Finite Group Databasebeta

← #13155432

Subgroups — #13155432

Order 1728.

810 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 122 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×#64306C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic, complemented
3#11723437C211yesmaximal, characteristic, complemented
4#11722043C212yesmaximal, complemented
5C3×#63569C212yesmaximal, complemented
6#10591670—32nomaximal
7#10590394—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235C2212yes—
10C3×#3811—21no—
11C3×#3805—21no—
12C3×#3802C411yescharacteristic, complemented
13C3×C4×D32—21no—
14C3×#3798—21no—
15C3×#3797C411yescharacteristic, complemented
16C3×#3796C412yescomplemented
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C3×#3748C2212yescomplemented
19C32×C4×Dic3C2211yescharacteristic, complemented
20#88809—21no—
21#88808—21no—
22#88807—21no—
23#88803—21no—
24C3×D3×#41—32no—
25#64306D311yescharacteristic
26#64081—31no—
27#63855—32no—
28#63855—61no—
29#63818—31no—
30D3×#208—32no—
31#63664—32no—
32C3×#837—32no—
33C3×#805—32no—
34#63569D312yescomplemented
35#63568—32no—
36#63568—32no—
37#63568—62no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×C3×D32—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3C2×C411yescharacteristic
44C3×C4×C3⋊D3C2×C411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3D411yescharacteristic
47C32×#67—22no—
48C2×C32×Dic3—21no—
49C32×D12C2×C412yes—
50C32×C4×D3—22no—
51C32×Dic6C2×C412yes—
52C3×#627—21no—
53C3×#626—22no—
54C3×#625—21no—
55C3×#624—21no—
56C3×D3×Dic3—22no—
57C3×C3⋊(C3⋊C8)—22no—
58#5236—91no—
59#4821—92no—
60C32×#41—32no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C3×#239—32no—
63C3×D3×Q8—32no—
64C3×#237—32no—
65C3×#237—62no—
66C3×D3×D4—32no—
67C3×D3×D4—62no—
68C3×#235—22no—
69C3×#235—62no—
70C3×#235C2×D312yes—
71C3×#235C2×D312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—62no—
74C2×C3×C4×D3—61no—
75#3811—21no—
76#3805—21no—
77#3802Dic311yescharacteristic, complemented
78C4×D32—21no—
79#3798—21no—
80#3797Dic311yescharacteristic, complemented
81#3796Dic312yescomplemented
82C32×C42—31no—
83#3752—61no—
84#3750—61no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—32no—
87C4×C3⋊Dic3—61no—
88C4×C3⋊Dic3—62no—
89C4×C3⋊Dic3—121no—
90#3748C2×D312yescomplemented
91C3×C4×Dic3—21no—
92C3×C4×Dic3C2×D312yescomplemented
93C3×#208—32no—
94C3×#208—32no—
95C3×#208—62no—
96Dic3⋊Dic3—62no—
97Dic3⋊Dic3—62no—
98(C2×D3)⋊Dic3—62no—
99(C2×D3)⋊Dic3—62no—
100#3712—61no—
101#3711—62no—
102#3710—61no—
103D3×C3⋊C8—62no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3C22⋊C411yescharacteristic
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C32×D3—22no—
109C3×D32—42no—
110C33×C4C22×C411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3⋊(C3⋊Dic3)—41no—
113C3×C3⋊Dic3C22⋊C411yescharacteristic
114C3×C3⋊Dic3C22⋊C411yescharacteristic
115C32×Dic3—22no—
116C32×Dic3—41no—
117C2×C3×#41—91no—
118D3×#41—32no—
119#926—91no—
120C2×#208—91no—
121D3×M16—92no—
122#872—92no—
123#872—181no—
124C3×#104—92no—
125#837—32no—
126#837—92no—
127#837—92no—
128#837—182no—
129#825—92no—
130#805—32no—
131C22×C3⋊D3#6711yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—62no—
135C2×D32—21no—
136C32×Q8—32no—
137C32×D4—32no—
138C32×D4—62no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—61no—
143C2×C32×C4—62no—
144C2×C32×C4C22×D312yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—122no—
150C2×C3⋊Dic3—122no—
151C2×C3⋊Dic3#6711yescharacteristic
152C4×C3⋊D3C2×Dic311yescharacteristic
153C4×C3⋊D3C2×Dic311yescharacteristic
154C4×C3⋊D3#6711yescharacteristic
155C4×C3⋊D3#6711yescharacteristic
156C3×#67—22no—
157C3×#67—22no—
158C3×#67—42no—
159C3×#67—62no—
160C3×#67—62no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—62no—
166C3×D12—22no—
167C3×D12—62no—
168C3×D12C4×D312yes—
169C3×D12C2×Dic312yes—
170C3×C4×D3—22no—
171C3×C4×D3—22no—
172C3×C4×D3—32no—
173C3×C4×D3—32no—
174C3×C4×D3—32no—
175C3×C4×D3—32no—
176C3×C4×D3—42no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—22no—
184C3×Dic6—62no—
185C3×Dic6C4×D312yes—
186C3×Dic6C2×Dic312yes—
187#627—21no—
188#626—22no—
189#625—21no—
190#624—21no—
191D3×Dic3—22no—
192C3⋊(C3⋊C8)—22no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—62no—
195C3×C3⋊C8—122no—
196#421—271noSylow
197C2×C33—21no—
198C2×C33C2×C22⋊C411yescharacteristic
199C3×C3⋊D3—21no—
200C3×C3⋊D3—21no—
201C32×D3—42no—
202C3×#41—32no—
203C3×#41—32no—
204C3×#41—62no—
205C3×#41—92no—
206C3×#41—181no—
207C3×#41—181no—
208C2×C3×Q8—91no—
209C2×C3×D4—91no—
210C2×C3×D4—181no—
211C22×C3×C4—91no—
212C22×C3×C4—181no—
213#239—32no—
214D3×Q8—32no—
215#237—32no—
216#237—62no—
217D3×D4—32no—
218D3×D4—62no—
219#235—62no—
220#235D3212yes—
221C2×C4×D3—32no—
222C2×C4×D3—61no—
223C2×C4×D3—62no—
224C3×M16—92no—
225C3×C42—32no—
226C3×C42—61no—
227#217—181no—
228(C2×D3)⋊C4—182no—
229(C2×D3)⋊C4—182no—
230C4⋊Dic3—181no—
231Dic3⋊C4—182no—
232Dic3⋊C4—182no—
233C4×Dic3—92no—
234C4×Dic3—91no—
235C4×Dic3—181no—
236C4×Dic3—182no—
237C4×Dic3—361no—
238C4×Dic3D3211yescharacteristic, complemented
239#208—32no—
240#208—32no—
241#208—62no—
242#208—181no—
243C2×C3⋊C8—181no—
244ΓL(1,𝔽25)—182no—
245D3×C8—182no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—41no—
249C22×C32—62no—
250C22×C32D3×D412yes—
251C22×C32C2×#6711yescharacteristic
252C2×C3⋊D3—41no—
253C2×C3⋊D3#21711yescharacteristic
254C2×C3⋊D3#21711yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—22no—
257C2×C3×D3—32no—
258C2×C3×D3—32no—
259C2×C3×D3—42no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—124no—
263C2×C3×D3—122no—
264C2×C3×D3—122no—
265D32—42no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—62no—
273C32×C4—121no—
274C32×C4C2×C4×D312yes—
275C32×C4D3×D412yes—
276C32×C4C22×Dic311yescharacteristic
277C32×C4C2×#6711yescharacteristic
278C3⋊Dic3—121no—
279C3⋊Dic3—122no—
280C3⋊Dic3—122no—
281C3⋊Dic3—124no—
282C3⋊Dic3—124no—
283C3⋊Dic3#21711yescharacteristic
284C3⋊Dic3#21711yescharacteristic
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—32no—
288C3×Dic3—32no—
289C3×Dic3—42no—
290C3×Dic3—42no—
291C3×Dic3—42no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C2×#41—91no—
295C2×M16—271no—
296#117—271no—
297#104—92no—
298#104—272no—
299C33#42111yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C23×C3—181no—
302C22×D3—32no—
303C22×D3—61no—
304C22×D3—62no—
305C3×Q8—32no—
306C3×Q8—32no—
307C3×Q8—62no—
308C3×Q8—181no—
309C3×D4—32no—
310C3×D4—32no—
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—122no—
315C3×D4—182no—
316C3×D4—181no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—41no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—122no—
330C2×C3×C4—181no—
331C2×C3×C4—182no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3212yes—
334#67—22no—
335#67—62no—
336#67—62no—
337C2×Dic3—21no—
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—62no—
341C2×Dic3—182no—
342C2×Dic3—182no—
343C2×Dic3—181no—
344C2×Dic3—362no—
345C2×Dic3—362no—
346D12—62no—
347D12D3×Dic312yes—
348C4×D3—22no—
349C4×D3—32no—
350C4×D3—32no—
351C4×D3—32no—
352C4×D3—32no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—62no—
358C4×D3—62no—
359Dic6—62no—
360Dic6D3×Dic312yes—
361C3×C8—182no—
362C3⋊C8—62no—
363C3⋊C8—62no—
364C3⋊C8—122no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—22no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—122no—
374C2×C32D3×C22⋊C412yes—
375C2×C32C2×#21711yescharacteristic
376C3⋊D3—21no—
377C3⋊D3—21no—
378C3×D3—42no—
379C3×D3—42no—
380C3×D3—62no—
381C3×D3—62no—
382C3×D3—82no—
383C3×D3—121no—
384C3×D3—121no—
385#41—32no—
386#41—92no—
387#41—181no—
388#41—181no—
389C2×Q8—91no—
390C2×D4—91no—
391C2×D4—181no—
392C22×C4—91no—
393C22×C4—181no—
394M16—92no—
395M16—541no—
396C2×C8—541no—
397C4⋊C4—541no—
398C22⋊C4—541no—
399C42—31no—
400C42—271no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—91no—
407C22×C3—91no—
408C22×C3—122no—
409C22×C3—361no—
410C22×C3—362no—
411C22×C3D3×#6712yes—
412C22×C3#381111yescharacteristic
413C2×D3—22no—
414C2×D3—32no—
415C2×D3—32no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—124no—
419C2×D3—122no—
420C2×D3—122no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—91no—
430C3×C4—91no—
431C3×C4—122no—
432C3×C4—122no—
433C3×C4—122no—
434C3×C4C4×D3211yescharacteristic
435C3×C4#380211yescharacteristic
436C3×C4C2×D3×Dic312yes—
437C3×C4D3×#6712yes—
438Dic3—22no—
439Dic3—32no—
440Dic3—32no—
441Dic3—41no—
442Dic3—61no—
443Dic3—61no—
444Dic3—362no—
445Dic3—362no—
446Dic3—361no—
447Dic3—364no—
448Dic3—364no—
449C32—22no—
450C32—21no—
451C32—41no—
452C32#482112yescomplemented
453C32#523611yescharacteristic, complemented
454C23—91no—
455C23—181no—
456Q8—32no—
457Q8—181no—
458D4—32no—
459D4—62no—
460D4—182no—
461D4—181no—
462C2×C4—62no—
463C2×C4—61no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4—181no—
470C2×C4—182no—
471C2×C4—541no—
472C2×C4D3311yescharacteristic
473C8—182no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—41no—
482C2×C3—122no—
483C2×C3—122no—
484C2×C3—181no—
485C2×C3—181no—
486C2×C3—242no—
487C2×C3#6392111yescharacteristic
488C2×C3D3×#21712yes—
489D3—42no—
490D3—62no—
491D3—62no—
492D3—121no—
493D3—121no—
494C22—62no—
495C22—91no—
496C22—91no—
497C22—361no—
498C22—362no—
499C22#8895211yescharacteristic
500C4—62no—
501C4—91no—
502C4—91no—
503C4—121no—
504C4—1081no—
505C4D32×Dic311yescharacteristic
506C4#8895311yescharacteristic
507C3—21no—
508C3—22no—
509C3—41no—
510C3#1059039411yescharacteristic, complemented
511C3#1059167012yescomplemented
512C2—21no—
513C2—122no—
514C2—181no—
515C2—181no—
516C2#1172290211yescharacteristic
subgroupG/H
centerC4#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723656C2semidirect
3. #11723437C2semidirect
4. #11722043C2semidirect
5. C3×#63569C2semidirect
12. C3×#3802C4semidirect
15. C3×#3797C4semidirect
16. C3×#3796C4semidirect
17. C4×C3⋊(C3⋊Dic3)C22semidirect
18. C3×#3748C22semidirect
19. C32×C4×Dic3C22semidirect
34. #63569D3semidirect
77. #3802Dic3semidirect
80. #3797Dic3semidirect
81. #3796Dic3semidirect
90. #3748C2×D3semidirect
92. C3×C4×Dic3C2×D3semidirect
238. C4×Dic3D32semidirect
299. C33#421semidirect
452. C32#4821semidirect
453. C32#5236semidirect
510. C3#10590394semidirect
511. C3#10591670semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117236562
#117234372
#117220432
C3×#635692
#105916703
#105903943