Finite Group Databasebeta

← C3×#10590431 #13157552

Subgroups — C3×#10590431 #13157552

Order 1728.

552 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64096C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3C32×#808C212yesmaximal
4C3×#63572C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6C3×#4854—32nomaximal
7#10590431C311yesmaximal, characteristic, direct product factor
8C2×C3×#634C2211yescharacteristic
9C2×C3×#627—21no—
10C33×C4⋊C4C2211yescharacteristic
11C3×#3750—21no—
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64096C2×C311yescharacteristic
18C3×#888—32no—
19C3×#887—32no—
20C3×#887—61no—
21#63919C2×C311yescharacteristic
22C32×#106—32no—
23C3×#816—32no—
24C3×Dic3⋊C8—32no—
25C3×#810—32no—
26C3×#808—22no—
27C3×#808D312yes—
28C3×#808C2×C312yes—
29#63572C2×C311yescharacteristic
30#63567C2×C312yes—
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×#634—21no—
37C3×#634Q812yescomplemented
38C2×C32×Dic3—22no—
39C3×#627—42no—
40C32×C3⋊C8—22no—
41C3×#475—91no—
42#4854—32no—
43C2×#634C22×C311yescharacteristic
44C2×C3×Dic6—32no—
45C2×C3×Dic6—61no—
46C2×C3×Dic6—62no—
47C2×#627—21no—
48C2×C32×C8—32no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4—22no—
51C32×C4⋊C4—32no—
52C32×C4⋊C4—41no—
53C32×C4⋊C4C2×D312yes—
54C32×C4⋊C4C22×C311yescharacteristic
55#3750—21no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C22×C312yes—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—62no—
66C3×C4×Dic3—61no—
67C3×C4×Dic3—62no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C33×C4—41no—
74C33×C4C2×Q811yescharacteristic
75C33×C4#4111yescharacteristic
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3SD1612yes—
79C32×Dic3—42no—
80C3×C4⋊Q8—91no—
81C3×C4×Q8—91no—
82#888—32no—
83#887—32no—
84#887—61no—
85C3×#106—32no—
86C3×#106—32no—
87C3×#106—62no—
88C3×C4⋊C8—91no—
89C3×#103—92no—
90#816—32no—
91Dic3⋊C8—32no—
92#810—32no—
93#808C3×D312yes—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—41no—
100C2×C32×C4—42no—
101C2×C32×C4—42no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3C3×D411yescharacteristic
107C2×C3⋊Dic3C3×D411yescharacteristic
108#634—21no—
109#634C3×Q812yescomplemented
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C3×Dic6—34no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—124no—
124#627—42no—
125C32×C8—62no—
126C3×C3⋊C8—24no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—22no—
129#475—91noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×SD1611yescharacteristic
132C2×C33#13711yescharacteristic
133C2×C3×Q8—91no—
134C2×C3×Q8—181no—
135C2×Dic6—32no—
136C2×Dic6—61no—
137C2×Dic6—62no—
138C2×C3×C8—32no—
139C2×C3×C8—32no—
140C2×C3×C8—62no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—32no—
144C3×C4⋊C4—32no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4D3211yescharacteristic
150C3×C4⋊C4C2×C3×D312yes—
151C3×C42—91no—
152C3×C42—181no—
153C4⋊Dic3C2×C3×D312yes—
154Dic3⋊C4—62no—
155Dic3⋊C4—62no—
156Dic3⋊C4—62no—
157C4×Dic3—32no—
158C4×Dic3—61no—
159C4×Dic3—62no—
160C4×Dic3—62no—
161C2×C3⋊C8C2×C3×D312yes—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—41no—
165C22×C32D3×D412yes—
166C22×C32C2×C3×D411yescharacteristic
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—41no—
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—82no—
177C32×C4—82no—
178C32×C4—122no—
179C32×C4#23512yes—
180C32×C4D3×Q812yes—
181C32×C4C2×C3×Q811yescharacteristic
182C32×C4C3×#4111yescharacteristic
183C3⋊Dic3—22no—
184C3⋊Dic3—21no—
185C3⋊Dic3C3×SD1612yes—
186C3×Dic3—34no—
187C3×Dic3—42no—
188C3×Dic3—42no—
189C3×Dic3—62no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C3×Dic3—62no—
193C3×Dic3—82no—
194C3×Dic3—121no—
195C4⋊Q8—91no—
196C4×Q8—91no—
197#106—32no—
198C4⋊C8—91no—
199#103—92no—
200C33#47511yescharacteristic, complemented, Sylow
201C3×Q8—92no—
202C3×Q8—181no—
203C3×Q8—362no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—122no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3211yescharacteristic
219C2×C3×C4C22×C3×D312yes—
220C2×Dic3—22no—
221C2×Dic3—32no—
222C2×Dic3—32no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227Dic6—34no—
228Dic6—62no—
229Dic6—62no—
230Dic6—62no—
231Dic6—124no—
232C3×C8—64no—
233C3×C8—62no—
234C3×C8—62no—
235C3⋊C8—22no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32#89612yes—
246C2×C32D3×SD1612yes—
247C2×C32#91512yes—
248C2×C32C3×C22⋊Q811yescharacteristic
249C2×C32C2×C3×SD1611yescharacteristic
250C2×C32C3×#13711yescharacteristic
251C2×Q8—91no—
252C2×Q8—181no—
253C2×C8—32no—
254C4⋊C4—32no—
255C4⋊C4—181no—
256C4⋊C4—181no—
257C4⋊C4C3×D3211yescharacteristic
258C42—91no—
259C42—181no—
260C22×C3—21no—
261C22×C3—22no—
262C22×C3—41no—
263C22×C3#381111yescharacteristic
264C22×C3C3×D3×D412yes—
265C3×C4—21no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—42no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—82no—
275C3×C4—82no—
276C3×C4—92no—
277C3×C4—122no—
278C3×C4—122no—
279C3×C4—182no—
280C3×C4—181no—
281C3×C4—242no—
282C3×C4#379711yescharacteristic
283C3×C4#379811yescharacteristic
284C3×C4C3×#23512yes—
285C3×C4C3×D3×Q812yes—
286Dic3—34no—
287Dic3—42no—
288Dic3—62no—
289Dic3—64no—
290Dic3—64no—
291Dic3—62no—
292Dic3—121no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#485412yescomplemented
297C32C3×#47511yescharacteristic, complemented
298Q8—92no—
299Q8—181no—
300Q8—362no—
301C2×C4—21no—
302C2×C4—62no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—181no—
306C2×C4C2×C3×D3211yescharacteristic
307C8—62no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3#6386511yescharacteristic
318C2×C3#6393111yescharacteristic
319C2×C3#6394011yescharacteristic
320C2×C3C3×#89612yes—
321C2×C3C3×D3×SD1612yes—
322C2×C3C3×#91512yes—
323C22C3×#381111yescharacteristic
324C4—41no—
325C4—92no—
326C4—122no—
327C4—182no—
328C4—181no—
329C4C3×#379711yescharacteristic
330C4C3×#379811yescharacteristic
331C3—21no—
332C3—22no—
333C3—41no—
334C3#1059043111yescharacteristic, direct product factor
335C3C3×#485412yescomplemented
336C2C3×#6386511yescharacteristic
337C2C3×#6393111yescharacteristic
338C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#475
LCSG ▷ C2×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
7. #10590431C3direct (N × H)
37. C3×#634Q8semidirect
109. #634C3×Q8semidirect
200. C33#475semidirect
296. C32#4854semidirect
297. C32C3×#475semidirect
334. C3#10590431direct (N × H)
335. C3C3×#4854semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#639192
C32×#8082
C3×#635722
C3×#635672
C3×#48543
#105904313