Finite Group Databasebeta

← C3×#10593419 #13166781

Subgroups — C3×#10593419 #13166781

Order 1728.

540 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3774C211yesmaximal, characteristic
2C3×#63902C211yesmaximal, characteristic, complemented
3C3×C4×#625C211yesmaximal, characteristic, complemented
4C3×#63777C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593419C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3774C414yescomplemented
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C3×#3724C2211yescharacteristic
19C3×#3724C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C2×#3774C2×C311yescharacteristic
22C3×C4×#67—61no—
23C3×#67⋊C4—61no—
24#63902C2×C311yescharacteristic, complemented
25C4×#625C2×C311yescharacteristic, complemented
26#63777C2×C311yescharacteristic, complemented
27#63742C2×C311yescharacteristic, complemented
28#63739C2×C311yescharacteristic, complemented
29#63739C2×C311yescharacteristic, complemented
30C22×C32×D3—21no—
31C22×C32×D3—21no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C32×C4×D3—42no—
38C3×#625—21no—
39C3×#625—21no—
40C3×#625C2×C412yescomplemented
41C3×#625C2×C412yescomplemented
42C3×D3×Dic3—42no—
43C3×#622—21no—
44C3×#622C2×C412yescomplemented
45C3×#442—91nomaximal
46C22×C32×C4—61no—
47C2×C3×#67—61no—
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—21no—
52C2×C3×C4×D3—41no—
53C2×C3×C4×D3—61no—
54C2×#625C22×C311yescharacteristic
55C2×#625C22×C311yescharacteristic
56C2×D3×Dic3—21no—
57C2×#622C22×C311yescharacteristic
58C2×#622C22×C311yescharacteristic
59#3774C3×C414yescomplemented
60C32×C22⋊C4—61no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×#217—61no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—41no—
66C3×(C2×D3)⋊C4—61no—
67C3×Dic3⋊C4—61no—
68C3×Dic3⋊C4—61no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—61no—
71#3724C22×C311yescharacteristic
72#3724C22×C311yescharacteristic
73(C2×D3)⋊Dic3—21no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—22no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4—21no—
82C3×C3⋊Dic3—21no—
83C3×C3⋊Dic3C22×C411yescharacteristic
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—41no—
86C32×Dic3—41no—
87C2×C3×D8—91no—
88C3×C4×D4—91no—
89C3×C4×D4—91no—
90C4×#67—61no—
91#67⋊C4—61no—
92C3×#106—91no—
93C3×#102—91no—
94C3×#102—91no—
95C3×#97—91no—
96C23×C32—61no—
97C23×C32—61no—
98C22×C3×D3—21no—
99C22×C3×D3—21no—
100C22×C3×D3—21no—
101C22×C3×D3—21no—
102C22×C3×D3—41no—
103C22×C3×D3—41no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—61no—
109C2×C32×C4—61no—
110C2×C32×C4—122no—
111C2×C32×C4C3×D411yescharacteristic
112C2×C3⋊Dic3C3×D411yescharacteristic
113C2×C3⋊Dic3C23×C311yescharacteristic
114C3×#67—62no—
115C3×#67—62no—
116C3×#67—62no—
117C3×#67—62no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—122no—
129C3×C4×D3—42no—
130C3×C4×D3—42no—
131C3×C4×D3—44no—
132C3×C4×D3—122no—
133#625—21no—
134#625—21no—
135#625C2×C3×C412yescomplemented
136#625C2×C3×C412yescomplemented
137D3×Dic3—42no—
138#622—21no—
139#622C2×C3×C412yescomplemented
140#442—91noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C32×D3—42no—
145C32×D3—42no—
146C2×C3×D4—91no—
147C2×C3×D4—91no—
148C22×C3×C4—61no—
149C22×C3×C4—61no—
150C22×C3×C4—121no—
151C22×C3×C4—181no—
152C2×#67—61no—
153C2×#67—61no—
154C22×Dic3—61no—
155C2×C4×D3—21no—
156C2×C4×D3—61no—
157C3×D8—94no—
158C2×C3×C8—91no—
159C2×C3×C8—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C22⋊C4—61no—
163C3×C22⋊C4—61no—
164C3×C22⋊C4—121no—
165C3×C22⋊C4—181no—
166C3×C42—91no—
167#217—61no—
168(C2×D3)⋊C4—21no—
169(C2×D3)⋊C4—61no—
170Dic3⋊C4—61no—
171Dic3⋊C4—61no—
172C4×Dic3—61no—
173C4×Dic3—61no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—62no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3×D3—22no—
186C2×C3×D3—22no—
187C2×C3×D3—22no—
188C2×C3×D3—22no—
189C2×C3×D3—42no—
190C2×C3×D3—42no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—81no—
202C2×C3×D3—81no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—41no—
206C32×C4—42no—
207C32×C4—81no—
208C32×C4—121no—
209C32×C4—121no—
210C3⋊Dic3—21no—
211C3⋊Dic3C22×C3×C411yescharacteristic
212C3⋊Dic3C3×#4111yescharacteristic
213C3×Dic3—41no—
214C3×Dic3—41no—
215C3×Dic3—42no—
216C3×Dic3—41no—
217C3×Dic3—41no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—62no—
223C3×Dic3—81no—
224C3×Dic3—121no—
225C2×D8—91no—
226C4×D4—91no—
227C4×D4—91no—
228#106—91no—
229#102—91no—
230#102—91no—
231#97—91no—
232C33#44211yescharacteristic, complemented, Sylow
233C23×C3—61no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—61no—
237C23×C3—121no—
238C23×C3—121no—
239C22×D3—21no—
240C22×D3—21no—
241C3×D4—92no—
242C3×D4—92no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—121no—
256C2×C3×C4—121no—
257C2×C3×C4—122no—
258C2×C3×C4—122no—
259C2×C3×C4—124no—
260C2×C3×C4—362no—
261C2×C3×C4GO+(4,𝔽2)11yescharacteristic
262#67—62no—
263#67—62no—
264#67—62no—
265#67—62no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—122no—
273C4×D3—42no—
274C4×D3—122no—
275C3×C8—92no—
276C3×C8—181no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—122no—
290C2×C32—122no—
291C2×C32C3×C4×D411yescharacteristic
292C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
293C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
294C3×D3—42no—
295C3×D3—42no—
296C3×D3—44no—
297C3×D3—42no—
298C3×D3—42no—
299C3×D3—82no—
300C2×D4—91no—
301C2×D4—91no—
302C22×C4—61no—
303C22×C4—181no—
304D8—94no—
305C2×C8—91no—
306C2×C8—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C22⋊C4—61no—
310C22⋊C4—181no—
311C42—91no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—122no—
321C22×C3—122no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—122no—
327C22×C3—122no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3C2×GO+(4,𝔽2)11yescharacteristic
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C3×C4—22no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—42no—
345C3×C4—81no—
346C3×C4—91no—
347C3×C4—91no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4—122no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—181no—
354C3×C4—241no—
355Dic3—41no—
356Dic3—41no—
357Dic3—61no—
358Dic3—61no—
359Dic3—61no—
360Dic3—61no—
361Dic3—62no—
362Dic3—121no—
363C32—21no—
364C32—21no—
365C32—41no—
366C32—41no—
367C32C3×#44211yescharacteristic, complemented
368C23—61no—
369C23—61no—
370D4—92no—
371D4—92no—
372D4—181no—
373D4—181no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—91no—
377C2×C4—91no—
378C2×C4—122no—
379C2×C4—362no—
380C2×C4C3×GO+(4,𝔽2)11yescharacteristic
381C8—92no—
382C8—181no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—122no—
396C2×C3—122no—
397C2×C3—124no—
398C2×C3—122no—
399C2×C3—122no—
400C2×C3—242no—
401C2×C3#6422411yescharacteristic
402C2×C3C4×GO+(4,𝔽2)11yescharacteristic
403C2×C3#6423611yescharacteristic
404D3—42no—
405D3—42no—
406C22—62no—
407C22—62no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22C2×C3×GO+(4,𝔽2)11yescharacteristic
413C4—21no—
414C4—91no—
415C4—91no—
416C4—121no—
417C4—121no—
418C4—181no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059341911yescharacteristic, direct product factor
424C2—122no—
425C2—122no—
426C2C3×#6422411yescharacteristic
427C2C3×C4×GO+(4,𝔽2)11yescharacteristic
428C2C3×#6423611yescharacteristic
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×#3774C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#442
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
2. C3×#63902C2semidirect
3. C3×C4×#625C2semidirect
4. C3×#63777C2semidirect
5. C3×#63742C2semidirect
6. C3×#63739C2semidirect
7. C3×#63739C2semidirect
8. #10593419C3direct (N × H)
15. C3×#3774C4semidirect
24. #63902C2×C3semidirect
25. C4×#625C2×C3semidirect
26. #63777C2×C3semidirect
27. #63742C2×C3semidirect
28. #63739C2×C3semidirect
29. #63739C2×C3semidirect
40. C3×#625C2×C4semidirect
41. C3×#625C2×C4semidirect
44. C3×#622C2×C4semidirect
59. #3774C3×C4semidirect
135. #625C2×C3×C4semidirect
136. #625C2×C3×C4semidirect
139. #622C2×C3×C4semidirect
232. C33#442semidirect
367. C32C3×#442semidirect
423. C3#10593419direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C3×#639022
C3×C4×#6252
C3×#637772
C3×#637422
C3×#637392
#105934193
C3×#4429