Finite Group Databasebeta

← C3×#10593440 #13166806

Subgroups — C3×#10593440 #13166806

Order 1728.

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

normal subgroups only

This group has 430 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
1C2×C3×#3776C211yesmaximal, characteristic, complemented
2C2×C3×#3775C211yesmaximal, characteristic
3C3×#63915C211yesmaximal, characteristic
4C3×#63838C211yesmaximal, characteristic, complemented
5C3×C4⋊#622C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic
8#10593440C311yesmaximal, characteristic, direct product factor
9C2×C3×#627C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×#3776—22no—
15C3×#3775—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C3×#3724C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C2×#3776C2×C311yescharacteristic, complemented
23C2×#3775C2×C311yescharacteristic
24C3×#887—61no—
25C3×#885—61no—
26#63915C2×C311yescharacteristic
27#63838C2×C311yescharacteristic, complemented
28C4⋊#622C2×C311yescharacteristic, complemented
29#63740C2×C311yescharacteristic, complemented
30#63739C2×C311yescharacteristic
31C22×C32×D3—21no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3×#627—21no—
41C3×#627D412yes—
42C3×#625—21no—
43C3×#625—21no—
44C3×#622—21no—
45C3×#622—21no—
46C3×#464—91nomaximal
47C2×C3×#67—61no—
48C2×C3×Dic6—61no—
49C2×C3×Dic6—61no—
50C2×#627C22×C311yescharacteristic
51C2×#627C22×C311yescharacteristic
52C2×#625C22×C311yescharacteristic
53C2×#622C22×C311yescharacteristic
54C2×#622C22×C311yescharacteristic
55#3776—22no—
56#3775—22no—
57C32×C4⋊C4—61no—
58C32×C22⋊C4—61no—
59C4×C3⋊Dic3C22×C311yescharacteristic
60C3×#217—61no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—41no—
64C3×(C2×D3)⋊C4—61no—
65C3×Dic3⋊C4—21no—
66C3×Dic3⋊C4—21no—
67C3×Dic3⋊C4—41no—
68C3×Dic3⋊C4—61no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—61no—
71C3×C4×Dic3—61no—
72C3×C4×Dic3—61no—
73#3724C22×C311yescharacteristic
74(C2×D3)⋊Dic3—21no—
75Dic32—21no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4—21no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3C2×D411yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84C32×Dic3—22no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87C2×C3×Q16—91no—
88C2×C3×SD16—91no—
89C3×#124—91no—
90C3×C4×Q8—91no—
91#887—61no—
92#885—61no—
93C3×C4⋊C8—91no—
94C3×#103—91no—
95C3×#102—91no—
96C23×C32—61no—
97C22×C3×D3—21no—
98C22×C3×D3—21no—
99C22×C3×D3—41no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4—61no—
107C2×C32×C4C3×D411yescharacteristic
108C2×C3⋊Dic3C3×D411yescharacteristic
109C2×C3⋊Dic3C23×C311yescharacteristic
110C3×#67—121no—
111C3×#67—121no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×Dic6—62no—
126C3×Dic6—62no—
127C3×Dic6—121no—
128C3×Dic6—121no—
129#627—21no—
130#627—21no—
131#627—21no—
132#627C3×D412yes—
133#625—21no—
134#625—21no—
135#622—21no—
136#622—21no—
137#464—91noSylow
138C2×C33#12111yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33#13711yescharacteristic
141C32×D3—81no—
142C2×C3×Q8—91no—
143C2×C3×Q8—91no—
144C2×C3×D4—91no—
145C2×#67—61no—
146C2×Dic6—61no—
147C2×Dic6—61no—
148C3×Q16—182no—
149C3×SD16—182no—
150C2×C3×C8—91no—
151C2×C3×C8—91no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—121no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—61no—
158C3×C22⋊C4—121no—
159C3×C22⋊C4—181no—
160C3×C42—91no—
161C3×C42—181no—
162#217—61no—
163(C2×D3)⋊C4—21no—
164(C2×D3)⋊C4—61no—
165Dic3⋊C4—21no—
166Dic3⋊C4—61no—
167C4×Dic3—61no—
168C4×Dic3—61no—
169C4×Dic3—61no—
170C4×Dic3—61no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—121no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32C2×C3×D411yescharacteristic
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—41no—
182C2×C3×D3—42no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C2×C3×D3—81no—
187C2×C3×D3—81no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—62no—
192C32×C4—81no—
193C32×C4—81no—
194C32×C4—121no—
195C32×C4—121no—
196C3⋊Dic3—21no—
197C3⋊Dic3C2×C3×D411yescharacteristic
198C3⋊Dic3C3×#4111yescharacteristic
199C3×Dic3—22no—
200C3×Dic3—22no—
201C3×Dic3—42no—
202C3×Dic3—41no—
203C3×Dic3—41no—
204C3×Dic3—41no—
205C3×Dic3—41no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—62no—
211C3×Dic3—81no—
212C3×Dic3—81no—
213C3×Dic3—121no—
214C2×Q16—91no—
215C2×SD16—91no—
216#124—91no—
217C4×Q8—91no—
218C4⋊C8—91no—
219#103—91no—
220#102—91no—
221C33#46411yescharacteristic, complemented, Sylow
222C23×C3—61no—
223C23×C3—61no—
224C23×C3—121no—
225C22×D3—21no—
226C3×Q8—92no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×D4—181no—
231C3×D4—181no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—121no—
245C2×C3×C4—121no—
246C2×C3×C4—121no—
247C2×C3×C4GO+(4,𝔽2)11yescharacteristic
248#67—121no—
249#67—121no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257Dic6—62no—
258Dic6—62no—
259Dic6—121no—
260Dic6—121no—
261C3×C8—181no—
262C3×C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—241no—
276C2×C32C3×#12111yescharacteristic
277C2×C32C3×#13511yescharacteristic
278C2×C32C3×#13711yescharacteristic
279C3×D3—81no—
280C3×D3—82no—
281C3×D3—81no—
282C2×Q8—91no—
283C2×Q8—91no—
284C2×D4—91no—
285Q16—182no—
286SD16—182no—
287C2×C8—91no—
288C2×C8—91no—
289C4⋊C4—61no—
290C4⋊C4—91no—
291C22⋊C4—61no—
292C22⋊C4—181no—
293C42—91no—
294C42—181no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—122no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—241no—
307C22×C3—241no—
308C22×C3C2×GO+(4,𝔽2)11yescharacteristic
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—62no—
316C3×C4—62no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4—91no—
320C3×C4—91no—
321C3×C4—122no—
322C3×C4—121no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—121no—
326C3×C4—181no—
327C3×C4—241no—
328C3×C4—241no—
329Dic3—22no—
330Dic3—41no—
331Dic3—41no—
332Dic3—61no—
333Dic3—61no—
334Dic3—61no—
335Dic3—61no—
336Dic3—62no—
337Dic3—121no—
338C32—21no—
339C32—21no—
340C32—41no—
341C32—41no—
342C32C3×#46411yescharacteristic, complemented
343C23—61no—
344Q8—92no—
345Q8—181no—
346Q8—181no—
347Q8—181no—
348D4—181no—
349D4—181no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4C3×GO+(4,𝔽2)11yescharacteristic
356C8—181no—
357C8—181no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—241no—
371C2×C3—242no—
372C2×C3—241no—
373C2×C3#6422611yescharacteristic
374C2×C3#6423011yescharacteristic
375C2×C3#6423711yescharacteristic
376D3—81no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22C2×C3×GO+(4,𝔽2)11yescharacteristic
381C4—21no—
382C4—62no—
383C4—91no—
384C4—91no—
385C4—121no—
386C4—121no—
387C4—181no—
388C3—21no—
389C3—21no—
390C3—41no—
391C3—41no—
392C3#1059344011yescharacteristic, direct product factor
393C2—241no—
394C2C3×#6422611yescharacteristic
395C2C3×#6423011yescharacteristic
396C2C3×#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×#464
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
1. C2×C3×#3776C2semidirect
4. C3×#63838C2semidirect
5. C3×C4⋊#622C2semidirect
6. C3×#63740C2semidirect
8. #10593440C3direct (N × H)
22. C2×#3776C2×C3semidirect
27. #63838C2×C3semidirect
28. C4⋊#622C2×C3semidirect
29. #63740C2×C3semidirect
221. C33#464semidirect
342. C32C3×#464semidirect
392. C3#10593440direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×C3×#37752
C3×#639152
C3×#638382
C3×C4⋊#6222
C3×#637402
C3×#637392
#105934403
C3×#4649