Finite Group Databasebeta

← #13152084

Subgroups — #13152084

Order 1728.

1419 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 219 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
1C22×#6235C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal, complemented
3#11721320C212yesmaximal, complemented
4#11721298C212yesmaximal, complemented
5#11721293C218yesmaximal, complemented
6#10595506—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×#272C2211yescharacteristic
9C2×#6235C418yescomplemented
10C2×#6235C2214yes—
11C22×#1084C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2214yescomplemented
13C2×#6199C2214yescomplemented
14#3757⋊C3C2214yes—
15#3757⋊C3C2214yescomplemented
16#88451C2214yescomplemented
17#88449C2214yescomplemented
18#88447C2214yescomplemented
19C22×C3×#67—32no—
20C22×C3×#67—61no—
21C2×#3757D312yescomplemented
22C2×C3×#217—32no—
23#67⋊Dic3—38no—
24Dic3×#67—38no—
25D3×#217—34no—
26#63955—34no—
27C23×H(3,𝔽3)—42no—
28C23×H(3,𝔽3)C2×C414yescomplemented
29C23×H(3,𝔽3)C2311yescharacteristic
30C23×H(3,𝔽3)C2312yes—
31C22×#272—42no—
32C22×#272C2×C414yescomplemented
33C22×#272C2312yes—
34#6235C2×C4116yescomplemented
35C2×#1084C2×C414yescomplemented
36C2×#1084C2312yes—
37C2×(C3⋊Dic3)⋊C3—44no—
38C2×(C3⋊Dic3)⋊C3C2318yes—
39#6199—44no—
40#5798—91no—
41#5486—92no—
42C24×C32—21no—
43C24×C32C2×D312yes—
44C23×C3×D3—32no—
45C23×C3×D3—61no—
46C22×C3⋊Dic3C2×D314yescomplemented
47C2×C3×#67—316no—
48C2×C3×#67—34no—
49C2×C3×#67—34no—
50C2×C3×#67—68no—
51C2×C3×#67—64no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—34no—
54C22×C3×Dic3—61no—
55C2×D3×Dic3—38no—
56#3757C2×D314yes—
57#3757C2×D314yescomplemented
58C3×#217—34no—
59C3×#217—34no—
60Dic3⋊Dic3—38no—
61(C2×D3)⋊Dic3—38no—
62Dic32—38no—
63C22×H(3,𝔽3)—22no—
64C22×H(3,𝔽3)—24no—
65C22×H(3,𝔽3)—44no—
66C22×H(3,𝔽3)C22×C412yes—
67C22×H(3,𝔽3)C22×C414yes—
68C22×H(3,𝔽3)C2411yescharacteristic
69C2×#272—24no—
70C2×#272—44no—
71C2×#272C22×C414yes—
72#1084C22×C414yes—
73(C3⋊Dic3)⋊C3—28no—
74C22×C3×D4—91no—
75C22×#67—32no—
76C22×#67—61no—
77C2×C3×C22⋊C4—92no—
78C2×#217—32no—
79C2×#217—92no—
80C2×#217—92no—
81D4×Dic3—98no—
82C4×#67—98no—
83#926—92no—
84#67⋊C4—98no—
85D3×C22⋊C4—94no—
86#877—94no—
87C23×C32—24no—
88C23×C32—21no—
89C23×C32—22no—
90C23×C32—44no—
91C23×C32—44no—
92C23×C32C4×D318yescomplemented
93C23×C32C22×D312yes—
94C23×C32C22×D312yes—
95C23×C32C22×D312yes—
96C22×C3×D3—34no—
97C22×C3×D3—38no—
98C22×C3×D3—64no—
99C22×C3×D3—62no—
100C22×C3×D3—124no—
101C22×C3×D3—124no—
102C2×C3⋊Dic3—44no—
103C2×C3⋊Dic3C22×D318yes—
104C3×#67—316no—
105C3×#67—316no—
106C3×#67—616no—
107C2×C3×Dic3—38no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—38no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—124no—
113D3×Dic3—128no—
114#518—271noSylow
115C2×H(3,𝔽3)—22no—
116C2×H(3,𝔽3)—24no—
117C2×H(3,𝔽3)C23×C411yescharacteristic
118C2×H(3,𝔽3)H(5,𝔽2)12yes—
119#272—24no—
120C24×C3—32no—
121C24×C3—61no—
122C24×C3—91no—
123C24×C3D3211yescharacteristic
124C23×D3—32no—
125C23×D3—61no—
126C2×C3×D4—98no—
127C2×C3×D4—94no—
128C22×C3×C4—91no—
129C22×C3×C4—94no—
130C2×#67—34no—
131C2×#67—34no—
132C2×#67—316no—
133C2×#67—68no—
134C2×#67—64no—
135C22×Dic3—32no—
136C22×Dic3—34no—
137C22×Dic3—61no—
138C22×Dic3—94no—
139C22×Dic3—94no—
140C2×C4×D3—98no—
141C3×C22⋊C4—94no—
142C3×C22⋊C4—94no—
143#217—34no—
144#217—34no—
145#217—94no—
146#217—94no—
147#217—94no—
148(C2×D3)⋊C4—98no—
149C4⋊Dic3—94no—
150Dic3⋊C4—98no—
151C4×Dic3—94no—
152C4×Dic3—98no—
153C22×C32—24no—
154C22×C32—24no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—21no—
158C22×C32—22no—
159C22×C32—24no—
160C22×C32—44no—
161C22×C32—48no—
162C22×C32—48no—
163C22×C32C2×C4×D314yes—
164C22×C32C2×C4×D314yes—
165C22×C32C2×C4×D314yes—
166C22×C32C23×D312yes—
167C2×C3×D3—38no—
168C2×C3×D3—64no—
169C2×C3×D3—68no—
170C2×C3×D3—124no—
171C2×C3×D3—128no—
172C2×C3×D3—128no—
173C3⋊Dic3—28no—
174C3×Dic3—38no—
175C3×Dic3—64no—
176C3×Dic3—68no—
177C22×D4—91no—
178C4×D4—278no—
179#117—272no—
180C2×C22⋊C4—92no—
181C2×C22⋊C4—272no—
182H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
183C23×C3—38no—
184C23×C3—32no—
185C23×C3—32no—
186C23×C3—32no—
187C23×C3—42no—
188C23×C3—64no—
189C23×C3—61no—
190C23×C3—62no—
191C23×C3—92no—
192C23×C3—94no—
193C23×C3—124no—
194C23×C3—124no—
195C23×C3—362no—
196C23×C3#62414yes—
197C23×C3C2×D3211yescharacteristic
198C23×C3C2×D3212yes—
199C22×D3—34no—
200C22×D3—38no—
201C22×D3—62no—
202C22×D3—64no—
203C22×D3—124no—
204C22×D3—124no—
205C3×D4—916no—
206C2×C3×C4—92no—
207C2×C3×C4—94no—
208C2×C3×C4—98no—
209C2×C3×C4—364no—
210#67—316no—
211#67—316no—
212#67—616no—
213C2×Dic3—34no—
214C2×Dic3—38no—
215C2×Dic3—38no—
216C2×Dic3—62no—
217C2×Dic3—64no—
218C2×Dic3—98no—
219C2×Dic3—124no—
220C2×Dic3—364no—
221C2×Dic3—364no—
222C4×D3—368no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32—44no—
230C2×C32C22×C4×D312yes—
231C2×C32#100414yes—
232C3×D3—68no—
233C3×D3—124no—
234C24—91no—
235C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C2×D4—98no—
237C2×D4—94no—
238C22×C4—91no—
239C22×C4—94no—
240C22×C4—274no—
241C4⋊C4—274no—
242C22⋊C4—94no—
243C22⋊C4—94no—
244C22⋊C4—274no—
245C42—274no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—32no—
249C22×C3—34no—
250C22×C3—34no—
251C22×C3—34no—
252C22×C3—44no—
253C22×C3—64no—
254C22×C3—64no—
255C22×C3—64no—
256C22×C3—64no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3—64no—
260C22×C3—94no—
261C22×C3—124no—
262C22×C3—128no—
263C22×C3—128no—
264C22×C3—184no—
265C22×C3—364no—
266C22×C3C2×#62412yes—
267C22×C3C2×#62414yes—
268C22×C3C22×D3211yescharacteristic
269C2×D3—38no—
270C2×D3—64no—
271C2×D3—68no—
272C2×D3—124no—
273C2×D3—128no—
274C2×D3—128no—
275C3×C4—94no—
276C3×C4—188no—
277Dic3—38no—
278Dic3—64no—
279Dic3—68no—
280Dic3—188no—
281C32—21no—
282C32#548612yescomplemented
283C23—42no—
284C23—92no—
285C23—94no—
286C23—362no—
287C23#619914yescomplemented
288C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C23C2×((C3⋊D3)⋊C3)⋊C212yes—
290D4—916no—
291C2×C4—92no—
292C2×C4—94no—
293C2×C4—98no—
294C2×C4—364no—
295C2×C4—1084no—
296C2×C3—22no—
297C2×C3—24no—
298C2×C3—32no—
299C2×C3—34no—
300C2×C3—64no—
301C2×C3—64no—
302C2×C3—64no—
303C2×C3—64no—
304C2×C3—61no—
305C2×C3—62no—
306C2×C3—124no—
307C2×C3—184no—
308C2×C3C22×#62411yescharacteristic
309C2×C3#6433012yes—
310D3—68no—
311D3—124no—
312C22—24no—
313C22—22no—
314C22—44no—
315C22—94no—
316C22—184no—
317C22—364no—
318C22C2×#619912yes—
319C22C2×#619914yes—
320C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C4—94no—
322C4—188no—
323C3—32no—
324C3—61no—
325C3#1059551011yescharacteristic
326C2—22no—
327C2—24no—
328C2—184no—
329C2C22×#619911yescharacteristic
330C2#1172254512yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
prime radicalC22×#6235C2
cosocleC22×H(3,𝔽3)C24
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#3757⋊C3C2semidirect
3. #11721320C2semidirect
4. #11721298C2semidirect
5. #11721293C2semidirect
9. C2×#6235C4semidirect
12. C22×(C3⋊Dic3)⋊C3C22semidirect
13. C2×#6199C22semidirect
15. #3757⋊C3C22semidirect
16. #88451C22semidirect
17. #88449C22semidirect
18. #88447C22semidirect
21. C2×#3757D3semidirect
28. C23×H(3,𝔽3)C2×C4semidirect
32. C22×#272C2×C4semidirect
34. #6235C2×C4semidirect
35. C2×#1084C2×C4semidirect
46. C22×C3⋊Dic3C2×D3semidirect
57. #3757C2×D3semidirect
92. C23×C32C4×D3semidirect
182. H(3,𝔽3)#518semidirect
282. C32#5486semidirect
287. C23#6199semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C2×#3757⋊C32
#117213202
#117212982
#117212932
#105955063