Finite Group Databasebeta

← #13147894

Subgroups — #13147894

Order 1728.

1263 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1263 conjugacy classes of subgroups — too many to draw. Only its 211 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C42×#272C211yesmaximal, characteristic, complemented
3#64081⋊C3C212yesmaximal, complemented
4#11721161C212yesmaximal
5#11721147C212yesmaximal, complemented
6C4×#6199C211yesmaximal, characteristic, complemented
7#11721121C214yesmaximal, complemented
8#11721105C212yesmaximal, complemented
9D3×#872—32nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
11C2×C4×#272C2212yes—
12C2×C4×#272C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#6199C2212yes—
15C2×#6199C2211yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
17C4×((C3⋊D3)⋊C3)⋊C2C418yescomplemented
18C42×H(3,𝔽3)C2211yescharacteristic, complemented
19C4×#1084C2211yescharacteristic, complemented
20C4×#1084C2212yescomplemented
21#3752⋊C3C2214yes—
22#3750⋊C3C2214yescomplemented
23C4×(C3⋊Dic3)⋊C3C2212yescomplemented
24#88450C2214yes—
25#88448C2214yescomplemented
26#88447C2212yescomplemented
27C2×C4×D32—32no—
28#64081D312yescomplemented
29C3×#872—32no—
30C3×C42×D3—32no—
31C3×C42×D3—61no—
32D3×(C2×D3)⋊C4—34no—
33#63877—34no—
34D3×Dic3⋊C4—34no—
35C4×D3×Dic3—32no—
36#63874—32no—
37#63855—32no—
38#63836—34no—
39C22×#272C2311yescharacteristic
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×((C3⋊D3)⋊C3)⋊C2—24no—
42C2×((C3⋊D3)⋊C3)⋊C2C2×C414yes—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084C2312yes—
46C2×#1084C2311yescharacteristic
47C4×#272—24no—
48C4×#272C2×C414yes—
49C2×(C3⋊Dic3)⋊C3C2314yes—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C4×(C3⋊D3)⋊C3C2×C418yescomplemented
52#6199—24no—
53#6199C2×C414yes—
54((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
55D3×#117—92no—
56C2×#872—91no—
57C22×D32—32no—
58C2×C4×C3⋊D3C2×D312yes—
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—34no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—62no—
64C2×#624—32no—
65C2×D3×Dic3—32no—
66C2×D3×Dic3—32no—
67C2×D3×Dic3—34no—
68C4×D32—316no—
69C32×C42—21no—
70C32×C42C2×D312yescomplemented
71#3752C2×D314yes—
72#3750C2×D314yescomplemented
73C4×C3⋊Dic3C2×D312yescomplemented
74C3×(C2×D3)⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×C4×Dic3—34no—
77C3×C4×Dic3—32no—
78C3×C4×Dic3—32no—
79C3×C4×Dic3—61no—
80C3×C4×Dic3—62no—
81#3724—34no—
82Dic3⋊Dic3—34no—
83#3722—34no—
84(C2×D3)⋊Dic3—34no—
85Dic32—34no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×#272C22×C412yes—
88C2×#272#4114yes—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3C22×C414yes—
91((C3⋊D3)⋊C3)⋊C2—28no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)C22×C412yes—
94#1084—22no—
95#1084C22×C412yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3C22×C414yes—
98C22×C4×D3—91no—
99C22×C4×D3—92no—
100C3×#117—92no—
101C2×C3×C42—91no—
102C2×(C2×D3)⋊C4—92no—
103#926—92no—
104C2×Dic3⋊C4—92no—
105C2×C4×Dic3—91no—
106#892—94no—
107D3×C4⋊C4—94no—
108D3×C22⋊C4—94no—
109#877—94no—
110#872—32no—
111#872—92no—
112#872—92no—
113#872—92no—
114#872—94no—
115C42×D3—32no—
116C42×D3—61no—
117C42×D3—92no—
118C22×C3⋊D3C22×D312yes—
119C22×C3×D3—32no—
120C22×C3×D3—32no—
121C22×C3×D3—61no—
122C2×D32—38no—
123C2×D32—68no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D312yes—
128C2×C3⋊Dic3C22×D314yes—
129C2×C3⋊Dic3C22×D312yes—
130C4×C3⋊D3C4×D318yescomplemented
131C2×C3×Dic3—32no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—62no—
137C3×C4×D3—38no—
138C3×C4×D3—38no—
139C3×C4×D3—64no—
140C3×C4×D3—68no—
141C3×C4×D3—68no—
142#624—38no—
143D3×Dic3—38no—
144D3×Dic3—38no—
145D3×Dic3—68no—
146C2×#117—271noSylow
147C2×H(3,𝔽3)C23×C411yescharacteristic
148C2×H(3,𝔽3)C2×#4112yes—
149#272#11714yescomplemented
150(C3⋊D3)⋊C3—24no—
151C23×D3—91no—
152C23×D3—92no—
153C22×C3×C4—92no—
154C22×C3×C4—91no—
155C22×C3×C4—92no—
156C22×Dic3—91no—
157C22×Dic3—92no—
158C22×Dic3—92no—
159C2×C4×D3—32no—
160C2×C4×D3—32no—
161C2×C4×D3—34no—
162C2×C4×D3—61no—
163C2×C4×D3—62no—
164C2×C4×D3—92no—
165C2×C4×D3—98no—
166C2×C4×D3—916no—
167C2×C4×D3—92no—
168C2×C4×D3—92no—
169C2×C4×D3—92no—
170C2×C4×D3—94no—
171C3×C4⋊C4—94no—
172C3×C22⋊C4—94no—
173C3×C42—32no—
174C3×C42—61no—
175C3×C42—92no—
176C3×C42—92no—
177C3×C42—91no—
178C3×C42D3211yescharacteristic
179#217—94no—
180(C2×D3)⋊C4—34no—
181(C2×D3)⋊C4—94no—
182(C2×D3)⋊C4—94no—
183(C2×D3)⋊C4—94no—
184C4⋊Dic3—94no—
185Dic3⋊C4—34no—
186Dic3⋊C4—94no—
187Dic3⋊C4—94no—
188Dic3⋊C4—94no—
189C4×Dic3—32no—
190C4×Dic3—34no—
191C4×Dic3—32no—
192C4×Dic3—62no—
193C4×Dic3—61no—
194C4×Dic3—92no—
195C4×Dic3—92no—
196C4×Dic3—94no—
197C22×C32—21no—
198C22×C32C23×D312yes—
199C2×C3⋊D3—24no—
200C2×C3⋊D3C2×C4×D314yes—
201C2×C3×D3—34no—
202C2×C3×D3—38no—
203C2×C3×D3—34no—
204C2×C3×D3—62no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207D32—616no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—22no—
211C32×C4C2×C4×D314yes—
212C3⋊Dic3—24no—
213C3⋊Dic3C2×C4×D314yes—
214C3×Dic3—34no—
215C3×Dic3—34no—
216C3×Dic3—62no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—64no—
220C23×C4—271no—
221#117—92no—
222#117—272no—
223#117—274no—
224C2×C4⋊C4—272no—
225C2×C22⋊C4—272no—
226C2×C42—91no—
227C2×C42—271no—
228H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
229C23×C3—92no—
230C23×C3—91no—
231C22×D3—32no—
232C22×D3—32no—
233C22×D3—61no—
234C22×D3—92no—
235C22×D3—94no—
236C22×D3—98no—
237C22×D3—188no—
238C22×D3—184no—
239C2×C3×C4—34no—
240C2×C3×C4—32no—
241C2×C3×C4—62no—
242C2×C3×C4—61no—
243C2×C3×C4—98no—
244C2×C3×C4—92no—
245C2×C3×C4—91no—
246C2×C3×C4—94no—
247C2×C3×C4—94no—
248C2×C3×C4—92no—
249C2×C3×C4—184no—
250C2×C3×C4C2×D3212yes—
251C2×C3×C4C2×D3211yescharacteristic
252C2×Dic3—32no—
253C2×Dic3—32no—
254C2×Dic3—34no—
255C2×Dic3—34no—
256C2×Dic3—61no—
257C2×Dic3—62no—
258C2×Dic3—92no—
259C2×Dic3—94no—
260C2×Dic3—94no—
261C2×Dic3—98no—
262C2×Dic3—184no—
263C4×D3—38no—
264C4×D3—38no—
265C4×D3—64no—
266C4×D3—68no—
267C4×D3—68no—
268C4×D3—98no—
269C4×D3—98no—
270C4×D3—98no—
271C4×D3—98no—
272C4×D3—188no—
273C2×C32—21no—
274C2×C32—22no—
275C2×C32C22×C4×D312yes—
276C2×C32D3×#4114yes—
277C3⋊D3—24no—
278C3×D3—38no—
279C3×D3—64no—
280C3×D3—64no—
281C24—271no—
282C22×C4—91no—
283C22×C4—92no—
284C22×C4—92no—
285C22×C4—278no—
286C22×C4—271no—
287C22×C4—272no—
288C22×C4—272no—
289C4⋊C4—94no—
290C4⋊C4—274no—
291C22⋊C4—94no—
292C22⋊C4—274no—
293C42—92no—
294C42—91no—
295C42—92no—
296C42—272no—
297C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
298C22×C3—32no—
299C22×C3—61no—
300C22×C3—94no—
301C22×C3—92no—
302C22×C3—94no—
303C22×C3—184no—
304C22×C3C22×D3211yescharacteristic
305C2×D3—34no—
306C2×D3—38no—
307C2×D3—34no—
308C2×D3—62no—
309C2×D3—64no—
310C2×D3—64no—
311C2×D3—94no—
312C2×D3—184no—
313C2×D3—1816no—
314C2×D3—188no—
315C3×C4—22no—
316C3×C4—34no—
317C3×C4—64no—
318C3×C4—64no—
319C3×C4—62no—
320C3×C4—94no—
321C3×C4—92no—
322C3×C4—184no—
323C3×C4—182no—
324C3×C4C4×D3212yes—
325Dic3—34no—
326Dic3—34no—
327Dic3—62no—
328Dic3—64no—
329Dic3—64no—
330Dic3—64no—
331Dic3—94no—
332Dic3—184no—
333C32—21no—
334C32D3×#11712yescomplemented
335C23—91no—
336C23—92no—
337C23—274no—
338C23—544no—
339C2×C4—91no—
340C2×C4—92no—
341C2×C4—94no—
342C2×C4—98no—
343C2×C4—92no—
344C2×C4—94no—
345C2×C4—184no—
346C2×C4—278no—
347C2×C4—274no—
348C2×C4—544no—
349C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
350C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C2×C3—32no—
352C2×C3—34no—
353C2×C3—61no—
354C2×C3—62no—
355C2×C3—94no—
356C2×C3—184no—
357C2×C3C2×C4×D3211yescharacteristic
358C2×C3#6430612yes—
359D3—38no—
360D3—64no—
361D3—64no—
362D3—184no—
363C22—92no—
364C22—94no—
365C22—94no—
366C22—184no—
367C22—548no—
368C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
369C4—22no—
370C4—92no—
371C4—94no—
372C4—182no—
373C4—184no—
374C4C4×((C3⋊D3)⋊C3)⋊C212yes—
375C3—32no—
376C3—61no—
377C3#1059399211yescharacteristic
378C2—94no—
379C2—184no—
380C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C2#1172249112yes—
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×#272C2semidirect
3. #64081⋊C3C2semidirect
5. #11721147C2semidirect
6. C4×#6199C2semidirect
7. #11721121C2semidirect
8. #11721105C2semidirect
17. C4×((C3⋊D3)⋊C3)⋊C2C4semidirect
18. C42×H(3,𝔽3)C22semidirect
19. C4×#1084C22semidirect
20. C4×#1084C22semidirect
22. #3750⋊C3C22semidirect
23. C4×(C3⋊Dic3)⋊C3C22semidirect
25. #88448C22semidirect
26. #88447C22semidirect
28. #64081D3semidirect
51. C4×(C3⋊D3)⋊C3C2×C4semidirect
54. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
70. C32×C42C2×D3semidirect
72. #3750C2×D3semidirect
73. C4×C3⋊Dic3C2×D3semidirect
130. C4×C3⋊D3C4×D3semidirect
149. #272#117semidirect
228. H(3,𝔽3)C2×#117semidirect
297. C42((C3⋊D3)⋊C3)⋊C2semidirect
334. C32D3×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C42×#2722
#64081⋊C32
#117211612
#117211472
C4×#61992
#117211212
#117211052
D3×#8723