Finite Group Databasebeta

← C3×#10593433 #13166797

Subgroups — C3×#10593433 #13166797

Order 1728.

468 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3775C211yesmaximal, characteristic
2C3×#63902C211yesmaximal, characteristic
3C3×#63838C211yesmaximal, characteristic, complemented
4C3×C4×#622C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic
8#10593433C311yesmaximal, characteristic, direct product factor
9C2×C3×#627C2211yescharacteristic
10C2×C3×#625C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×#3775C414yescomplemented
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×Dic3⋊C4—21no—
18C3×#3724C2211yescharacteristic
19C3×#3724C2211yescharacteristic
20C3×Dic32—21no—
21C2×#3775C2×C311yescharacteristic
22C3×#887—61no—
23C3×#67⋊C4—61no—
24#63902C2×C311yescharacteristic
25#63838C2×C311yescharacteristic, complemented
26C4×#622C2×C311yescharacteristic, complemented
27#63742C2×C311yescharacteristic, complemented
28#63740C2×C311yescharacteristic, complemented
29#63739C2×C311yescharacteristic
30C22×C32×D3—21no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C3×#627—21no—
38C3×#627C2×C412yescomplemented
39C3×#625—21no—
40C3×#625C2×C412yes—
41C3×D3×Dic3—42no—
42C3×#622—21no—
43C3×#622C2×C412yescomplemented
44C3×C4×SD16—91nomaximal
45C2×C3×#67—61no—
46C22×C3×Dic3—61no—
47C2×C3×C4×D3—61no—
48C2×C3×Dic6—61no—
49C2×#627C22×C311yescharacteristic
50C2×#625C22×C311yescharacteristic
51C2×D3×Dic3—21no—
52C2×#622C22×C311yescharacteristic
53C2×#622C22×C311yescharacteristic
54#3775C3×C414yescomplemented
55C32×C4⋊C4—61no—
56C32×C22⋊C4—61no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×(C2×D3)⋊C4—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—41no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×Dic3⋊C4—41no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—61no—
66C3×C4×Dic3—61no—
67C3×C4×Dic3—61no—
68C3×C4×Dic3—61no—
69C3×C4×Dic3—61no—
70#3724C22×C311yescharacteristic
71#3724C22×C311yescharacteristic
72Dic32—21no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C33×C4—21no—
78C3×C3⋊Dic3SD1612yescomplemented
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C32×Dic3—22no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84C2×C3×SD16—91no—
85C3×C4×Q8—91no—
86C3×C4×D4—91no—
87#887—61no—
88#67⋊C4—61no—
89C3×#106—91no—
90C3×#103—91no—
91C3×#102—91no—
92C3×C4×C8—91no—
93C23×C32—61no—
94C22×C3×D3—21no—
95C22×C3×D3—21no—
96C22×C3×D3—41no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—61no—
102C2×C32×C4—61no—
103C2×C32×C4—61no—
104C2×C32×C4C3×D411yescharacteristic
105C2×C3⋊Dic3C3×D411yescharacteristic
106C2×C3⋊Dic3C23×C311yescharacteristic
107C3×#67—62no—
108C3×#67—62no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—122no—
123C3×C4×D3—122no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126#627—21no—
127#627C2×C3×C412yescomplemented
128#625—21no—
129#625C2×C3×C412yes—
130D3×Dic3—42no—
131#622—21no—
132#622C2×C3×C412yescomplemented
133C4×SD16—91noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33#13511yescharacteristic
137C32×D3—42no—
138C2×C3×Q8—91no—
139C2×C3×D4—91no—
140C22×C3×C4—181no—
141C2×#67—61no—
142C22×Dic3—61no—
143C2×C4×D3—61no—
144C2×Dic6—61no—
145C3×SD16—94no—
146C2×C3×C8—91no—
147C2×C3×C8—91no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—121no—
153C3×C22⋊C4—61no—
154C3×C22⋊C4—61no—
155C3×C22⋊C4—121no—
156C3×C42—91no—
157C3×C42—181no—
158(C2×D3)⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—61no—
161Dic3⋊C4—61no—
162C4×Dic3—61no—
163C4×Dic3—61no—
164C4×Dic3—61no—
165C4×Dic3—61no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—62no—
171C22×C32—121no—
172C22×C32—121no—
173C22×C32C2×C3×D411yescharacteristic
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—42no—
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—81no—
182C2×C3×D3—81no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—62no—
187C32×C4—81no—
188C32×C4—81no—
189C32×C4—121no—
190C32×C4—121no—
191C3⋊Dic3C3×SD1612yescomplemented
192C3⋊Dic3C22×C3×C411yescharacteristic
193C3⋊Dic3C3×#4111yescharacteristic
194C3×Dic3—22no—
195C3×Dic3—22no—
196C3×Dic3—42no—
197C3×Dic3—41no—
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—81no—
208C3×Dic3—81no—
209C2×SD16—91no—
210C4×Q8—91no—
211C4×D4—91no—
212#106—91no—
213#103—91no—
214#102—91no—
215C4×C8—91no—
216C33C4×SD1611yescharacteristic, complemented, Sylow
217C23×C3—61no—
218C23×C3—61no—
219C23×C3—121no—
220C22×D3—21no—
221C3×Q8—92no—
222C3×Q8—181no—
223C3×D4—92no—
224C3×D4—181no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—121no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4—362no—
241C2×C3×C4GO+(4,𝔽2)11yescharacteristic
242#67—62no—
243#67—62no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—122no—
252C4×D3—122no—
253Dic6—62no—
254Dic6—62no—
255C3×C8—92no—
256C3×C8—181no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—122no—
270C2×C32C3×C4×D411yescharacteristic
271C2×C32C2×C3×SD1611yescharacteristic
272C2×C32C3×#13511yescharacteristic
273C3×D3—42no—
274C3×D3—42no—
275C3×D3—82no—
276C2×Q8—91no—
277C2×D4—91no—
278C22×C4—181no—
279SD16—94no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—61no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C22⋊C4—61no—
286C42—91no—
287C42—181no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—122no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—241no—
300C22×C3—241no—
301C22×C3C2×GO+(4,𝔽2)11yescharacteristic
302C2×D3—22no—
303C2×D3—41no—
304C2×D3—41no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—62no—
309C3×C4—62no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—92no—
313C3×C4—91no—
314C3×C4—91no—
315C3×C4—122no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—121no—
320C3×C4—241no—
321C3×C4—241no—
322Dic3—22no—
323Dic3—41no—
324Dic3—41no—
325Dic3—61no—
326Dic3—61no—
327Dic3—62no—
328Dic3—61no—
329Dic3—61no—
330Dic3—62no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32C3×C4×SD1611yescharacteristic, complemented
336C23—61no—
337Q8—92no—
338Q8—181no—
339D4—92no—
340D4—181no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—362no—
347C2×C4C3×GO+(4,𝔽2)11yescharacteristic
348C8—92no—
349C8—181no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—122no—
363C2×C3—122no—
364C2×C3—242no—
365C2×C3#6422711yescharacteristic
366C2×C3C4×GO+(4,𝔽2)11yescharacteristic
367C2×C3#6423711yescharacteristic
368D3—42no—
369C22—62no—
370C22—121no—
371C22—121no—
372C22C2×C3×GO+(4,𝔽2)11yescharacteristic
373C4—21no—
374C4—62no—
375C4—92no—
376C4—91no—
377C4—91no—
378C4—121no—
379C4—121no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3—41no—
384C3#1059343311yescharacteristic, direct product factor
385C2—122no—
386C2C3×#6422711yescharacteristic
387C2C3×C4×GO+(4,𝔽2)11yescharacteristic
388C2C3×#6423711yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×C4×SD16
LCSG ▷ 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
3. C3×#63838C2semidirect
4. C3×C4×#622C2semidirect
5. C3×#63742C2semidirect
6. C3×#63740C2semidirect
8. #10593433C3direct (N × H)
14. C3×#3775C4semidirect
25. #63838C2×C3semidirect
26. C4×#622C2×C3semidirect
27. #63742C2×C3semidirect
28. #63740C2×C3semidirect
38. C3×#627C2×C4semidirect
43. C3×#622C2×C4semidirect
54. #3775C3×C4semidirect
78. C3×C3⋊Dic3SD16semidirect
127. #627C2×C3×C4semidirect
132. #622C2×C3×C4semidirect
191. C3⋊Dic3C3×SD16semidirect
216. C33C4×SD16semidirect
335. C32C3×C4×SD16semidirect
384. C3#10593433direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C3×#639022
C3×#638382
C3×C4×#6222
C3×#637422
C3×#637402
C3×#637392
#105934333
C3×C4×SD169