Finite Group Databasebeta

← C3×#10593422 #13166786

Subgroups — C3×#10593422 #13166786

Order 1728.

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

normal subgroups only

This group has 506 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×#3775C211yesmaximal, characteristic, complemented
2C2×C3×#3774C211yesmaximal, characteristic
3C3×#63915C211yesmaximal, characteristic, complemented
4C3×C4×#625C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593422C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×#625C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3775—22no—
16C3×#3774—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C3×#3724C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C2×#3775C2×C311yescharacteristic, complemented
23C2×#3774C2×C311yescharacteristic
24C3×C4×#67—61no—
25C3×#885—61no—
26#63915C2×C311yescharacteristic, complemented
27C4×#625C2×C311yescharacteristic, complemented
28#63779C2×C311yescharacteristic, complemented
29#63740C2×C311yescharacteristic, complemented
30#63739C2×C311yescharacteristic, complemented
31C22×C32×D3—21no—
32C22×C32×D3—21no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C32×C4×D3—42no—
39C3×#627—21no—
40C3×#627—21no—
41C3×#625—21no—
42C3×#625—21no—
43C3×#625—21no—
44C3×#625D412yes—
45C3×#622—21no—
46C3×#622—21no—
47C3×#463—91nomaximal
48C22×C32×C4—61no—
49C2×C3×#67—61no—
50C2×C3×#67—61no—
51C2×C3×C4×D3—21no—
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3—41no—
54C2×C3×Dic6—61no—
55C2×#627C22×C311yescharacteristic
56C2×#625C22×C311yescharacteristic
57C2×#625C22×C311yescharacteristic
58C2×#622C22×C311yescharacteristic
59C2×#622C22×C311yescharacteristic
60#3775—22no—
61#3774—22no—
62C32×C22⋊C4—61no—
63C4×C3⋊Dic3C22×C311yescharacteristic
64C3×#217—61no—
65C3×#217—61no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—41no—
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×C4×Dic3—61no—
73C3×C4×Dic3—61no—
74#3724C22×C311yescharacteristic
75(C2×D3)⋊Dic3—21no—
76(C2×D3)⋊Dic3—21no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C2×C32×D3—41no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3C2×D411yescharacteristic
87C3×C3⋊Dic3#4111yescharacteristic
88C32×Dic3—41no—
89C32×Dic3—41no—
90C2×C3×SD16—91no—
91C2×C3×D8—91no—
92C3×#124—91no—
93C3×C4×D4—91no—
94C4×#67—61no—
95#885—61no—
96C3×C4⋊C8—91no—
97C3×#103—91no—
98C3×#102—91no—
99C23×C32—61no—
100C23×C32—61no—
101C22×C3×D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—41no—
106C22×C3×D3—41no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—61no—
112C2×C32×C4—61no—
113C2×C32×C4—122no—
114C2×C32×C4C3×D411yescharacteristic
115C2×C3⋊Dic3C3×D411yescharacteristic
116C2×C3⋊Dic3C23×C311yescharacteristic
117C3×#67—62no—
118C3×#67—62no—
119C3×#67—121no—
120C3×#67—121no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×C4×D3—42no—
132C3×C4×D3—42no—
133C3×C4×D3—44no—
134C3×Dic6—121no—
135C3×Dic6—121no—
136#627—21no—
137#627—21no—
138#625—21no—
139#625—21no—
140#625—21no—
141#625C3×D412yes—
142#622—21no—
143#622—21no—
144#463—91noSylow
145C2×C33#12111yescharacteristic
146C2×C33#13511yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C32×D3—42no—
149C32×D3—81no—
150C2×C3×Q8—91no—
151C2×C3×D4—91no—
152C2×C3×D4—91no—
153C22×C3×C4—61no—
154C22×C3×C4—61no—
155C22×C3×C4—121no—
156C2×#67—61no—
157C2×#67—61no—
158C2×C4×D3—21no—
159C2×Dic6—61no—
160C3×SD16—182no—
161C3×D8—182no—
162C2×C3×C8—91no—
163C2×C3×C8—91no—
164C3×C4⋊C4—91no—
165C3×C22⋊C4—61no—
166C3×C22⋊C4—61no—
167C3×C22⋊C4—121no—
168C3×C22⋊C4—181no—
169C3×C22⋊C4—181no—
170C3×C42—91no—
171#217—61no—
172#217—61no—
173(C2×D3)⋊C4—21no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—61no—
176Dic3⋊C4—61no—
177C4×Dic3—61no—
178C4×Dic3—61no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32—62no—
184C22×C32—121no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32C2×C3×D411yescharacteristic
190C2×C3×D3—22no—
191C2×C3×D3—22no—
192C2×C3×D3—42no—
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—41no—
202C2×C3×D3—42no—
203C2×C3×D3—41no—
204C2×C3×D3—41no—
205C2×C3×D3—41no—
206C2×C3×D3—81no—
207C2×C3×D3—81no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—42no—
212C32×C4—81no—
213C32×C4—121no—
214C32×C4—121no—
215C3⋊Dic3—21no—
216C3⋊Dic3C2×C3×D411yescharacteristic
217C3⋊Dic3C3×#4111yescharacteristic
218C3×Dic3—41no—
219C3×Dic3—41no—
220C3×Dic3—42no—
221C3×Dic3—41no—
222C3×Dic3—41no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—61no—
226C3×Dic3—61no—
227C3×Dic3—81no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C2×SD16—91no—
231C2×D8—91no—
232#124—91no—
233C4×D4—91no—
234C4⋊C8—91no—
235#103—91no—
236#102—91no—
237C33#46311yescharacteristic, complemented, Sylow
238C23×C3—61no—
239C23×C3—61no—
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—121no—
243C23×C3—121no—
244C22×D3—21no—
245C22×D3—21no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×D4—92no—
249C3×D4—181no—
250C3×D4—181no—
251C3×D4—181no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—121no—
263C2×C3×C4—121no—
264C2×C3×C4—122no—
265C2×C3×C4—122no—
266C2×C3×C4—124no—
267C2×C3×C4GO+(4,𝔽2)11yescharacteristic
268#67—62no—
269#67—62no—
270#67—121no—
271#67—121no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C4×D3—42no—
279Dic6—121no—
280Dic6—121no—
281C3×C8—181no—
282C3×C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—122no—
296C2×C32—241no—
297C2×C32C3×#12111yescharacteristic
298C2×C32C3×#13511yescharacteristic
299C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
300C3×D3—42no—
301C3×D3—44no—
302C3×D3—42no—
303C3×D3—81no—
304C3×D3—82no—
305C3×D3—81no—
306C2×Q8—91no—
307C2×D4—91no—
308C2×D4—91no—
309C22×C4—61no—
310SD16—182no—
311D8—182no—
312C2×C8—91no—
313C2×C8—91no—
314C4⋊C4—91no—
315C22⋊C4—61no—
316C22⋊C4—181no—
317C22⋊C4—181no—
318C42—91no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—62no—
324C22×C3—62no—
325C22×C3—122no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—241no—
340C22×C3—241no—
341C22×C3C2×GO+(4,𝔽2)11yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—42no—
352C3×C4—81no—
353C3×C4—91no—
354C3×C4—91no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—121no—
359C3×C4—121no—
360C3×C4—181no—
361C3×C4—241no—
362Dic3—41no—
363Dic3—41no—
364Dic3—61no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—121no—
369Dic3—121no—
370C32—21no—
371C32—21no—
372C32—41no—
373C32—41no—
374C32C3×#46311yescharacteristic, complemented
375C23—61no—
376C23—61no—
377Q8—181no—
378Q8—181no—
379D4—92no—
380D4—181no—
381D4—181no—
382D4—181no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—122no—
388C2×C4C3×GO+(4,𝔽2)11yescharacteristic
389C8—181no—
390C8—181no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—122no—
404C2×C3—124no—
405C2×C3—122no—
406C2×C3—241no—
407C2×C3—242no—
408C2×C3—241no—
409C2×C3#6422311yescharacteristic
410C2×C3#6423011yescharacteristic
411C2×C3#6423611yescharacteristic
412D3—42no—
413D3—81no—
414C22—62no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22C2×C3×GO+(4,𝔽2)11yescharacteristic
421C4—21no—
422C4—91no—
423C4—91no—
424C4—121no—
425C4—121no—
426C4—181no—
427C3—21no—
428C3—21no—
429C3—41no—
430C3—41no—
431C3#1059342211yescharacteristic, direct product factor
432C2—122no—
433C2—241no—
434C2C3×#6422311yescharacteristic
435C2C3×#6423011yescharacteristic
436C2C3×#6423611yescharacteristic
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×#3774C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#463
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×#3775C2semidirect
3. C3×#63915C2semidirect
4. C3×C4×#625C2semidirect
5. C3×#63779C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593422C3direct (N × H)
22. C2×#3775C2×C3semidirect
26. #63915C2×C3semidirect
27. C4×#625C2×C3semidirect
28. #63779C2×C3semidirect
29. #63740C2×C3semidirect
30. #63739C2×C3semidirect
237. C33#463semidirect
374. C32C3×#463semidirect
431. C3#10593422direct (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
C2×C3×#37742
C3×#639152
C3×C4×#6252
C3×#637792
C3×#637402
C3×#637392
#105934223
C3×#4639