Finite Group Databasebeta

← C3×#10593364 #13166705

Subgroups — C3×#10593364 #13166705

Order 1728.

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

normal subgroups only

This group has 414 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×#63902C211yesmaximal, characteristic
2C3×#63870C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic
7C3×#63739C211yesmaximal, characteristic
8#10593364C311yesmaximal, characteristic, direct product factor
9C2×C3×#625C2211yescharacteristic
10C2×C3×D3×Dic3—21no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×(C2×D3)⋊C4—21no—
15C32×Dic3⋊C4—21no—
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×#889—61no—
21C3×#67⋊C4—61no—
22#63902C2×C311yescharacteristic
23#63870C2×C311yescharacteristic, complemented
24C4⋊#622C2×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—
36C3×#625—21no—
37C3×#625Q812yes—
38C3×D3×Dic3—42no—
39C3×#622—21no—
40C3×#622—21no—
41C3×#478—91nomaximal
42C2×C3×#67—61no—
43C22×C3×Dic3—61no—
44C2×C3×C4×D3—61no—
45C2×#625C22×C311yescharacteristic
46C2×D3×Dic3—21no—
47C2×#622C22×C311yescharacteristic
48C2×#622C22×C311yescharacteristic
49C32×C4⋊C4—61no—
50C32×C22⋊C4—61no—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C3×(C2×D3)⋊C4—21no—
53C3×(C2×D3)⋊C4—21no—
54C3×(C2×D3)⋊C4—41no—
55C3×C4⋊Dic3—61no—
56C3×Dic3⋊C4—21no—
57C3×Dic3⋊C4—21no—
58C3×Dic3⋊C4—41no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65#3724C22×C311yescharacteristic
66#3724C22×C311yescharacteristic
67#3724C22×C311yescharacteristic
68Dic3⋊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—
82#889—61no—
83#67⋊C4—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×C4C3×D411yescharacteristic
101C2×C3⋊Dic3C3×D411yescharacteristic
102C2×C3⋊Dic3C23×C311yescharacteristic
103C3×#67—62no—
104C3×#67—62no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—122no—
119C3×C4×D3—122no—
120#625—21no—
121#625C3×Q812yes—
122D3×Dic3—42no—
123#622—21no—
124#622—21no—
125#478—91noSylow
126C2×C33C22⋊Q811yescharacteristic
127C2×C33#13511yescharacteristic
128C2×C33AGL(1,ℤ/8)11yescharacteristic
129C32×D3—42no—
130C2×C3×D4—91no—
131C22×C3×C4—181no—
132C2×#67—61no—
133C22×Dic3—61no—
134C2×C4×D3—61no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—121no—
143C3×C4⋊C4—181no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—121no—
147C3×C42—91no—
148(C2×D3)⋊C4—21no—
149C4⋊Dic3—61no—
150Dic3⋊C4—21no—
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—61no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—62no—
162C22×C32—121no—
163C22×C32—121no—
164C22×C32C2×C3×D411yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—22no—
167C2×C3×D3—42no—
168C2×C3×D3—41no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—21no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4—121no—
180C32×C4—121no—
181C32×C4—121no—
182C3⋊Dic3—21no—
183C3⋊Dic3C2×C3×Q811yescharacteristic
184C3⋊Dic3C3×#4111yescharacteristic
185C3×Dic3—41no—
186C3×Dic3—41no—
187C3×Dic3—41no—
188C3×Dic3—42no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—62no—
197C3×Dic3—81no—
198C3×Dic3—81no—
199C3×Dic3—121no—
200#125—91no—
201C4×D4—91no—
202#107—91no—
203#106—91no—
204C4⋊C8—91no—
205#102—91no—
206#102—91no—
207C33#47811yescharacteristic, complemented, Sylow
208C23×C3—61no—
209C23×C3—61no—
210C23×C3—121no—
211C22×D3—21no—
212C3×D4—92no—
213C3×D4—181no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—121no—
227C2×C3×C4—121no—
228C2×C3×C4—121no—
229C2×C3×C4—362no—
230C2×C3×C4GO+(4,𝔽2)11yescharacteristic
231#67—62no—
232#67—62no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—122no—
241C4×D3—122no—
242C3×C8—181no—
243C3×C8—181no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—122no—
257C2×C32C3×C22⋊Q811yescharacteristic
258C2×C32C3×#13511yescharacteristic
259C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
260C3×D3—42no—
261C3×D3—42no—
262C3×D3—82no—
263C2×D4—91no—
264C22×C4—181no—
265C2×C8—91no—
266C2×C8—91no—
267C4⋊C4—61no—
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C22⋊C4—61no—
273C42—91no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—122no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—241no—
286C22×C3—241no—
287C22×C3C2×GO+(4,𝔽2)11yescharacteristic
288C2×D3—22no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4—91no—
297C3×C4—91no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—122no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4—181no—
306C3×C4—241no—
307C3×C4—241no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3—61no—
312Dic3—61no—
313Dic3—61no—
314Dic3—61no—
315Dic3—62no—
316Dic3—121no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32C3×#47811yescharacteristic, complemented
322C23—61no—
323D4—92no—
324D4—181no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—362no—
331C2×C4C3×GO+(4,𝔽2)11yescharacteristic
332C8—181no—
333C8—181no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—122no—
347C2×C3—122no—
348C2×C3—242no—
349C2×C3D32⋊Q811yescharacteristic
350C2×C3#6422411yescharacteristic
351C2×C3#6423711yescharacteristic
352D3—42no—
353C22—62no—
354C22—121no—
355C22—121no—
356C22C2×C3×GO+(4,𝔽2)11yescharacteristic
357C4—21no—
358C4—91no—
359C4—91no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—181no—
364C3—21no—
365C3—21no—
366C3—41no—
367C3—41no—
368C3#1059336411yescharacteristic, direct product factor
369C2—122no—
370C2C3×D32⋊Q811yescharacteristic
371C2C3×#6422411yescharacteristic
372C2C3×#6423711yescharacteristic
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×C4⋊#622C2semidirect
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
8. #10593364C3direct (N × H)
23. #63870C2×C3semidirect
24. C4⋊#622C2×C3semidirect
25. #63743C2×C3semidirect
26. #63742C2×C3semidirect
207. C33#478semidirect
321. C32C3×#478semidirect
368. C3#10593364direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639022
C3×#638702
C3×C4⋊#6222
C3×#637432
C3×#637422
C3×#637392
#105933643
C3×#4789