Finite Group Databasebeta

← C3×#10590552 #13157535

Subgroups — C3×#10590552 #13157535

Order 1728.

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

normal subgroups only

This group has 666 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, complemented
2C3×#63914C211yesmaximal, characteristic
3C32×#809C212yesmaximal, complemented
4C3×#63563C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#4760—32nomaximal
7#10590552C311yesmaximal, characteristic, direct product factor
8C2×C3×#634—21no—
9C2×C32×D12C2212yes—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C33×C4⋊C4C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17#64097C2×C311yescharacteristic, complemented
18C3×#940—32no—
19C3×#888—32no—
20C3×#888—61no—
21#63914C2×C311yescharacteristic
22C32×#102—32no—
23C3×#832—32no—
24C3×#821—32no—
25C3×C8×Dic3—32no—
26C3×#809—22no—
27C3×#809D312yescomplemented
28C3×#809C2×C312yescomplemented
29#63563C2×C312yescomplemented
30#63553C2×C311yescharacteristic, complemented
31C22×C32×D3—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×#634—42no—
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—42no—
41C3×#625—42no—
42C32×C3⋊C8—22no—
43C3×#486—91no—
44#4760—32no—
45C2×C32×D4—32no—
46C2×#634—21no—
47C2×C3×#67—62no—
48C2×C3×#67—62no—
49C2×C3×D12—22no—
50C2×C3×D12C2×D312yes—
51C2×C3×D12C22×C312yes—
52C2×C3×Dic6—62no—
53C2×C3×Dic6—62no—
54C2×#625—21no—
55C2×#625—21no—
56C2×C32×C8—32no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—22no—
59C32×C4⋊C4—41no—
60C32×C4⋊C4C2×D312yes—
61C32×C4⋊C4C22×C311yescharacteristic
62C4⋊(C3⋊Dic3)C22×C311yescharacteristic
63#3750—21no—
64C4×C3⋊Dic3C22×C311yescharacteristic
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8C2×D312yes—
73C2×C3×C3⋊C8C22×C312yes—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—42no—
76C2×C32×D3—42no—
77C2×C32×D3—42no—
78C33×C4—41no—
79C33×C4#4112yes—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3D812yes—
82C3×C3⋊Dic3SD1612yes—
83C3×C4⋊Q8—91no—
84C3×#127—91no—
85#940—32no—
86#888—32no—
87#888—61no—
88C3×#102—32no—
89C3×#102—32no—
90C3×#102—62no—
91C3×#102—92no—
92C3×C4×C8—91no—
93#832—32no—
94#821—32no—
95C8×Dic3—32no—
96#809C3×D312yescomplemented
97C23×C32—62no—
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C22×C3×D3—42no—
101C32×D4—62no—
102C32×D4—62no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—41no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3C3×D411yescharacteristic
115C2×C3⋊Dic3C3×D411yescharacteristic
116#634—42no—
117C3×#67—124no—
118C3×#67—124no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C3×D12—24no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—22no—
129C3×D12—22no—
130C3×D12—42no—
131C3×Dic6—124no—
132C3×Dic6—124no—
133#625—42no—
134#625—42no—
135C32×C8—62no—
136C3×C3⋊C8—22no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—42no—
139#486—91noSylow
140C2×C33#12411yescharacteristic
141C2×C33C2×D811yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C32×D3—82no—
144C2×C3×Q8—181no—
145C2×C3×D4—32no—
146C2×C3×D4—32no—
147C2×C3×D4—62no—
148C2×C3×D4—181no—
149C2×C3×D4—181no—
150C2×#67—62no—
151C2×#67—62no—
152C2×D12C2×C3×D312yes—
153C2×Dic6—62no—
154C2×Dic6—62no—
155C2×C3×C8—32no—
156C2×C3×C8—32no—
157C2×C3×C8—62no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4D3211yescharacteristic
164C3×C4⋊C4C2×C3×D312yes—
165C3×C42—91no—
166C4⋊Dic3—32no—
167C4⋊Dic3—61no—
168Dic3⋊C4—62no—
169Dic3⋊C4—62no—
170C4×Dic3—32no—
171C4×Dic3—61no—
172C2×C3⋊C8C2×C3×D312yes—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32—122no—
177C22×C32—122no—
178C22×C32—122no—
179C22×C32D3×D412yes—
180C22×C32C2×C3×D411yescharacteristic
181C2×C3×D3—44no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—82no—
189C2×C3×D3—82no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—42no—
194C32×C4—41no—
195C32×C4—42no—
196C32×C4—42no—
197C32×C4—82no—
198C32×C4—82no—
199C32×C4#23512yes—
200C32×C4#23712yes—
201C32×C4C3×#4112yes—
202C3⋊Dic3—41no—
203C3⋊Dic3C3×D812yes—
204C3⋊Dic3C3×SD1612yes—
205C3×Dic3—34no—
206C3×Dic3—34no—
207C3×Dic3—62no—
208C3×Dic3—62no—
209C3×Dic3—122no—
210C3×Dic3—122no—
211C4⋊Q8—91no—
212#127—91no—
213#102—32no—
214#102—92no—
215C4×C8—91no—
216C33#48611yescharacteristic, complemented, Sylow
217C23×C3—62no—
218C23×C3—62no—
219C23×C3—122no—
220C22×D3—22no—
221C3×Q8—362no—
222C3×D4—64no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—122no—
228C3×D4—362no—
229C3×D4—362no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×C3×C4C22×C3×D312yes—
243#67—124no—
244#67—124no—
245C2×Dic3—32no—
246C2×Dic3—32no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251D12—22no—
252D12—22no—
253Dic6—124no—
254Dic6—124no—
255C3×C8—62no—
256C3×C8—62no—
257C3×C8—122no—
258C3⋊C8—22no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—242no—
269C2×C32#88512yes—
270C2×C32D3×D812yes—
271C2×C32D3×SD1612yes—
272C2×C32C3×#12411yescharacteristic
273C2×C32C2×C3×D811yescharacteristic
274C2×C32C2×C3×SD1611yescharacteristic
275C3×D3—84no—
276C3×D3—82no—
277C3×D3—82no—
278C2×Q8—181no—
279C2×D4—32no—
280C2×D4—181no—
281C2×D4—181no—
282C2×C8—32no—
283C4⋊C4—91no—
284C4⋊C4—181no—
285C4⋊C4C3×D3211yescharacteristic
286C42—91no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—124no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—242no—
298C22×C3—242no—
299C22×C3#381111yescharacteristic
300C22×C3C3×D3×D412yes—
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—42no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—41no—
310C3×C4—42no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4—92no—
314C3×C4—92no—
315C3×C4—361no—
316C3×C4#379512yes—
317C3×C4C3×#23512yes—
318C3×C4C3×#23712yes—
319Dic3—34no—
320Dic3—34no—
321Dic3—62no—
322Dic3—62no—
323Dic3—122no—
324Dic3—122no—
325C32—21no—
326C32—22no—
327C32—41no—
328C32#476012yescomplemented
329C32C3×#48611yescharacteristic, complemented
330C23—62no—
331Q8—362no—
332D4—62no—
333D4—62no—
334D4—362no—
335D4—362no—
336C2×C4—21no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—181no—
340C2×C4C2×C3×D3211yescharacteristic
341C8—62no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—244no—
352C2×C3—242no—
353C2×C3—242no—
354C2×C3#6391311yescharacteristic
355C2×C3#6392611yescharacteristic
356C2×C3#6394011yescharacteristic
357C2×C3C3×#88512yes—
358C2×C3C3×D3×D812yes—
359C2×C3C3×D3×SD1612yes—
360D3—82no—
361C22—122no—
362C22—122no—
363C22—122no—
364C22C3×#381111yescharacteristic
365C4—41no—
366C4—92no—
367C4—92no—
368C4—361no—
369C4C3×#379512yes—
370C3—22no—
371C3—21no—
372C3—41no—
373C3#1059055211yescharacteristic, direct product factor
374C3C3×#476012yescomplemented
375C2—242no—
376C2C3×#6391311yescharacteristic
377C2C3×#6392611yescharacteristic
378C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#486
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
1. C3×#64097C2semidirect
3. C32×#809C2semidirect
4. C3×#63563C2semidirect
5. C3×#63553C2semidirect
7. #10590552C3direct (N × H)
17. #64097C2×C3semidirect
27. C3×#809D3semidirect
28. C3×#809C2×C3semidirect
29. #63563C2×C3semidirect
30. #63553C2×C3semidirect
96. #809C3×D3semidirect
216. C33#486semidirect
328. C32#4760semidirect
329. C32C3×#486semidirect
373. C3#10590552direct (N × H)
374. C3C3×#4760semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
C3×#639142
C32×#8092
C3×#635632
C3×#635532
C3×#47603
#105905523