Finite Group Databasebeta

← C3×#10590351 #13157367

Subgroups — C3×#10590351 #13157367

Order 1728.

830 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C3×#64083C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic, complemented
3C2×C3×#3717C212yesmaximal, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C32×#803C211yesmaximal, characteristic, complemented
6C3×#5058—31nomaximal
7C3×#4712—31nomaximal
8#10590351C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)—22no—
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C32×(C2×D3)⋊C4—22no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3717—24no—
17C3×#3714—24no—
18C32×#124—31no—
19#64083C2×C311yescharacteristic, complemented
20C2×C3×#215—32no—
21C2×C3×#213—32no—
22C2×C3×D24—32no—
23C2×C3×#204—32no—
24C3×#875—21no—
25C3×#875D311yescharacteristic, complemented
26C3×#875C2×C311yescharacteristic, complemented
27C3×#874—31no—
28C3×#874—31no—
29C3×#874—61no—
30C2×#3717C2×C312yescomplemented
31C2×#3714C2×C312yescomplemented
32C32×#97—31no—
33C3×#803—21no—
34C3×#803D311yescharacteristic, complemented
35C3×#803C2×C311yescharacteristic, complemented
36C22×C3×C3⋊D3—22no—
37C22×C32×D3—21no—
38C2×C33×C4D412yes—
39C2×C33×C4C2311yescharacteristic
40C3×C4⋊(C3⋊D3)—22no—
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)—44no—
43C2×C32×Dic3—21no—
44C32×D12—22no—
45C32×Dic6—22no—
46C32×C3⋊C8D414yescomplemented
47C3×#496—91no—
48#5058—31no—
49#4712—31no—
50C2×C32×Q8—31no—
51C2×C32×D4—31no—
52C2×C4⋊(C3⋊D3)—22no—
53C2×C4⋊(C3⋊D3)C22×C312yes—
54C2×C3×D12—21no—
55C2×C3×D12—32no—
56C2×C3×D12—32no—
57C2×C3×D12—62no—
58C2×C3×D12—64no—
59C2×C3×D12—62no—
60C2×C3×D12—62no—
61C2×C3×D12C2×D311yescharacteristic
62C2×C3×D12C22×C311yescharacteristic
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×C3×Dic6C22×C311yescharacteristic
66C2×C32×C8—32no—
67C32×C22⋊C4—62no—
68C32×C42—21no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42—41no—
72C32×C42C2×D311yescharacteristic
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75C3×#215—64no—
76C3×#213—64no—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—22no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8C2×D312yes—
82C2×C3×C3⋊C8C22×C312yes—
83C3×D24—64no—
84C3×#204—64no—
85#3717—24no—
86#3714—24no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—42no—
89C2×C3×C3⋊D3—44no—
90C2×C32×D3—41no—
91C2×C32×D3—42no—
92C33×C4—22no—
93C33×C4C2×D412yes—
94C32×Dic3—41no—
95C2×C3×SD16—92no—
96C2×C3×D8—92no—
97C3×#127—91no—
98C3×#124—31no—
99C3×#124—31no—
100C3×#124—61no—
101C2×#215—32no—
102C2×#213—32no—
103C2×D24—32no—
104C2×#204—32no—
105#875C3×D311yescharacteristic, complemented
106#874—31no—
107#874—31no—
108#874—61no—
109C3×#97—31no—
110C3×#97—31no—
111C3×#97—61no—
112#803C3×D311yescharacteristic, complemented
113C23×C32—61no—
114C22×C3⋊D3—22no—
115C22×C3×D3—22no—
116C22×C3×D3—21no—
117C22×C3×D3—21no—
118C22×C3×D3—64no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C32×Q8—62no—
122C32×D4—62no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—42no—
130C2×C32×C4—41no—
131C2×C32×C4—61no—
132C2×C32×C4D1212yes—
133C2×C32×C4#6712yes—
134C2×C32×C4C3×D412yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C23×C311yescharacteristic
138C4⋊(C3⋊D3)—22no—
139C4⋊(C3⋊D3)—22no—
140C4⋊(C3⋊D3)—44no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C3×D12—22no—
145C3×D12—22no—
146C3×D12—42no—
147C3×D12—62no—
148C3×D12—62no—
149C3×D12—62no—
150C3×D12—62no—
151C3×D12—122no—
152C3×D12—122no—
153C3×D12—128no—
154C3×D12—124no—
155C3×D12—124no—
156C3×Dic6—22no—
157C3×Dic6—22no—
158C3×Dic6—42no—
159C32×C8—34no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8D1214yescomplemented
162C3×C3⋊C8C3×D414yescomplemented
163#496—91noSylow
164C2×C33#12711yescharacteristic
165C2×C33AGL(1,ℤ/8)12yes—
166C3×C3⋊D3—82no—
167C32×D3—81no—
168C2×C3×Q8—31no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×D4—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—92no—
175C2×C3×D4—182no—
176C2×D12—32no—
177C2×D12—32no—
178C2×D12—62no—
179C2×D12—64no—
180C2×D12—62no—
181C2×D12—62no—
182C2×D12C2×C3×D311yescharacteristic
183C2×Dic6C2×C3×D311yescharacteristic
184C3×SD16—184no—
185C3×D8—184no—
186C2×C3×C8—32no—
187C2×C3×C8—32no—
188C2×C3×C8—62no—
189C3×C22⋊C4—64no—
190C3×C22⋊C4—62no—
191C3×C22⋊C4—62no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—41no—
196C3×C42D3211yescharacteristic
197C3×C42C2×C3×D311yescharacteristic
198C3×C42C2×C3×D311yescharacteristic
199#215—64no—
200#213—64no—
201(C2×D3)⋊C4—22no—
202C2×C3⋊C8C2×C3×D312yes—
203D24—64no—
204#204—64no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—41no—
209C22×C32—121no—
210C22×C32—122no—
211C22×C32C2×D1211yescharacteristic
212C22×C32C2×#6711yescharacteristic
213C22×C32C2×C3×D411yescharacteristic
214C2×C3⋊D3—42no—
215C2×C3⋊D3—44no—
216C2×C3×D3—42no—
217C2×C3×D3—44no—
218C2×C3×D3—41no—
219C2×C3×D3—42no—
220C2×C3×D3—41no—
221C2×C3×D3—42no—
222C2×C3×D3—128no—
223C2×C3×D3—124no—
224C2×C3×D3—122no—
225C2×C3×D3—124no—
226C2×C3×D3—122no—
227C2×C3×D3—124no—
228C32×C4—22no—
229C32×C4—24no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—42no—
236C32×C4—42no—
237C32×C4—42no—
238C32×C4—44no—
239C32×C4—121no—
240C32×C4C2×D1212yes—
241C32×C4D3×D412yes—
242C32×C4C2×C3×D412yes—
243C3×Dic3—42no—
244C3×Dic3—41no—
245C3×Dic3—41no—
246C2×SD16—92no—
247C2×D8—92no—
248#127—91no—
249#124—31no—
250#97—31no—
251C33#49611yescharacteristic, complemented, Sylow
252C23×C3—62no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—182no—
256C22×D3—21no—
257C22×D3—64no—
258C22×D3—62no—
259C22×D3—62no—
260C3×Q8—62no—
261C3×Q8—62no—
262C3×Q8—122no—
263C3×D4—62no—
264C3×D4—62no—
265C3×D4—122no—
266C3×D4—182no—
267C3×D4—182no—
268C3×D4—364no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—62no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4#62612yes—
281C2×C3×C4C3×D1212yes—
282C2×C3×C4C3×#6712yes—
283C2×C3×C4C2×D3211yescharacteristic
284C2×C3×C4C22×C3×D311yescharacteristic
285C2×C3×C4C22×C3×D311yescharacteristic
286C2×Dic3—21no—
287D12—22no—
288D12—62no—
289D12—62no—
290D12—62no—
291D12—62no—
292D12—122no—
293D12—122no—
294D12—128no—
295D12—124no—
296D12—124no—
297Dic6—22no—
298C3×C8—34no—
299C3×C8—34no—
300C3×C8—64no—
301C3⋊C8C3×D1214yescomplemented
302C2×C32—22no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—42no—
310C2×C32—241no—
311C2×C32#87411yescharacteristic
312C2×C32#90812yes—
313C2×C32#94011yescharacteristic
314C2×C32#94912yes—
315C2×C32C3×#12711yescharacteristic
316C2×C32C3×AGL(1,ℤ/8)12yes—
317C3⋊D3—82no—
318C3×D3—82no—
319C3×D3—81no—
320C3×D3—81no—
321C3×D3—244no—
322C3×D3—242no—
323C3×D3—242no—
324C2×Q8—31no—
325C2×D4—31no—
326C2×D4—92no—
327C2×D4—182no—
328SD16—184no—
329D8—184no—
330C2×C8—32no—
331C22⋊C4—62no—
332C42C3×D3211yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—122no—
338C22×C3—124no—
339C22×C3—122no—
340C22×C3—122no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3—364no—
344C22×C3—362no—
345C22×C3C2×#62611yescharacteristic
346C22×C3C2×C3×D1211yescharacteristic
347C22×C3C2×C3×#6711yescharacteristic
348C2×D3—41no—
349C2×D3—42no—
350C2×D3—128no—
351C2×D3—124no—
352C2×D3—124no—
353C2×D3—122no—
354C2×D3—124no—
355C2×D3—122no—
356C3×C4—24no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—44no—
366C3×C4—42no—
367C3×C4—122no—
368C3×C4—121no—
369C3×C4—121no—
370C3×C4D3×D1212yes—
371C3×C4C2×C3×D1212yes—
372C3×C4C3×D3×D412yes—
373Dic3—41no—
374C32—21no—
375C32—21no—
376C32—21no—
377C32—41no—
378C32#471211yescharacteristic, complemented
379C32#505811yescharacteristic, complemented
380C32C3×#49611yescharacteristic, complemented
381C23—61no—
382C23—182no—
383Q8—62no—
384D4—62no—
385D4—182no—
386D4—182no—
387D4—364no—
388C2×C4—61no—
389C2×C4C3×#62612yes—
390C2×C4C2×C3×D3211yescharacteristic
391C8—34no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—42no—
399C2×C3—41no—
400C2×C3—242no—
401C2×C3—241no—
402C2×C3—241no—
403C2×C3—722no—
404C2×C3#6382512yes—
405C2×C3#6391111yescharacteristic
406C2×C3C3×#87411yescharacteristic
407C2×C3C3×#90812yes—
408C2×C3C3×#94011yescharacteristic
409C2×C3C3×#94912yes—
410D3—81no—
411D3—244no—
412D3—242no—
413D3—242no—
414C22—122no—
415C22—121no—
416C22—364no—
417C22—362no—
418C22C2×C3×#62611yescharacteristic
419C4—22no—
420C4—121no—
421C4C3×D3×D1212yes—
422C3—21no—
423C3—21no—
424C3—21no—
425C3—41no—
426C3#1059035111yescharacteristic, direct product factor
427C3C3×#471211yescharacteristic, complemented
428C3C3×#505811yescharacteristic, complemented
429C2—241no—
430C2—722no—
431C2C3×#6382512yes—
432C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64083C2semidirect
2. C32×#875C2semidirect
3. C2×C3×#3717C2semidirect
4. C2×C3×#3714C2semidirect
5. C32×#803C2semidirect
8. #10590351C3direct (N × H)
19. #64083C2×C3semidirect
25. C3×#875D3semidirect
26. C3×#875C2×C3semidirect
30. C2×#3717C2×C3semidirect
31. C2×#3714C2×C3semidirect
34. C3×#803D3semidirect
35. C3×#803C2×C3semidirect
46. C32×C3⋊C8D4semidirect
105. #875C3×D3semidirect
112. #803C3×D3semidirect
161. C3×C3⋊C8D12semidirect
162. C3×C3⋊C8C3×D4semidirect
251. C33#496semidirect
301. C3⋊C8C3×D12semidirect
378. C32#4712semidirect
379. C32#5058semidirect
380. C32C3×#496semidirect
426. C3#10590351direct (N × H)
427. C3C3×#4712semidirect
428. C3C3×#5058semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C32×#8752
C2×C3×#37172
C2×C3×#37142
C32×#8032
C3×#50583
C3×#47123
#105903513