Finite Group Databasebeta

← #126×(C3⋊D3)⋊C3 #13153234

Subgroups — #126×(C3⋊D3)⋊C3 #13153234

Order 1728.

921 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 921 conjugacy classes of subgroups — too many to draw. Only its 222 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
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#64107⋊C3C213yesmaximal, complemented
3C4⋊C4×(C3⋊D3)⋊C3C213yesmaximal, complemented
4C22⋊C4×(C3⋊D3)⋊C3C213yesmaximal
5#64088⋊C3C213yesmaximal, complemented
6#11721387C211yesmaximal, characteristic, complemented
7C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8C3⋊D3×#126C311yesmaximal, characteristic, complemented
9C3×D3×#126—31nomaximal
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2213yes—
12C2×C4×(C3⋊D3)⋊C3C2213yes—
13C4⋊C4×H(3,𝔽3)C2213yescomplemented
14C22⋊C4×H(3,𝔽3)C2213yes—
15C42×H(3,𝔽3)C2211yescharacteristic, complemented
16#3757⋊C3C2213yes—
17#3752⋊C3C2213yes—
18#3752⋊C3C2213yes—
19C4⋊((C3⋊Dic3)⋊C3)C2213yescomplemented
20#3750⋊C3C2213yescomplemented
21#3750⋊C3C2213yescomplemented
22C4×(C3⋊Dic3)⋊C3C2213yescomplemented
23C32×#126—21no—
24C32×#126D311yescharacteristic, complemented
25C32×#126C2×C311yescharacteristic, complemented
26#64107C2×C313yescomplemented
27C4⋊C4×C3⋊D3C2×C313yescomplemented
28C22⋊C4×C3⋊D3C2×C313yes—
29#64088C2×C313yescomplemented
30#64085C2×C311yescharacteristic, complemented
31C42×C3⋊D3C2×C311yescharacteristic, complemented
32C3×#898—33no—
33C3×D3×C4⋊C4—33no—
34C3×D3×C22⋊C4—33no—
35C3×#879—33no—
36C3×#876—31no—
37C3×C42×D3—31no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—26no—
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
41C22×(C3⋊D3)⋊C3C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2313yes—
43C2×C4×H(3,𝔽3)C2313yes—
44C2×(C3⋊Dic3)⋊C3C2313yes—
45C2×(C3⋊Dic3)⋊C3C2313yes—
46C4×(C3⋊D3)⋊C3—26no—
47C4×(C3⋊D3)⋊C3—26no—
48C2×C3×#126—91no—
49D3×#126—31no—
50D3×#126—91no—
51C23×C3⋊D3C22×C311yescharacteristic
52C23×C3×D3—31no—
53C2×C4×C3⋊D3C22×C313yes—
54C2×C4×C3⋊D3C22×C313yes—
55C2×C3×C4×D3—33no—
56C2×C3×C4×D3—33no—
57C32×C4⋊C4—23no—
58C32×C4⋊C4C2×D313yescomplemented
59C32×C4⋊C4C22×C313yescomplemented
60C32×C22⋊C4—23no—
61C32×C22⋊C4C2×D313yes—
62C32×C22⋊C4C22×C313yes—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic, complemented
65C32×C42C22×C311yescharacteristic, complemented
66#3757C22×C313yes—
67#3752C22×C313yes—
68#3752C22×C313yes—
69C4⋊(C3⋊Dic3)C22×C313yescomplemented
70#3750C22×C313yescomplemented
71#3750C22×C313yescomplemented
72C4×C3⋊Dic3C22×C313yescomplemented
73C3×#217—33no—
74C3×(C2×D3)⋊C4—33no—
75C3×(C2×D3)⋊C4—33no—
76C3×C4⋊Dic3—33no—
77C3×Dic3⋊C4—33no—
78C3×Dic3⋊C4—33no—
79C3×C4×Dic3—33no—
80C22×H(3,𝔽3)—23no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×(C3⋊D3)⋊C3—23no—
83C2×(C3⋊D3)⋊C3—44no—
84C2×(C3⋊D3)⋊C3#4116yes—
85C4×H(3,𝔽3)—23no—
86C4×H(3,𝔽3)—23no—
87(C3⋊Dic3)⋊C3—23no—
88(C3⋊Dic3)⋊C3—23no—
89C3×#126—31no—
90C3×#126—31no—
91C3×#126—61no—
92C3×#126—91no—
93C3×#126—93no—
94C3×#126—93no—
95C3×#126C3×D311yescharacteristic
96C2×C3×C4⋊C4—93no—
97C2×C3×C22⋊C4—93no—
98C2×C3×C42—91no—
99#898—33no—
100#898—93no—
101D3×C4⋊C4—33no—
102D3×C4⋊C4—93no—
103D3×C22⋊C4—33no—
104D3×C22⋊C4—93no—
105#879—33no—
106#879—93no—
107#876—31no—
108#876—91no—
109C42×D3—31no—
110C42×D3—91no—
111C23×C32—21no—
112C23×C32C22×D311yescharacteristic
113C23×C32C23×C311yescharacteristic
114C22×C3⋊D3—26no—
115C22×C3⋊D3C23×C311yescharacteristic
116C22×C3⋊D3C23×C311yescharacteristic
117C22×C3×D3—31no—
118C22×C3×D3—31no—
119C22×C3×D3—66no—
120C2×C32×C4—23no—
121C2×C32×C4—23no—
122C2×C32×C4C22×D313yes—
123C2×C32×C4C22×D313yes—
124C2×C32×C4C23×C313yes—
125C2×C32×C4C23×C313yes—
126C2×C3⋊Dic3C23×C313yes—
127C2×C3⋊Dic3C23×C313yes—
128C4×C3⋊D3—26no—
129C4×C3⋊D3—26no—
130C2×C3×Dic3—33no—
131C2×C3×Dic3—33no—
132C3×C4×D3—66no—
133C3×C4×D3—66no—
134C2×#126—91noSylow
135C2×H(3,𝔽3)—41no—
136C2×H(3,𝔽3)C2×#4113yes—
137(C3⋊D3)⋊C3—41no—
138(C3⋊D3)⋊C3#12614yesdirect product factor
139C24×C3—91no—
140C23×D3—31no—
141C23×D3—91no—
142C22×C3×C4—93no—
143C22×C3×C4—93no—
144C2×C4×D3—33no—
145C2×C4×D3—33no—
146C2×C4×D3—93no—
147C2×C4×D3—93no—
148C3×C4⋊C4—33no—
149C3×C4⋊C4—33no—
150C3×C4⋊C4—63no—
151C3×C4⋊C4—93no—
152C3×C4⋊C4—93no—
153C3×C4⋊C4—93no—
154C3×C4⋊C4C2×C3×D313yes—
155C3×C22⋊C4—33no—
156C3×C22⋊C4—33no—
157C3×C22⋊C4—63no—
158C3×C22⋊C4—93no—
159C3×C22⋊C4—93no—
160C3×C22⋊C4—93no—
161C3×C22⋊C4C2×C3×D313yes—
162C3×C42—31no—
163C3×C42—31no—
164C3×C42—61no—
165C3×C42—93no—
166C3×C42C2×C3×D311yescharacteristic
167#217—33no—
168#217—93no—
169(C2×D3)⋊C4—33no—
170(C2×D3)⋊C4—33no—
171(C2×D3)⋊C4—93no—
172(C2×D3)⋊C4—93no—
173C4⋊Dic3—33no—
174C4⋊Dic3—93no—
175Dic3⋊C4—33no—
176Dic3⋊C4—33no—
177Dic3⋊C4—93no—
178Dic3⋊C4—93no—
179C4×Dic3—33no—
180C4×Dic3—93no—
181C22×C32—23no—
182C22×C32—23no—
183C22×C32—21no—
184C22×C32—43no—
185C22×C32C23×D311yescharacteristic
186C22×C32C24×C311yescharacteristic
187C2×C3⋊D3—23no—
188C2×C3⋊D3—44no—
189C2×C3⋊D3C3×#4116yes—
190C2×C3×D3—36no—
191C2×C3×D3—63no—
192C2×C3×D3—124no—
193C32×C4—23no—
194C32×C4—23no—
195C32×C4—23no—
196C32×C4—23no—
197C32×C4—43no—
198C32×C4—43no—
199C3⋊Dic3—23no—
200C3⋊Dic3—23no—
201C3×Dic3—63no—
202C3×Dic3—63no—
203#126—93no—
204#126—93no—
205#126—91no—
206#126(C3⋊D3)⋊C311yescharacteristic, direct product factor
207C2×C4⋊C4—93no—
208C2×C22⋊C4—93no—
209C2×C42—91no—
210H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
211C23×C3—31no—
212C23×C3—31no—
213C23×C3—61no—
214C23×C3—91no—
215C23×C3—91no—
216C23×C3—186no—
217C23×C3C22×C3×D311yescharacteristic
218C22×D3—31no—
219C22×D3—31no—
220C22×D3—66no—
221C22×D3—91no—
222C22×D3—91no—
223C22×D3—186no—
224C2×C3×C4—33no—
225C2×C3×C4—33no—
226C2×C3×C4—33no—
227C2×C3×C4—33no—
228C2×C3×C4—63no—
229C2×C3×C4—63no—
230C2×C3×C4—93no—
231C2×C3×C4—93no—
232C2×C3×C4—186no—
233C2×C3×C4—186no—
234C2×C3×C4C22×C3×D313yes—
235C2×C3×C4C22×C3×D313yes—
236C2×Dic3—33no—
237C2×Dic3—33no—
238C2×Dic3—93no—
239C2×Dic3—93no—
240C4×D3—66no—
241C4×D3—66no—
242C4×D3—186no—
243C4×D3—186no—
244C2×C32—23no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—81no—
248C2×C32D3×#4113yes—
249C2×C32C2×C3×#4113yes—
250C3⋊D3—41no—
251C3⋊D3C3×#12614yescomplemented
252C3×D3—34no—
253C3×D3—121no—
254C24—91no—
255C22×C4—93no—
256C22×C4—93no—
257C4⋊C4—93no—
258C4⋊C4—93no—
259C4⋊C4—93no—
260C4⋊C4C2×(C3⋊D3)⋊C313yescomplemented
261C22⋊C4—93no—
262C22⋊C4—93no—
263C22⋊C4—93no—
264C22⋊C4C2×(C3⋊D3)⋊C313yes—
265C42—93no—
266C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
267C22×C3—23no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—63no—
271C22×C3—63no—
272C22×C3—61no—
273C22×C3—96no—
274C22×C3—123no—
275C22×C3—183no—
276C22×C3—364no—
277C22×C3C23×C3×D311yescharacteristic
278C2×D3—36no—
279C2×D3—63no—
280C2×D3—96no—
281C2×D3—124no—
282C2×D3—183no—
283C2×D3—364no—
284C3×C4—23no—
285C3×C4—23no—
286C3×C4—63no—
287C3×C4—63no—
288C3×C4—63no—
289C3×C4—63no—
290C3×C4—123no—
291C3×C4—123no—
292C3×C4—183no—
293C3×C4—183no—
294Dic3—63no—
295Dic3—63no—
296Dic3—183no—
297Dic3—183no—
298C32—21no—
299C32D3×#12611yescharacteristic, complemented
300C32C2×C3×#12611yescharacteristic, complemented
301C23—91no—
302C23—91no—
303C23—186no—
304C23C22×(C3⋊D3)⋊C311yescharacteristic
305C2×C4—93no—
306C2×C4—93no—
307C2×C4—186no—
308C2×C4—186no—
309C2×C4C22×(C3⋊D3)⋊C313yes—
310C2×C4C22×(C3⋊D3)⋊C313yes—
311C2×C3—33no—
312C2×C3—33no—
313C2×C3—41no—
314C2×C3—63no—
315C2×C3—94no—
316C2×C3—121no—
317C2×C3—121no—
318C2×C3—241no—
319C2×C3—361no—
320C2×C3C3×D3×#4113yes—
321D3—34no—
322D3—94no—
323D3—121no—
324D3—361no—
325C22—23no—
326C22—96no—
327C22—183no—
328C22—364no—
329C22C23×(C3⋊D3)⋊C311yescharacteristic
330C4—23no—
331C4—23no—
332C4—183no—
333C4—183no—
334C3—31no—
335C3—31no—
336C3—61no—
337C3C3×D3×#12611yescharacteristic
338C2—41no—
339C2—94no—
340C2—361no—
341C2#41×(C3⋊D3)⋊C313yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#126C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenter#126(C3⋊D3)⋊C3
hypocenterC32C2×C3×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #64107⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
5. #64088⋊C3C2semidirect
6. #11721387C2semidirect
7. C42×(C3⋊D3)⋊C3C2semidirect
8. C3⋊D3×#126C3semidirect
13. C4⋊C4×H(3,𝔽3)C22semidirect
15. C42×H(3,𝔽3)C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #3750⋊C3C22semidirect
21. #3750⋊C3C22semidirect
22. C4×(C3⋊Dic3)⋊C3C22semidirect
24. C32×#126D3semidirect
25. C32×#126C2×C3semidirect
26. #64107C2×C3semidirect
27. C4⋊C4×C3⋊D3C2×C3semidirect
29. #64088C2×C3semidirect
30. #64085C2×C3semidirect
31. C42×C3⋊D3C2×C3semidirect
58. C32×C4⋊C4C2×D3semidirect
59. C32×C4⋊C4C22×C3semidirect
64. C32×C42C2×D3semidirect
65. C32×C42C22×C3semidirect
69. C4⋊(C3⋊Dic3)C22×C3semidirect
70. #3750C22×C3semidirect
71. #3750C22×C3semidirect
72. C4×C3⋊Dic3C22×C3semidirect
138. (C3⋊D3)⋊C3#126direct (N × H)
206. #126(C3⋊D3)⋊C3direct (N × H)
210. H(3,𝔽3)C2×#126semidirect
251. C3⋊D3C3×#126semidirect
260. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
266. C42C2×(C3⋊D3)⋊C3semidirect
299. C32D3×#126semidirect
300. C32C2×C3×#126semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#64107⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C22⋊C4×(C3⋊D3)⋊C32
#64088⋊C32
#117213872
C42×(C3⋊D3)⋊C32
C3⋊D3×#1263
C3×D3×#1263