Finite Group Databasebeta

← C3×#10593435 #13166807

Subgroups — C3×#10593435 #13166807

Order 1728.

476 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3787C211yesmaximal, characteristic, complemented
2C2×C3×#3775C211yesmaximal, characteristic, complemented
3C3×#63912C211yesmaximal, characteristic
4C3×#63865C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593435C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#626—21no—
12C2×C3×#625C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3787—22no—
16C3×#3775—22no—
17C32×(C2×D3)⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C3×#3724C2211yescharacteristic
20C3×#3724C2211yescharacteristic
21C3×#3722—21no—
22C2×#3787C2×C311yescharacteristic, complemented
23C2×#3775C2×C311yescharacteristic, complemented
24C3×#896—61no—
25C3×#884—61no—
26#63912C2×C311yescharacteristic
27#63865C2×C311yescharacteristic, complemented
28#63742C2×C311yescharacteristic, complemented
29#63740C2×C311yescharacteristic, complemented
30#63739C2×C311yescharacteristic, complemented
31C22×C3×C3⋊D3D411yescharacteristic
32C22×C32×D3—21no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—21no—
39C2×C32×Dic3—21no—
40C3×#627—21no—
41C3×#627—21no—
42C3×#626—42no—
43C3×#625—21no—
44C3×#625—21no—
45C3×#622—21no—
46C3×#622D412yescomplemented
47C3×#465—91nomaximal
48C2×C4×C3⋊D3C22×C311yescharacteristic
49C2×C3×#67—61no—
50C2×C3×#67—61no—
51C2×C3×D12—61no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—61no—
55C2×#627C22×C311yescharacteristic
56C2×#626—21no—
57C2×#625C22×C311yescharacteristic
58C2×#622C22×C311yescharacteristic
59C2×#622C22×C311yescharacteristic
60#3787—22no—
61#3775—22no—
62C32×C4⋊C4—61no—
63C32×C22⋊C4—61no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—41no—
67C3×(C2×D3)⋊C4—61no—
68C3×(C2×D3)⋊C4—61no—
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—41no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74#3724C22×C311yescharacteristic
75#3724C22×C311yescharacteristic
76#3722—21no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3SD1612yescomplemented
81C2×C32×D3—41no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—21no—
85C3×C3⋊Dic3C2×D411yescharacteristic
86C3×C3⋊Dic3#4111yescharacteristic
87C32×Dic3—41no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C2×C3×SD16—91no—
91C22×C3×C8—91no—
92C3×C22⋊Q8—91no—
93C3×#121—91no—
94#896—61no—
95#884—61no—
96C3×#106—91no—
97C3×#103—91no—
98C3×#102—91no—
99C23×C32—61no—
100C22×C3⋊D3C3×D411yescharacteristic
101C22×C3×D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—41no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—61no—
111C2×C32×C4—61no—
112C2×C32×C4—61no—
113C2×C32×C4C3×D411yescharacteristic
114C2×C3⋊Dic3C23×C311yescharacteristic
115C4×C3⋊D3—21no—
116C4×C3⋊D3—21no—
117C3×#67—121no—
118C3×#67—121no—
119C3×#67—122no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×D12—122no—
132C3×C4×D3—121no—
133C3×C4×D3—121no—
134C3×C4×D3—121no—
135C3×C4×D3—121no—
136C3×Dic6—121no—
137C3×Dic6—121no—
138#627—21no—
139#627—21no—
140#626—42no—
141#625—21no—
142#625—21no—
143#622—21no—
144#622C3×D412yescomplemented
145#465—91noSylow
146C2×C33#12111yescharacteristic
147C2×C33C2×SD1611yescharacteristic
148C2×C33#13511yescharacteristic
149C3×C3⋊D3—22no—
150C32×D3—81no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×C3×D4—181no—
154C22×C3×C4—91no—
155C2×#67—61no—
156C2×#67—61no—
157C2×D12—61no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C2×Dic6—61no—
161C3×SD16—182no—
162C2×C3×C8—91no—
163C2×C3×C8—91no—
164C2×C3×C8—182no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—121no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—61no—
172C3×C22⋊C4—121no—
173C3×C22⋊C4—181no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4—61no—
177Dic3⋊C4—21no—
178Dic3⋊C4—61no—
179Dic3⋊C4—61no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—121no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3⋊D3C3×SD1612yescomplemented
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—42no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—81no—
201C2×C3×D3—81no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—21no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—81no—
210C32×C4—81no—
211C32×C4—121no—
212C32×C4—121no—
213C32×C4—121no—
214C3⋊Dic3C2×C3×D411yescharacteristic
215C3⋊Dic3C3×#4111yescharacteristic
216C3×Dic3—41no—
217C3×Dic3—41no—
218C3×Dic3—41no—
219C3×Dic3—42no—
220C3×Dic3—41no—
221C3×Dic3—41no—
222C3×Dic3—41no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—61no—
226C3×Dic3—61no—
227C3×Dic3—81no—
228C3×Dic3—81no—
229C2×SD16—91no—
230C22×C8—91no—
231C22⋊Q8—91no—
232#121—91no—
233#106—91no—
234#103—91no—
235#102—91no—
236C33#46511yescharacteristic, complemented, Sylow
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—91no—
240C23×C3—121no—
241C22×D3—21no—
242C22×D3—61no—
243C22×D3—61no—
244C3×Q8—181no—
245C3×Q8—181no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4—362no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—121no—
261C2×C3×C4—121no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4GO+(4,𝔽2)11yescharacteristic
266#67—121no—
267#67—121no—
268#67—122no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274D12—122no—
275C4×D3—121no—
276C4×D3—121no—
277C4×D3—121no—
278C4×D3—121no—
279Dic6—121no—
280Dic6—121no—
281C3×C8—92no—
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—241no—
296C2×C32C3×#12111yescharacteristic
297C2×C32C2×C3×SD1611yescharacteristic
298C2×C32C3×#13511yescharacteristic
299C3⋊D3—22no—
300C3×D3—81no—
301C3×D3—82no—
302C3×D3—81no—
303C3×D3—122no—
304C3×D3—122no—
305C2×Q8—91no—
306C2×D4—91no—
307C2×D4—181no—
308C22×C4—91no—
309SD16—182no—
310C2×C8—91no—
311C2×C8—91no—
312C2×C8—182no—
313C4⋊C4—61no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C22⋊C4—61no—
317C22⋊C4—181no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—92no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—122no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3C2×GO+(4,𝔽2)11yescharacteristic
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—62no—
339C2×D3—62no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4—91no—
350C3×C4—91no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—121no—
354C3×C4—122no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—121no—
358C3×C4—241no—
359C3×C4—241no—
360Dic3—41no—
361Dic3—41no—
362Dic3—41no—
363Dic3—61no—
364Dic3—61no—
365Dic3—61no—
366Dic3—61no—
367C32—21no—
368C32—21no—
369C32—41no—
370C32—41no—
371C32C3×#46511yescharacteristic, complemented
372C23—61no—
373C23—91no—
374Q8—181no—
375Q8—181no—
376D4—181no—
377D4—181no—
378D4—362no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—91no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4C3×GO+(4,𝔽2)11yescharacteristic
386C8—92no—
387C8—181no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—182no—
401C2×C3—241no—
402C2×C3—242no—
403C2×C3—241no—
404C2×C3#6422311yescharacteristic
405C2×C3#6422711yescharacteristic
406C2×C3#6424211yescharacteristic
407D3—81no—
408D3—122no—
409D3—122no—
410C22—92no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—181no—
415C22—181no—
416C22C2×C3×GO+(4,𝔽2)11yescharacteristic
417C4—21no—
418C4—91no—
419C4—91no—
420C4—121no—
421C4—121no—
422C4—121no—
423C3—21no—
424C3—21no—
425C3—41no—
426C3—41no—
427C3#1059343511yescharacteristic, direct product factor
428C2—182no—
429C2—241no—
430C2C3×#6422311yescharacteristic
431C2C3×#6422711yescharacteristic
432C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63912C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#465
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×#3787C2semidirect
2. C2×C3×#3775C2semidirect
4. C3×#63865C2semidirect
5. C3×#63742C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593435C3direct (N × H)
22. C2×#3787C2×C3semidirect
23. C2×#3775C2×C3semidirect
27. #63865C2×C3semidirect
28. #63742C2×C3semidirect
29. #63740C2×C3semidirect
30. #63739C2×C3semidirect
46. C3×#622D4semidirect
80. C2×C3×C3⋊D3SD16semidirect
144. #622C3×D4semidirect
190. C2×C3⋊D3C3×SD16semidirect
236. C33#465semidirect
371. C32C3×#465semidirect
427. C3#10593435direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×C3×#37752
C3×#639122
C3×#638652
C3×#637422
C3×#637402
C3×#637392
#105934353
C3×#4659