Finite Group Databasebeta

← #13147914

Subgroups — #13147914

Order 1728.

921 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 921 conjugacy classes of subgroups — too many to draw. Only its 149 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
1#11721722C211yesmaximal, characteristic, complemented
2#11721387C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11721160C213yesmaximal
5#11721119C213yesmaximal, complemented
6#11721109C213yesmaximal, complemented
7#11721092C213yesmaximal, complemented
8#10594020—31nomaximal
9D3×#876—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2213yes—
12C2×((C3⋊D3)⋊C3)⋊C4C2213yes—
13C42×H(3,𝔽3)C2211yescharacteristic, complemented
14#88542C2213yes—
15#88540C2213yescomplemented
16#3752⋊C3C2213yes—
17#3750⋊C3C2213yescomplemented
18C4×(C3⋊Dic3)⋊C3C2213yescomplemented
19#88451C2213yes—
20#88450C2213yes—
21#88449C2213yescomplemented
22#88448C2213yescomplemented
23#64085D311yescharacteristic, complemented
24C42×C3⋊D3D311yescharacteristic, complemented
25C3×#876—31no—
26C3×#876—31no—
27C3×#876—31no—
28C3×#876—61no—
29C3×C42×D3—31no—
30#63921—33no—
31D3×(C2×D3)⋊C4—33no—
32#63883—33no—
33D3×Dic3⋊C4—33no—
34#63868—33no—
35#63862—33no—
36#63862—33no—
37#63859—33no—
38C22×#272C2311yescharacteristic
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
41C2×((C3⋊D3)⋊C3)⋊C2—26no—
42C2×C4×H(3,𝔽3)C2313yes—
43C2×#1084C2313yes—
44C2×(C3⋊Dic3)⋊C3C2313yes—
45C2×(C3⋊Dic3)⋊C3C2313yes—
46C4×(C3⋊D3)⋊C3—26no—
47((C3⋊D3)⋊C3)⋊C4—26no—
48D3×#126—91no—
49D3×#126—91no—
50C2×#876—91no—
51C22×D32—31no—
52C22×D32—31no—
53C2×C4×C3⋊D3C2×D313yes—
54C2×C3×C4×D3—33no—
55C2×#624—33no—
56C2×D3×Dic3—33no—
57C32×C42—21no—
58C32×C42C2×D311yescharacteristic, complemented
59C32×C42C2×D311yescharacteristic, complemented
60#3752C2×D313yes—
61#3750C2×D313yescomplemented
62C4×C3⋊Dic3C2×D313yescomplemented
63C3×(C2×D3)⋊C4—33no—
64C3×(C2×D3)⋊C4—33no—
65C3×(C2×D3)⋊C4—33no—
66C3×(C2×D3)⋊C4—63no—
67C3×Dic3⋊C4—33no—
68C3×Dic3⋊C4—33no—
69C3×Dic3⋊C4—33no—
70C3×Dic3⋊C4—63no—
71C3×C4×Dic3—33no—
72#3724—33no—
73Dic3⋊Dic3—33no—
74Dic3⋊Dic3—33no—
75Dic3⋊Dic3—33no—
76#3722—33no—
77(C2×D3)⋊Dic3—33no—
78(C2×D3)⋊Dic3—33no—
79(C2×D3)⋊Dic3—33no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×#272—23no—
82C2×(C3⋊D3)⋊C3—23no—
83C2×(C3⋊D3)⋊C3#4116yes—
84((C3⋊D3)⋊C3)⋊C2—44no—
85C4×H(3,𝔽3)—23no—
86#1084—23no—
87(C3⋊Dic3)⋊C3—23no—
88(C3⋊Dic3)⋊C3—23no—
89C3×#126—91no—
90C3×#126—91no—
91C2×C3×C42—91no—
92C2×(C2×D3)⋊C4—93no—
93C2×Dic3⋊C4—93no—
94#898—93no—
95#898—93no—
96D3×C4⋊C4—93no—
97D3×C4⋊C4—93no—
98D3×C22⋊C4—93no—
99D3×C22⋊C4—93no—
100#879—93no—
101#879—93no—
102#876—31no—
103#876—31no—
104#876—31no—
105#876—61no—
106#876—91no—
107#876—93no—
108#876—93no—
109C42×D3—31no—
110C42×D3—91no—
111C22×C3⋊D3C22×D311yescharacteristic
112C22×C3⋊D3C22×D311yescharacteristic
113C22×C3×D3—31no—
114C22×C3×D3—31no—
115C22×C3×D3—31no—
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C2×D32—66no—
119C2×D32—66no—
120C2×C32×C4—23no—
121C2×C32×C4C22×D313yes—
122C2×C32×C4C22×D313yes—
123C2×C3⋊Dic3C22×D313yes—
124C2×C3⋊Dic3C22×D313yes—
125C4×C3⋊D3—26no—
126C2×C3×Dic3—33no—
127C2×C3×Dic3—33no—
128C2×C3×Dic3—33no—
129C2×C3×Dic3—33no—
130C2×C3×Dic3—63no—
131C3×C4×D3—66no—
132#624—66no—
133D3×Dic3—66no—
134C2×#126—271noSylow
135C2×H(3,𝔽3)C2×#4113yes—
136#272—41no—
137(C3⋊D3)⋊C3—41no—
138(C3⋊D3)⋊C3#12614yescomplemented
139C23×D3—91no—
140C23×D3—91no—
141C23×D3—91no—
142C22×C3×C4—93no—
143C22×Dic3—93no—
144C2×C4×D3—33no—
145C2×C4×D3—93no—
146C2×C4×D3—93no—
147C2×C4×D3—93no—
148C3×C4⋊C4—93no—
149C3×C4⋊C4—93no—
150C3×C22⋊C4—93no—
151C3×C22⋊C4—93no—
152C3×C42—31no—
153C3×C42—31no—
154C3×C42—61no—
155C3×C42—93no—
156C3×C42D3211yescharacteristic
157#217—93no—
158#217—93no—
159(C2×D3)⋊C4—33no—
160(C2×D3)⋊C4—33no—
161(C2×D3)⋊C4—33no—
162(C2×D3)⋊C4—63no—
163(C2×D3)⋊C4—93no—
164(C2×D3)⋊C4—93no—
165(C2×D3)⋊C4—93no—
166(C2×D3)⋊C4—93no—
167(C2×D3)⋊C4—93no—
168C4⋊Dic3—93no—
169C4⋊Dic3—93no—
170Dic3⋊C4—33no—
171Dic3⋊C4—33no—
172Dic3⋊C4—33no—
173Dic3⋊C4—63no—
174Dic3⋊C4—93no—
175Dic3⋊C4—93no—
176Dic3⋊C4—93no—
177Dic3⋊C4—93no—
178Dic3⋊C4—93no—
179C4×Dic3—33no—
180C4×Dic3—93no—
181C22×C32—21no—
182C22×C32C23×D311yescharacteristic
183C22×C32C23×D311yescharacteristic
184C2×C3⋊D3—23no—
185C2×C3⋊D3#23516yes—
186C2×C3×D3—36no—
187C2×C3×D3—63no—
188C2×C3×D3—63no—
189C2×C3×D3—63no—
190C2×C3×D3—123no—
191D32—124no—
192D32—124no—
193C32×C4—23no—
194C32×C4—23no—
195C32×C4—43no—
196C3⋊Dic3—23no—
197C3⋊Dic3—23no—
198C3×Dic3—63no—
199C3×Dic3—63no—
200C3×Dic3—63no—
201C3×Dic3—63no—
202C3×Dic3—123no—
203#126—91no—
204#126—91no—
205#126—273no—
206#126—273no—
207C2×C4⋊C4—273no—
208C2×C22⋊C4—273no—
209C2×C42—91no—
210H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C23×C3—91no—
213C23×C3—91no—
214C22×D3—31no—
215C22×D3—31no—
216C22×D3—31no—
217C22×D3—31no—
218C22×D3—61no—
219C22×D3—91no—
220C22×D3—91no—
221C22×D3—186no—
222C22×D3—186no—
223C22×D3—186no—
224C2×C3×C4—33no—
225C2×C3×C4—33no—
226C2×C3×C4—63no—
227C2×C3×C4—93no—
228C2×C3×C4—93no—
229C2×C3×C4—93no—
230C2×C3×C4—186no—
231C2×C3×C4C2×D3213yes—
232C2×Dic3—33no—
233C2×Dic3—33no—
234C2×Dic3—33no—
235C2×Dic3—33no—
236C2×Dic3—63no—
237C2×Dic3—93no—
238C2×Dic3—93no—
239C2×Dic3—186no—
240C4×D3—66no—
241C4×D3—186no—
242C4×D3—186no—
243C4×D3—186no—
244C2×C32—23no—
245C2×C32C2×#23513yes—
246C2×C32D3×#4113yes—
247C3⋊D3—41no—
248C3⋊D3#87614yescomplemented
249C3×D3—34no—
250C3×D3—121no—
251C3×D3—121no—
252C3×D3—121no—
253C3×D3—241no—
254C24—271no—
255C22×C4—93no—
256C22×C4—273no—
257C4⋊C4—93no—
258C4⋊C4—93no—
259C4⋊C4—273no—
260C4⋊C4—273no—
261C22⋊C4—93no—
262C22⋊C4—93no—
263C22⋊C4—273no—
264C22⋊C4—273no—
265C42—93no—
266C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—61no—
270C22×C3—96no—
271C22×C3—183no—
272C22×C3—183no—
273C22×C3C22×D3211yescharacteristic
274C2×D3—36no—
275C2×D3—63no—
276C2×D3—63no—
277C2×D3—63no—
278C2×D3—96no—
279C2×D3—123no—
280C2×D3—183no—
281C2×D3—364no—
282C2×D3—364no—
283C2×D3—364no—
284C3×C4—23no—
285C3×C4—63no—
286C3×C4—63no—
287C3×C4—123no—
288C3×C4—183no—
289C3×C4—183no—
290C3×C4—183no—
291Dic3—63no—
292Dic3—63no—
293Dic3—63no—
294Dic3—63no—
295Dic3—123no—
296Dic3—183no—
297Dic3—183no—
298C32—21no—
299C32C2×#87611yescharacteristic, complemented
300C32D3×#12611yescharacteristic, complemented
301C23—91no—
302C23—91no—
303C23—91no—
304C23—546no—
305C2×C4—93no—
306C2×C4—93no—
307C2×C4—93no—
308C2×C4—186no—
309C2×C4—546no—
310C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
311C2×C3—33no—
312C2×C3—33no—
313C2×C3—63no—
314C2×C3—94no—
315C2×C3—361no—
316C2×C3—361no—
317C2×C3D3×#23513yes—
318D3—34no—
319D3—94no—
320D3—121no—
321D3—121no—
322D3—121no—
323D3—241no—
324D3—361no—
325C22—96no—
326C22—183no—
327C22—183no—
328C22—1084no—
329C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
330C4—23no—
331C4—183no—
332C4—183no—
333C4—183no—
334C3—31no—
335C3—31no—
336C3—61no—
337C3D3×#87611yescharacteristic
338C2—94no—
339C2—361no—
340C2—361no—
341C2#1172249013yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721722C2semidirect
2. #11721387C2semidirect
3. C42×(C3⋊D3)⋊C3C2semidirect
5. #11721119C2semidirect
6. #11721109C2semidirect
7. #11721092C2semidirect
13. C42×H(3,𝔽3)C22semidirect
15. #88540C22semidirect
17. #3750⋊C3C22semidirect
18. C4×(C3⋊Dic3)⋊C3C22semidirect
21. #88449C22semidirect
22. #88448C22semidirect
23. #64085D3semidirect
24. C42×C3⋊D3D3semidirect
58. C32×C42C2×D3semidirect
59. C32×C42C2×D3semidirect
61. #3750C2×D3semidirect
62. C4×C3⋊Dic3C2×D3semidirect
138. (C3⋊D3)⋊C3#126semidirect
210. H(3,𝔽3)C2×#126semidirect
248. C3⋊D3#876semidirect
266. C42((C3⋊D3)⋊C3)⋊C2semidirect
299. C32C2×#876semidirect
300. C32D3×#126semidirect

Maximal subgroups

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

subgroupindex
#117217222
#117213872
C42×(C3⋊D3)⋊C32
#117211602
#117211192
#117211092
#117210922
#105940203
D3×#8763