Finite Group Databasebeta

← C3×#10590584 #13157579

Subgroups — C3×#10590584 #13157579

Order 1728.

646 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C3×#64101C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4C32×#808C212yesmaximal, complemented
5C3×#63575C212yesmaximal, complemented
6C3×#4861—32nomaximal
7#10590584C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#624—21no—
10C33×C4⋊C4C2211yescharacteristic
11C3×#3752—21no—
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4×C3⋊Dic3—21no—
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17C3×#3712C414yescomplemented
18#64101C2×C311yescharacteristic
19C2×C3×ΓL(1,𝔽25)—32no—
20C3×#892—32no—
21C3×#892—61no—
22C3×D3×C4⋊C4—32no—
23#63900C2×C311yescharacteristic
24C2×#3712C2×C311yescharacteristic
25C32×#106—32no—
26C3×#818—32no—
27C3×#808—22no—
28C3×#808D312yescomplemented
29C3×#808C2×C312yescomplemented
30C3×#807—32no—
31#63575C2×C312yescomplemented
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4×C3⋊D3C2×C412yes—
38C3×C4×C3⋊D3Q812yes—
39C2×C32×Dic3—22no—
40C3×#624—42no—
41C32×C3⋊C8C2×C414yescomplemented
42C3×#428—91no—
43#4861—32no—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—62no—
48C2×#624—21no—
49C2×C32×C8—32no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—22no—
52C32×C4⋊C4—32no—
53C32×C4⋊C4—41no—
54C32×C4⋊C4C2×D312yes—
55C32×C4⋊C4C22×C311yescharacteristic
56#3752—21no—
57C4⋊(C3⋊Dic3)C22×C311yescharacteristic
58C4×C3⋊Dic3—21no—
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—62no—
61C3×C4⋊Dic3—22no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—62no—
68C3×C4×Dic3—62no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C22×C312yes—
72C3×ΓL(1,𝔽25)—38no—
73#3724—21no—
74#3712C3×C414yescomplemented
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—21no—
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3C4⋊C412yes—
79C33×C4—41no—
80C33×C4C22×C411yescharacteristic
81C33×C4C2×Q811yescharacteristic
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3C4⋊C412yes—
84C32×Dic3—42no—
85C2×C3×M16—91no—
86C3×#117—91no—
87C2×C3×C4⋊C4—91no—
88C2×ΓL(1,𝔽25)—32no—
89#892—32no—
90#892—61no—
91D3×C4⋊C4—32no—
92C3×#107—92no—
93C3×#106—32no—
94C3×#106—32no—
95C3×#106—62no—
96#818—32no—
97#808C3×D312yescomplemented
98#807—32no—
99C22×C3⋊D3C3×D411yescharacteristic
100C22×C3×D3—32no—
101C22×C3×D3—61no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—22no—
107C2×C32×C4—41no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4—62no—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3C3×D411yescharacteristic
115C4×C3⋊D3C2×C3×C412yes—
116C4×C3⋊D3C3×Q812yes—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—42no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C3×C4×D3—34no—
125C3×C4×D3—34no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—124no—
129#624—42no—
130C32×C8—34no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8C4×D314yescomplemented
133C3×C3⋊C8C2×C3×C414yescomplemented
134#428—91noSylow
135C2×C33C2×C4⋊C411yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C33#13711yescharacteristic
138C3×C3⋊D3—22no—
139C22×C3×C4—91no—
140C22×C3×C4—181no—
141C2×C4×D3—32no—
142C2×C4×D3—61no—
143C2×C4×D3—62no—
144C3×M16—94no—
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—62no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—62no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—181no—
156C3×C4⋊C4D3211yescharacteristic
157C3×C4⋊C4C2×C3×D312yes—
158C3×C22⋊C4—181no—
159C3×C42—181no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
162C4⋊Dic3—32no—
163C4⋊Dic3—61no—
164C4⋊Dic3C2×C3×D312yes—
165Dic3⋊C4—62no—
166C4×Dic3—62no—
167C4×Dic3—62no—
168C2×C3⋊C8C2×C3×D312yes—
169ΓL(1,𝔽25)—38no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—41no—
173C22×C32D3×D412yes—
174C22×C32C2×C3×D411yescharacteristic
175C2×C3⋊D3—21no—
176C2×C3⋊D3—21no—
177C2×C3⋊D3C3×C4⋊C412yes—
178C2×C3×D3—34no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—42no—
192C32×C4—42no—
193C32×C4—82no—
194C32×C4—82no—
195C32×C4—122no—
196C32×C4C2×C4×D312yes—
197C32×C4D3×Q812yes—
198C32×C4C22×C3×C411yescharacteristic
199C32×C4C2×C3×Q811yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3C3×C4⋊C412yes—
202C3×Dic3—34no—
203C3×Dic3—42no—
204C3×Dic3—42no—
205C3×Dic3—62no—
206C3×Dic3—82no—
207C3×Dic3—122no—
208C3×Dic3—122no—
209C2×M16—91no—
210#117—91no—
211C2×C4⋊C4—91no—
212#107—92no—
213#106—32no—
214C33#42811yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—32no—
217C22×D3—61no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—91no—
229C2×C3×C4—92no—
230C2×C3×C4—92no—
231C2×C3×C4—122no—
232C2×C3×C4—181no—
233C2×C3×C4—362no—
234C2×C3×C4C2×D3211yescharacteristic
235C2×C3×C4C22×C3×D312yes—
236C2×Dic3—22no—
237C2×Dic3—32no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C4×D3—34no—
242C4×D3—34no—
243C4×D3—62no—
244C4×D3—62no—
245C4×D3—124no—
246C3×C8—34no—
247C3×C8—34no—
248C3×C8—64no—
249C3⋊C8C3×C4×D314yescomplemented
250C2×C32—22no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32D3×C4⋊C412yes—
260C2×C32#91412yes—
261C2×C32#91512yes—
262C2×C32C2×C3×C4⋊C411yescharacteristic
263C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
264C2×C32C3×#13711yescharacteristic
265C3⋊D3—22no—
266C3×D3—64no—
267C3×D3—122no—
268C22×C4—91no—
269C22×C4—181no—
270M16—94no—
271C2×C8—32no—
272C4⋊C4—32no—
273C4⋊C4—91no—
274C4⋊C4—181no—
275C4⋊C4C3×D3211yescharacteristic
276C22⋊C4—181no—
277C42—181no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—92no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3#381111yescharacteristic
285C22×C3C3×D3×D412yes—
286C2×D3—34no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—21no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—42no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4—92no—
304C3×C4—122no—
305C3×C4—122no—
306C3×C4—242no—
307C3×C4—361no—
308C3×C4#379711yescharacteristic
309C3×C4C4×D3211yescharacteristic
310C3×C4C2×C3×C4×D312yes—
311C3×C4C3×D3×Q812yes—
312Dic3—34no—
313Dic3—42no—
314Dic3—62no—
315Dic3—122no—
316Dic3—122no—
317C32—21no—
318C32—22no—
319C32—41no—
320C32#486112yescomplemented
321C32C3×#42811yescharacteristic, complemented
322C23—91no—
323C2×C4—21no—
324C2×C4—62no—
325C2×C4—91no—
326C2×C4—92no—
327C2×C4—92no—
328C2×C4—181no—
329C2×C4—362no—
330C2×C4C2×C3×D3211yescharacteristic
331C8—34no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—182no—
342C2×C3#6388311yescharacteristic
343C2×C3#6393111yescharacteristic
344C2×C3#6394111yescharacteristic
345C2×C3C3×D3×C4⋊C412yes—
346C2×C3C3×#91412yes—
347C2×C3C3×#91512yes—
348D3—64no—
349D3—122no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22C3×#381111yescharacteristic
354C4—41no—
355C4—92no—
356C4—122no—
357C4—361no—
358C4C3×#379711yescharacteristic
359C4C3×C4×D3211yescharacteristic
360C3—21no—
361C3—22no—
362C3—41no—
363C3#1059058411yescharacteristic, direct product factor
364C3C3×#486112yescomplemented
365C2—182no—
366C2C3×#6388311yescharacteristic
367C2C3×#6393111yescharacteristic
368C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C32×#808C2semidirect
5. C3×#63575C2semidirect
7. #10590584C3direct (N × H)
17. C3×#3712C4semidirect
28. C3×#808D3semidirect
29. C3×#808C2×C3semidirect
31. #63575C2×C3semidirect
41. C32×C3⋊C8C2×C4semidirect
74. #3712C3×C4semidirect
97. #808C3×D3semidirect
132. C3×C3⋊C8C4×D3semidirect
133. C3×C3⋊C8C2×C3×C4semidirect
214. C33#428semidirect
249. C3⋊C8C3×C4×D3semidirect
320. C32#4861semidirect
321. C32C3×#428semidirect
363. C3#10590584direct (N × H)
364. C3C3×#4861semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
C3×#639002
C2×C3×#37122
C32×#8082
C3×#635752
C3×#48613
#105905843