Finite Group Databasebeta

← C3×#10590274 #13157289

Subgroups — C3×#10590274 #13157289

Order 1728.

590 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 128 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×#64079C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic
3C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63566C212yesmaximal, complemented
5C3×#63563C212yesmaximal
6C3×#5056—31nomaximal
7C3×#4697—31nomaximal
8#10590274C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C2211yescharacteristic
10C2×C32×Dic6C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×#3750—22no—
14C32×(C2×D3)⋊C4—22no—
15C2×C32×C3⋊C8C2212yes—
16C32×#124—31no—
17#64079C2×C311yescharacteristic, complemented
18C3×#875—21no—
19C3×#875D311yescharacteristic
20C3×#875C2×C311yescharacteristic
21C3×#870—31no—
22C3×#870—31no—
23C3×#870—61no—
24C32×C4×C8—31no—
25C3×#835—32no—
26C3×#832—32no—
27C3×#821—32no—
28C3×#816—32no—
29C3×C4×C3⋊C8—21no—
30C3×C4×C3⋊C8D311yescharacteristic, complemented
31C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
32#63566C2×C312yescomplemented
33#63563C2×C312yes—
34C22×C32×D3—21no—
35C2×C33×C4D412yes—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C32×Dic3—21no—
39C32×D12—22no—
40C32×Dic6—22no—
41C32×C3⋊C8—22no—
42C3×#488—91no—
43#5056—31no—
44#4697—31no—
45C2×C32×Q8—31no—
46C2×C32×D4—31no—
47C2×C3×D12—21no—
48C2×C3×D12C2×D311yescharacteristic
49C2×C3×D12C22×C311yescharacteristic
50C2×C3×Dic6—21no—
51C2×C3×Dic6C2×D311yescharacteristic
52C2×C3×Dic6C22×C311yescharacteristic
53C2×C32×C8—32no—
54C32×C22⋊C4—62no—
55C32×C42—21no—
56C32×C42—21no—
57C32×C42—21no—
58C32×C42—41no—
59C32×C42C2×D311yescharacteristic
60C32×C42C2×D311yescharacteristic
61C32×C42C22×C311yescharacteristic
62C4⋊(C3⋊Dic3)C22×C312yes—
63#3750—22no—
64C3×(C2×D3)⋊C4—24no—
65C3×(C2×D3)⋊C4—22no—
66C3×(C2×D3)⋊C4—22no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—62no—
70C3×Dic3⋊C4—64no—
71C3×Dic3⋊C4—62no—
72C3×Dic3⋊C4—62no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2×D312yes—
75C2×C3×C3⋊C8C22×C312yes—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—41no—
78C2×C32×D3—42no—
79C33×C4—22no—
80C33×C4#4112yes—
81C3×C3⋊Dic3—42no—
82C32×Dic3—41no—
83C3×#125—91no—
84C3×#124—31no—
85C3×#124—31no—
86C3×#124—61no—
87#875C3×D311yescharacteristic
88#870—31no—
89#870—31no—
90#870—61no—
91C3×#103—92no—
92C3×#102—92no—
93C3×C4×C8—31no—
94C3×C4×C8—31no—
95C3×C4×C8—61no—
96#835—32no—
97#832—32no—
98#821—32no—
99#816—32no—
100C4×C3⋊C8C3×D311yescharacteristic, complemented
101C23×C32—61no—
102C22×C3×D3—22no—
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C32×Q8—62no—
106C32×D4—62no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—42no—
114C2×C32×C4—41no—
115C2×C32×C4—61no—
116C2×C32×C4D1212yes—
117C2×C32×C4#6712yes—
118C2×C32×C4C3×D412yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C23×C311yescharacteristic
122C2×C3⋊Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C3×D12—22no—
130C3×D12—22no—
131C3×D12—42no—
132C3×Dic6—22no—
133C3×Dic6—22no—
134C3×Dic6—42no—
135C32×C8—62no—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139#488—91noSylow
140C2×C33#12411yescharacteristic
141C2×C33#13512yes—
142C32×D3—81no—
143C2×C3×Q8—31no—
144C2×C3×Q8—31no—
145C2×C3×Q8—61no—
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—61no—
149C2×D12C2×C3×D311yescharacteristic
150C2×Dic6C2×C3×D311yescharacteristic
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—62no—
154C3×C4⋊C4—92no—
155C3×C4⋊C4—182no—
156C3×C22⋊C4—64no—
157C3×C22⋊C4—62no—
158C3×C22⋊C4—62no—
159C3×C42—21no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42—41no—
163C3×C42D3211yescharacteristic
164C3×C42C2×C3×D311yescharacteristic
165C3×C42C2×C3×D311yescharacteristic
166(C2×D3)⋊C4—22no—
167C4⋊Dic3—32no—
168C4⋊Dic3—32no—
169C4⋊Dic3—62no—
170Dic3⋊C4—64no—
171Dic3⋊C4—62no—
172Dic3⋊C4—62no—
173C2×C3⋊C8C2×C3×D312yes—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32—121no—
179C22×C32—122no—
180C22×C32C2×D1211yescharacteristic
181C22×C32C2×#6711yescharacteristic
182C22×C32C2×C3×D411yescharacteristic
183C2×C3×D3—42no—
184C2×C3×D3—44no—
185C2×C3×D3—41no—
186C2×C3×D3—42no—
187C2×C3×D3—41no—
188C2×C3×D3—42no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—22no—
196C32×C4—42no—
197C32×C4—42no—
198C32×C4—42no—
199C32×C4—44no—
200C32×C4—121no—
201C32×C4#23512yes—
202C32×C4#23712yes—
203C32×C4C3×#4112yes—
204C3⋊Dic3—42no—
205C3×Dic3—42no—
206C3×Dic3—41no—
207C3×Dic3—41no—
208C3×Dic3—124no—
209C3×Dic3—122no—
210C3×Dic3—122no—
211#125—91no—
212#124—31no—
213#103—92no—
214#102—92no—
215C4×C8—31no—
216C33#48811yescharacteristic, complemented, Sylow
217C23×C3—62no—
218C23×C3—61no—
219C23×C3—61no—
220C22×D3—21no—
221C3×Q8—62no—
222C3×Q8—62no—
223C3×Q8—122no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—122no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—182no—
239C2×C3×C4#62612yes—
240C2×C3×C4C3×D1212yes—
241C2×C3×C4C3×#6712yes—
242C2×C3×C4C2×D3211yescharacteristic
243C2×C3×C4C22×C3×D311yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—64no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249D12—22no—
250Dic6—22no—
251C3×C8—64no—
252C3×C8—62no—
253C3×C8—62no—
254C3⋊C8—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32—241no—
264C2×C32#87511yescharacteristic
265C2×C32#90412yes—
266C2×C32#93611yescharacteristic
267C2×C32#95012yes—
268C2×C32C3×#12411yescharacteristic
269C2×C32C3×#13512yes—
270C3×D3—82no—
271C3×D3—81no—
272C3×D3—81no—
273C2×Q8—31no—
274C2×D4—31no—
275C2×C8—32no—
276C4⋊C4—92no—
277C4⋊C4—182no—
278C22⋊C4—62no—
279C42C3×D3211yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—122no—
285C22×C3—124no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3—121no—
289C22×C3—121no—
290C22×C3C2×#62611yescharacteristic
291C22×C3C2×C3×D1211yescharacteristic
292C22×C3C2×C3×#6711yescharacteristic
293C2×D3—41no—
294C2×D3—42no—
295C3×C4—24no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—44no—
305C3×C4—42no—
306C3×C4—122no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4—362no—
310C3×C4#379512yes—
311C3×C4C3×#23512yes—
312C3×C4C3×#23712yes—
313Dic3—41no—
314Dic3—124no—
315Dic3—122no—
316Dic3—122no—
317C32—21no—
318C32—21no—
319C32—21no—
320C32—41no—
321C32#469711yescharacteristic, complemented
322C32#505611yescharacteristic, complemented
323C32C3×#48811yescharacteristic, complemented
324C23—61no—
325Q8—62no—
326D4—62no—
327C2×C4—61no—
328C2×C4—182no—
329C2×C4C3×#62612yes—
330C2×C4C2×C3×D3211yescharacteristic
331C8—62no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—42no—
339C2×C3—41no—
340C2×C3—242no—
341C2×C3—241no—
342C2×C3—241no—
343C2×C3#6382912yes—
344C2×C3#6386011yescharacteristic
345C2×C3C3×#87511yescharacteristic
346C2×C3C3×#90412yes—
347C2×C3C3×#93611yescharacteristic
348C2×C3C3×#95012yes—
349D3—81no—
350C22—122no—
351C22—121no—
352C22C2×C3×#62611yescharacteristic
353C4—22no—
354C4—121no—
355C4—362no—
356C4C3×#379512yes—
357C3—21no—
358C3—21no—
359C3—21no—
360C3—41no—
361C3#1059027411yescharacteristic, direct product factor
362C3C3×#469711yescharacteristic, complemented
363C3C3×#505611yescharacteristic, complemented
364C2—241no—
365C2C3×#6382912yes—
366C2C3×#6386011yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C42D32
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64079C2semidirect
3. C32×C4×C3⋊C8C2semidirect
4. C3×#63566C2semidirect
8. #10590274C3direct (N × H)
17. #64079C2×C3semidirect
30. C3×C4×C3⋊C8D3semidirect
31. C3×C4×C3⋊C8C2×C3semidirect
32. #63566C2×C3semidirect
100. C4×C3⋊C8C3×D3semidirect
216. C33#488semidirect
321. C32#4697semidirect
322. C32#5056semidirect
323. C32C3×#488semidirect
361. C3#10590274direct (N × H)
362. C3C3×#4697semidirect
363. C3C3×#5056semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×#8752
C32×C4×C3⋊C82
C3×#635662
C3×#635632
C3×#50563
C3×#46973
#105902743