Finite Group Databasebeta

← C3×#10590567 #13157560

Subgroups — C3×#10590567 #13157560

Order 1728.

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

normal subgroups only

This group has 570 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×#64097C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C32×#810C212yesmaximal
4C3×#63566C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6C3×#4792—32nomaximal
7#10590567C311yesmaximal, characteristic, direct product factor
8C2×C3×#634—21no—
9C2×C32×Dic6C2212yes—
10C33×C4⋊C4C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×#3750—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—21no—
16C3×#3724—21no—
17#64097C2×C311yescharacteristic
18C3×#944—32no—
19C3×#888—32no—
20C3×#888—61no—
21#63873C2×C311yescharacteristic
22C32×#103—32no—
23C3×#835—32no—
24C3×#816—32no—
25C3×C8×Dic3—32no—
26C3×#810—22no—
27C3×#810D312yes—
28C3×#810C2×C312yes—
29#63566C2×C312yes—
30#63553C2×C311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×#634—42no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C32×C3⋊C8—22no—
41C3×#487—91no—
42#4792—32no—
43C2×C32×Q8—32no—
44C2×#634—21no—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—62no—
47C2×C3×Dic6—62no—
48C2×C3×Dic6C2×D312yes—
49C2×C3×Dic6C22×C312yes—
50C2×C32×C8—32no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—22no—
53C32×C4⋊C4—41no—
54C32×C4⋊C4C2×D312yes—
55C32×C4⋊C4C22×C311yescharacteristic
56C4⋊(C3⋊Dic3)C22×C311yescharacteristic
57#3750—21no—
58C4×C3⋊Dic3C22×C311yescharacteristic
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8C2×D312yes—
69C2×C3×C3⋊C8C22×C312yes—
70#3724—21no—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C33×C4—41no—
74C33×C4#4112yes—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3SD1612yes—
77C3×C3⋊Dic3Q1612yes—
78C32×Dic3—42no—
79C3×C4⋊Q8—91no—
80C3×C4⋊Q8—91no—
81#944—32no—
82#888—32no—
83#888—61no—
84C3×#103—32no—
85C3×#103—32no—
86C3×#103—62no—
87C3×#103—92no—
88C3×C4×C8—91no—
89#835—32no—
90#816—32no—
91C8×Dic3—32no—
92#810C3×D312yes—
93C32×Q8—62no—
94C32×Q8—62no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C32×C4—42no—
103C2×C32×C4—62no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C23×C311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3C3×D411yescharacteristic
108C2×C3⋊Dic3C3×D411yescharacteristic
109#634—42no—
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—24no—
120C3×Dic6—22no—
121C3×Dic6—22no—
122C3×Dic6—22no—
123C3×Dic6—22no—
124C3×Dic6—42no—
125C3×Dic6—124no—
126C3×Dic6—124no—
127C32×C8—62no—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—42no—
131#487—91noSylow
132C2×C33#12411yescharacteristic
133C2×C33C2×SD1611yescharacteristic
134C2×C33C2×Q1611yescharacteristic
135C2×C3×Q8—32no—
136C2×C3×Q8—32no—
137C2×C3×Q8—62no—
138C2×C3×Q8—181no—
139C2×Dic6—62no—
140C2×Dic6—62no—
141C2×Dic6C2×C3×D312yes—
142C2×C3×C8—32no—
143C2×C3×C8—32no—
144C2×C3×C8—62no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4D3211yescharacteristic
153C3×C4⋊C4C2×C3×D312yes—
154C3×C42—91no—
155C4⋊Dic3—32no—
156C4⋊Dic3—61no—
157Dic3⋊C4—62no—
158Dic3⋊C4—62no—
159Dic3⋊C4—62no—
160Dic3⋊C4—62no—
161C4×Dic3—32no—
162C4×Dic3—61no—
163C2×C3⋊C8C2×C3×D312yes—
164C22×C32—21no—
165C22×C32—22no—
166C22×C32—41no—
167C22×C32D3×D412yes—
168C22×C32C2×C3×D411yescharacteristic
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—42no—
173C32×C4—41no—
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—82no—
177C32×C4—82no—
178C32×C4—122no—
179C32×C4#23512yes—
180C32×C4#23712yes—
181C32×C4C3×#4112yes—
182C3⋊Dic3—41no—
183C3⋊Dic3C3×SD1612yes—
184C3⋊Dic3C3×Q1612yes—
185C3×Dic3—34no—
186C3×Dic3—34no—
187C3×Dic3—44no—
188C3×Dic3—42no—
189C3×Dic3—42no—
190C3×Dic3—62no—
191C3×Dic3—62no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C4⋊Q8—91no—
195C4⋊Q8—91no—
196#103—32no—
197#103—92no—
198C4×C8—91no—
199C33#48711yescharacteristic, complemented, Sylow
200C3×Q8—64no—
201C3×Q8—62no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—62no—
205C3×Q8—122no—
206C3×Q8—362no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—122no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3211yescharacteristic
222C2×C3×C4C22×C3×D312yes—
223C2×Dic3—22no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230Dic6—22no—
231Dic6—22no—
232Dic6—124no—
233Dic6—124no—
234C3×C8—62no—
235C3×C8—62no—
236C3×C8—122no—
237C3⋊C8—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32#88512yes—
248C2×C32D3×SD1612yes—
249C2×C32D3×Q1612yes—
250C2×C32C3×#12411yescharacteristic
251C2×C32C2×C3×SD1611yescharacteristic
252C2×C32C2×C3×Q1611yescharacteristic
253C2×Q8—32no—
254C2×Q8—181no—
255C2×C8—32no—
256C4⋊C4—91no—
257C4⋊C4—181no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C4⋊C4C3×D3211yescharacteristic
261C42—91no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—41no—
265C22×C3#381111yescharacteristic
266C22×C3C3×D3×D412yes—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—42no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—42no—
274C3×C4—82no—
275C3×C4—82no—
276C3×C4—92no—
277C3×C4—92no—
278C3×C4—124no—
279C3×C4—122no—
280C3×C4—122no—
281C3×C4—361no—
282C3×C4#379512yes—
283C3×C4C3×#23512yes—
284C3×C4C3×#23712yes—
285Dic3—34no—
286Dic3—34no—
287Dic3—42no—
288Dic3—62no—
289Dic3—62no—
290Dic3—122no—
291Dic3—122no—
292C32—21no—
293C32—22no—
294C32—41no—
295C32#479212yescomplemented
296C32C3×#48711yescharacteristic, complemented
297Q8—62no—
298Q8—62no—
299Q8—362no—
300C2×C4—21no—
301C2×C4—62no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4C2×C3×D3211yescharacteristic
306C8—62no—
307C2×C3—21no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3#6391311yescharacteristic
317C2×C3#6393011yescharacteristic
318C2×C3#6394411yescharacteristic
319C2×C3C3×#88512yes—
320C2×C3C3×D3×SD1612yes—
321C2×C3C3×D3×Q1612yes—
322C22C3×#381111yescharacteristic
323C4—41no—
324C4—92no—
325C4—92no—
326C4—122no—
327C4—361no—
328C4C3×#379512yes—
329C3—22no—
330C3—21no—
331C3—41no—
332C3#1059056711yescharacteristic, direct product factor
333C3C3×#479212yescomplemented
334C2C3×#6391311yescharacteristic
335C2C3×#6393011yescharacteristic
336C2C3×#6394411yescharacteristic
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×#487
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. #10590567C3direct (N × H)
199. C33#487semidirect
295. C32#4792semidirect
296. C32C3×#487semidirect
332. C3#10590567direct (N × H)
333. C3C3×#4792semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
C3×#638732
C32×#8102
C3×#635662
C3×#635532
C3×#47923
#105905673