no—
278C2×D3—64no—
279C2×D3—98no—
280C2×D3—1216no—
281C2×D3—184no—
282C2×D3—184no—
283C2×D3—3616no—
284C3×C4—21no—
285C3×C4—34no—
286C3×C4—34no—
287C3×C4—61no—
288C3×C4—61no—
289C3×C4—62no—
290C3×C4—64no—
291C3×C4—184no—
292C3×C4C3×D3×D414yes—
293Dic3—64no—
294Dic3—184no—
295C32—21no—
296C32#571711yescharacteristic, complemented
297C32C3×D4211yescharacteristic, complemented
298C23—94no—
299C23—1816no—
300C23—364no—
301C23C22×(C3⋊D3)⋊C314yes—
302D4—28no—
303D4—42no—
304D4—98no—
305D4—1816no—
306C2×C4—94no—
307C2×C4—188no—
308C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
309C2×C4C22×(C3⋊D3)⋊C312yes—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—44no—
315C2×C3—62no—
316C2×C3—61no—
317C2×C3—124no—
318C2×C3—124no—
319C2×C3—188no—
320C2×C3—244no—
321C2×C3C2×C3×D3×D412yes—
322C2×C3C3×#100411yescharacteristic
323D3—68no—
324D3—188no—
325C22—28no—
326C22—24no—
327C22—98no—
328C22—184no—
329C22—184no—
330C22—3616no—
331C22C23×(C3⋊D3)⋊C311yescharacteristic
332C4—21no—
333C4—184no—
334C4D4×(C3⋊D3)⋊C314yescomplemented
335C3—31no—
336C3—31no—
337C3—61no—
338C3C3×#571711yescharacteristic
339C2—44no—
340C2—188no—
341C2C2×D4×(C3⋊D3)⋊C312yes—
342C2#64361⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×D4×(C3⋊D3)⋊C3C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #64147⋊C3C2semidirect
4. #64146⋊C3C2semidirect
5. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
6. #10596188C3semidirect
8. C23×(C3⋊D3)⋊C3C22semidirect
9. C2×D4×H(3,𝔽3)C22semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
11. C2×#638⋊C3C22semidirect
14. C2×C4×(C3⋊D3)⋊C3C22semidirect
15. C42×H(3,𝔽3)C22semidirect
16. #3757⋊C3C22semidirect
17. #3752⋊C3C22semidirect
18. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
19. C2×D4×C3⋊D3C2×C3semidirect
22. C32×#127D3semidirect
23. C32×#127C2×C3semidirect
24. #64147C2×C3semidirect
25. #64146C2×C3semidirect
26. C4×C4⋊(C3⋊D3)C2×C3semidirect
40. C4⋊((C3⋊D3)⋊C3)D4semidirect
45. C23×C3⋊D3C22×C3semidirect
49. C2×C32×D4C2×D3semidirect
50. C2×C32×D4C2×D3semidirect
51. C2×C32×D4C22×C3semidirect
52. C2×C32×D4C22×C3semidirect
53. C2×#638C22×C3semidirect
56. C2×C4×C3⋊D3C22×C3semidirect
62. C32×C42C2×D3semidirect
63. C32×C42C22×C3semidirect
64. #3757C22×C3semidirect
65. #3752C22×C3semidirect
66. C4⋊(C3⋊Dic3)C22×C3semidirect
76. C2×(C3⋊D3)⋊C3C2×D4semidirect
78. C4×H(3,𝔽3)C2×D4semidirect
119. C4⋊(C3⋊D3)C3×D4semidirect
178. C2×C3⋊D3C2×C3×D4semidirect
187. C32×C4D3×D4semidirect
188. C32×C4C2×C3×D4semidirect
192. #127(C3⋊D3)⋊C3semidirect
196. H(3,𝔽3)D42semidirect
253. C2×D4C2×(C3⋊D3)⋊C3semidirect
254. C2×D4C2×(C3⋊D3)⋊C3semidirect
259. C42C2×(C3⋊D3)⋊C3semidirect
296. C32#5717semidirect
297. C32C3×D42semidirect
334. C4D4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1272
#64147⋊C32
#64146⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105961883
C3×#57173