Finite Group Databasebeta

← C3×#10593350 #13166695

Subgroups — C3×#10593350 #13166695

Order 1728.

434 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 66 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×C4×#625C211yesmaximal, characteristic
2C3×#63870C211yesmaximal, characteristic, complemented
3C3×#63779C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic
7C3×#63739C211yesmaximal, characteristic
8#10593350C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×Dic3⋊C4—21no—
15C3×#3724C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×(C2×D3)⋊Dic3—21no—
20C3×C4×#67—61no—
21C3×#889—61no—
22C4×#625C2×C311yescharacteristic
23#63870C2×C311yescharacteristic, complemented
24#63779C2×C311yescharacteristic, complemented
25#63743C2×C311yescharacteristic, complemented
26#63742C2×C311yescharacteristic, complemented
27#63739C2×C311yescharacteristic
28#63739C2×C311yescharacteristic
29C22×C32×D3—21no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C32×C4×D3—42no—
37C3×#625—21no—
38C3×#625Q812yes—
39C3×#622—21no—
40C3×#622—21no—
41C3×#478—91nomaximal
42C22×C32×C4—61no—
43C2×C3×#67—61no—
44C2×C3×C4×D3—21no—
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—41no—
47C2×#625C22×C311yescharacteristic
48C2×#622C22×C311yescharacteristic
49C2×#622C22×C311yescharacteristic
50C32×C4⋊C4—61no—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C3×#217—61no—
53C3×(C2×D3)⋊C4—61no—
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—21no—
56C3×Dic3⋊C4—21no—
57C3×Dic3⋊C4—41no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×C4×Dic3—61no—
63C3×C4×Dic3—61no—
64#3724C22×C311yescharacteristic
65#3724C22×C311yescharacteristic
66#3724C22×C311yescharacteristic
67Dic3⋊Dic3—21no—
68(C2×D3)⋊Dic3—21no—
69C22×C33C2×D411yescharacteristic
70C2×C32×D3—22no—
71C2×C32×D3—41no—
72C2×C32×D3—41no—
73C33×C4—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3C2×Q811yescharacteristic
76C3×C3⋊Dic3#4111yescharacteristic
77C32×Dic3—41no—
78C32×Dic3—41no—
79C32×Dic3—41no—
80C3×#125—91no—
81C3×C4×D4—91no—
82C4×#67—61no—
83#889—61no—
84C3×#107—91no—
85C3×#106—91no—
86C3×C4⋊C8—91no—
87C3×#102—91no—
88C3×#102—91no—
89C23×C32—61no—
90C22×C3×D3—21no—
91C22×C3×D3—21no—
92C22×C3×D3—41no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99C2×C32×C4—61no—
100C2×C32×C4—122no—
101C2×C32×C4C3×D411yescharacteristic
102C2×C3⋊Dic3C3×D411yescharacteristic
103C2×C3⋊Dic3C23×C311yescharacteristic
104C3×#67—62no—
105C3×#67—62no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×C4×D3—42no—
120C3×C4×D3—42no—
121C3×C4×D3—44no—
122#625—21no—
123#625C3×Q812yes—
124#622—21no—
125#622—21no—
126#478—91noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33AGL(1,ℤ/8)11yescharacteristic
130C32×D3—42no—
131C2×C3×D4—91no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—121no—
135C2×#67—61no—
136C2×C4×D3—21no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—121no—
145C3×C4⋊C4—181no—
146C3×C22⋊C4—181no—
147C3×C42—91no—
148#217—61no—
149(C2×D3)⋊C4—61no—
150C4⋊Dic3—61no—
151Dic3⋊C4—21no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—62no—
163C22×C32—121no—
164C22×C32—121no—
165C22×C32C2×C3×D411yescharacteristic
166C2×C3×D3—22no—
167C2×C3×D3—22no—
168C2×C3×D3—42no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4—22no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—42no—
179C32×C4—81no—
180C32×C4—121no—
181C32×C4—121no—
182C32×C4—121no—
183C3⋊Dic3—21no—
184C3⋊Dic3C2×C3×Q811yescharacteristic
185C3⋊Dic3C3×#4111yescharacteristic
186C3×Dic3—41no—
187C3×Dic3—41no—
188C3×Dic3—41no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—41no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—81no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201#125—91no—
202C4×D4—91no—
203#107—91no—
204#106—91no—
205C4⋊C8—91no—
206#102—91no—
207#102—91no—
208C33#47811yescharacteristic, complemented, Sylow
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—121no—
212C22×D3—21no—
213C3×D4—92no—
214C3×D4—181no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—121no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—122no—
231C2×C3×C4—122no—
232C2×C3×C4—124no—
233C2×C3×C4GO+(4,𝔽2)11yescharacteristic
234#67—62no—
235#67—62no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C4×D3—42no—
244C3×C8—181no—
245C3×C8—181no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—122no—
259C2×C32C3×C22⋊Q811yescharacteristic
260C2×C32C3×#13511yescharacteristic
261C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
262C3×D3—42no—
263C3×D3—44no—
264C3×D3—42no—
265C2×D4—91no—
266C22×C4—61no—
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—61no—
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—181no—
274C22⋊C4—181no—
275C42—91no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—122no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3C2×GO+(4,𝔽2)11yescharacteristic
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4—81no—
298C3×C4—91no—
299C3×C4—91no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—121no—
303C3×C4—122no—
304C3×C4—122no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4—181no—
309C3×C4—241no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—61no—
314Dic3—61no—
315Dic3—61no—
316Dic3—61no—
317Dic3—121no—
318Dic3—121no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32C3×#47811yescharacteristic, complemented
324C23—61no—
325D4—92no—
326D4—181no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4—122no—
333C2×C4C3×GO+(4,𝔽2)11yescharacteristic
334C8—181no—
335C8—181no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—122no—
349C2×C3—124no—
350C2×C3—122no—
351C2×C3D32⋊Q811yescharacteristic
352C2×C3#6422311yescharacteristic
353C2×C3#6423611yescharacteristic
354D3—42no—
355C22—62no—
356C22—121no—
357C22—121no—
358C22C2×C3×GO+(4,𝔽2)11yescharacteristic
359C4—21no—
360C4—91no—
361C4—91no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4—181no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3#1059335011yescharacteristic, direct product factor
371C2—122no—
372C2C3×D32⋊Q811yescharacteristic
373C2C3×#6422311yescharacteristic
374C2C3×#6423611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#478
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63870C2semidirect
3. C3×#63779C2semidirect
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
8. #10593350C3direct (N × H)
23. #63870C2×C3semidirect
24. #63779C2×C3semidirect
25. #63743C2×C3semidirect
26. #63742C2×C3semidirect
208. C33#478semidirect
323. C32C3×#478semidirect
370. C3#10593350direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#6252
C3×#638702
C3×#637792
C3×#637432
C3×#637422
C3×#637392
#105933503
C3×#4789