Finite Group Databasebeta

← C3×#10593323 #13166727

Subgroups — C3×#10593323 #13166727

Order 1728.

614 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3798C211yesmaximal, characteristic
2C3×#64286C211yesmaximal, characteristic, complemented
3C3×#63818C211yesmaximal, characteristic, complemented
4C3×#63731C214yesmaximal, complemented
5#10593323C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×C4×D3—21no—
8C2×C3×#627C411yescharacteristic, complemented
9C2×C3×#626—21no—
10C2×C3×#625C411yescharacteristic, complemented
11C3×#3798—21no—
12C3×#3798—21no—
13C3×#3798C412yescomplemented
14C3×#3798C2212yes—
15C3×C22⋊#157—21no—
16C3×C4⋊#157—21no—
17C3×C4×#157C2212yescomplemented
18C32×#208—21no—
19C3×#3712C2212yescomplemented
20C3×#3710—21no—
21C2×C3×#235—61no—
22C2×#3798C2×C311yescharacteristic
23#64286C2×C311yescharacteristic, complemented
24C3×D3×M16—61no—
25#63818C2×C311yescharacteristic, complemented
26#63731C2×C314yescomplemented
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C32×D3—21no—
29C2×C3×#157—22no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C32×C4×D3—22no—
38C32×C4×D3—22no—
39C3×#627—21no—
40C3×#627C2×C412yes—
41C3×#626—22no—
42C3×#626—22no—
43C3×#625—21no—
44C3×#625C2×C412yes—
45C32×C3⋊C8—22no—
46C3×#421—91nomaximal
47C22×C32×C4—61no—
48C2×C4×C3⋊D3C22×C311yescharacteristic
49C2×C3×#67—61no—
50C2×C3×#67—61no—
51C3×#235—62no—
52C3×#235—62no—
53C3×#235—62no—
54C3×#235—62no—
55C2×C3×D12—61no—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—41no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×C3×Dic6—61no—
62C2×#627C3×C411yescharacteristic, complemented
63C2×#626—21no—
64C2×#625C3×C411yescharacteristic, complemented
65#3798—21no—
66#3798—21no—
67#3798C3×C412yescomplemented
68#3798C22×C312yes—
69C22⋊#157—21no—
70C4⋊#157—21no—
71C4×#157C22×C312yescomplemented
72C32×M16—61no—
73C3×#208—21no—
74C3×#208—21no—
75C3×#208—41no—
76C3×ΓL(1,𝔽25)—62no—
77C3×D3×C8—62no—
78#3712C22×C312yescomplemented
79#3710—21no—
80C22×C33C22⋊C411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3C22⋊C411yescharacteristic
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C32×D3—22no—
85C2×C32×D3—42no—
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C22×C411yescharacteristic
91C32×Dic3—22no—
92C2×C3×#41—91no—
93C2×#235—61no—
94C2×C3×M16—91no—
95C3×#117—91no—
96D3×M16—61no—
97C3×#104—94no—
98C23×C32—61no—
99C22×C3⋊D3C3×D411yescharacteristic
100C22×C3×D3—21no—
101C22×C3×D3—21no—
102C22×C3×D3—41no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C2×#157—22no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4—62no—
112C2×C32×C4—61no—
113C2×C32×C4C3×D411yescharacteristic
114C2×C3⋊Dic3C2×C3×C411yescharacteristic
115C4×C3⋊D3C2×C3×C411yescharacteristic
116C4×C3⋊D3C3×D411yescharacteristic
117C4×C3⋊D3C3×D411yescharacteristic
118C4×C3⋊D3C23×C311yescharacteristic
119C3×#67—62no—
120C3×#67—62no—
121C3×#67—62no—
122C3×#67—62no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—62no—
129C3×D12—62no—
130C3×C4×D3—22no—
131C3×C4×D3—22no—
132C3×C4×D3—22no—
133C3×C4×D3—22no—
134C3×C4×D3—42no—
135C3×C4×D3—42no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×Dic6—62no—
145C3×Dic6—62no—
146#627—21no—
147#627C2×C3×C412yes—
148#626—22no—
149#626—22no—
150#625—21no—
151#625C2×C3×C412yes—
152C32×C8—62no—
153C3×C3⋊C8—22no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—42no—
156#421—91noSylow
157C2×C33—21no—
158C2×C33C2×C22⋊C411yescharacteristic
159C3×C3⋊D3—21no—
160C3×C3⋊D3—21no—
161C32×D3—42no—
162C3×#41—92no—
163C3×#41—92no—
164C3×#41—181no—
165C3×#41—181no—
166C2×C3×Q8—91no—
167C2×C3×D4—91no—
168C2×C3×D4—181no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4—91no—
172C22×C3×C4—121no—
173C2×#67—61no—
174C2×#67—61no—
175#235—62no—
176#235—62no—
177#235—62no—
178#235—62no—
179C2×D12—61no—
180C2×C4×D3—21no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×Dic6—61no—
184C3×M16—61no—
185C3×M16—61no—
186C3×M16—92no—
187C3×M16—121no—
188C2×C3×C8—181no—
189C3×C4⋊C4—181no—
190C3×C22⋊C4—181no—
191C3×C42—92no—
192#208—21no—
193ΓL(1,𝔽25)—62no—
194D3×C8—62no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—62no—
200C22×C32—122no—
201C22×C32C3×C22⋊C411yescharacteristic
202C2×C3⋊D3—41no—
203C2×C3⋊D3C3×C22⋊C411yescharacteristic
204C2×C3⋊D3C2×C3×D411yescharacteristic
205C2×C3×D3—22no—
206C2×C3×D3—22no—
207C2×C3×D3—42no—
208C2×C3×D3—42no—
209C2×C3×D3—42no—
210C2×C3×D3—44no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217#157—42no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—62no—
227C32×C4C3×C22⋊C411yescharacteristic
228C32×C4C2×C3×D411yescharacteristic
229C3⋊Dic3C3×C22⋊C411yescharacteristic
230C3⋊Dic3C22×C3×C411yescharacteristic
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C2×#41—91no—
239C2×M16—91no—
240#117—91no—
241#104—94no—
242C33#42111yescharacteristic, complemented, Sylow
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—91no—
246C23×C3—121no—
247C22×D3—21no—
248C22×D3—61no—
249C22×D3—61no—
250C3×Q8—92no—
251C3×Q8—181no—
252C3×D4—92no—
253C3×D4—182no—
254C3×D4—182no—
255C3×D4—181no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—122no—
272C2×C3×C4—122no—
273C2×C3×C4—121no—
274C2×C3×C4—182no—
275C2×C3×C4GO+(4,𝔽2)11yescharacteristic
276#67—62no—
277#67—62no—
278#67—62no—
279#67—62no—
280C2×Dic3—21no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283D12—62no—
284D12—62no—
285C4×D3—22no—
286C4×D3—22no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295Dic6—62no—
296Dic6—62no—
297C3×C8—62no—
298C3×C8—62no—
299C3×C8—122no—
300C3⋊C8—22no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—81no—
310C2×C32—122no—
311C2×C32C2×C3×C22⋊C411yescharacteristic
312C3⋊D3—21no—
313C3⋊D3—21no—
314C3×D3—42no—
315C3×D3—44no—
316C3×D3—42no—
317C3×D3—62no—
318C3×D3—121no—
319C3×D3—121no—
320C3×D3—121no—
321#41—92no—
322#41—92no—
323#41—181no—
324#41—181no—
325C2×Q8—91no—
326C2×D4—91no—
327C2×D4—181no—
328C22×C4—61no—
329C22×C4—91no—
330M16—61no—
331M16—92no—
332C2×C8—181no—
333C4⋊C4—181no—
334C22⋊C4—181no—
335C42—92no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—91no—
343C22×C3—91no—
344C22×C3—122no—
345C22×C3—122no—
346C22×C3—122no—
347C22×C3—124no—
348C22×C3—361no—
349C22×C3D32⋊C411yescharacteristic
350C2×D3—22no—
351C2×D3—42no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—122no—
357C2×D3—122no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—62no—
367C3×C4—62no—
368C3×C4—91no—
369C3×C4—91no—
370C3×C4—122no—
371C3×C4—362no—
372C3×C4D32⋊C411yescharacteristic
373C3×C4C2×GO+(4,𝔽2)11yescharacteristic
374Dic3—22no—
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378Dic3—61no—
379C32—21no—
380C32—21no—
381C32—41no—
382C32—41no—
383C32C3×#42111yescharacteristic, complemented
384C23—61no—
385C23—91no—
386Q8—92no—
387Q8—181no—
388D4—92no—
389D4—182no—
390D4—182no—
391D4—181no—
392C2×C4—61no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—182no—
401C2×C4C3×GO+(4,𝔽2)11yescharacteristic
402C8—62no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—81no—
412C2×C3—122no—
413C2×C3—124no—
414C2×C3—122no—
415C2×C3—181no—
416C2×C3—181no—
417C2×C3C2×D32⋊C411yescharacteristic
418D3—42no—
419D3—62no—
420D3—121no—
421D3—121no—
422D3—121no—
423C22—62no—
424C22—91no—
425C22—91no—
426C22—122no—
427C22—361no—
428C22C3×D32⋊C411yescharacteristic
429C4—62no—
430C4—91no—
431C4—91no—
432C4—362no—
433C4C3×D32⋊C411yescharacteristic
434C4C2×C3×GO+(4,𝔽2)11yescharacteristic
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059332311yescharacteristic, direct product factor
440C2—21no—
441C2—122no—
442C2—181no—
443C2—181no—
444C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3798C2
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#421
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64286C2semidirect
3. C3×#63818C2semidirect
4. C3×#63731C2semidirect
5. #10593323C3direct (N × H)
8. C2×C3×#627C4semidirect
10. C2×C3×#625C4semidirect
13. C3×#3798C4semidirect
17. C3×C4×#157C22semidirect
19. C3×#3712C22semidirect
23. #64286C2×C3semidirect
25. #63818C2×C3semidirect
26. #63731C2×C3semidirect
62. C2×#627C3×C4semidirect
64. C2×#625C3×C4semidirect
67. #3798C3×C4semidirect
71. C4×#157C22×C3semidirect
78. #3712C22×C3semidirect
242. C33#421semidirect
383. C32C3×#421semidirect
439. C3#10593323direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C3×#642862
C3×#638182
C3×#637312
#105933233
C3×#4219