Finite Group Databasebeta

← #13181077

Subgroups — #13181077

Order 1728.

914 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#11724618C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11724326C214yesmaximal, complemented
5#11724311C212yesmaximal, complemented
6#11724308C212yesmaximal, complemented
7#11724307C214yesmaximal, complemented
8#10596505—31nomaximal
9C3×#3828—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6—21no—
12#89049—21no—
13#89046C2212yes—
14C4⋊#1113C2211yescharacteristic
15C4×#1113—21no—
16#89042C2212yes—
17#89041—21no—
18#89040C2211yescharacteristic
19C2×C3×#627C2212yescomplemented
20C2×C3×#624—21no—
21C3×#3797—21no—
22C3×#3797—21no—
23C3×#3797C2212yes—
24C3×#3797C2212yescomplemented
25#88993C2212yescomplemented
26#88989C2212yescomplemented
27#88988C2212yes—
28#88938C2218yescomplemented
29#88936C2218yescomplemented
30C2×C3×D3×Q8—61no—
31#64306—31no—
32D3×#235—61no—
33C2×#3797D311yescharacteristic, complemented
34#64283—31no—
35#64239—34no—
36#64228—32no—
37#64227—32no—
38#64226—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×#1113—21no—
41C2×C33×C4D411yescharacteristic
42C3×#638—22no—
43C2×C3×C3⋊Dic3C2311yescharacteristic
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4×C3⋊D3—22no—
46C3×C4×C3⋊D3D411yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C3×C4×C3⋊D3C2311yescharacteristic
49C3×C4×C3⋊D3C2311yescharacteristic
50C3×#634—21no—
51C2×C32×Dic3—22no—
52C32×Dic6—22no—
53C32×Dic6—22no—
54#6296C2312yes—
55#6295—22no—
56#6295C2312yes—
57#6294—21no—
58#6294—22no—
59C3×#627—22no—
60C3×#627C2314yes—
61C3×#624—22no—
62C3×#624—22no—
63C3⋊#622C2314yes—
64C2×#942—91nomaximal
65C2×C32×Q8—61no—
66#3828—21no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C3×D3×Q8—62no—
69C3×D3×Q8—62no—
70C3×D3×Q8—62no—
71C3×D3×Q8—62no—
72C3×#235—61no—
73C3×#235—61no—
74C3×#235—121no—
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—41no—
81C2×C3×Dic6—62no—
82#3811—61no—
83D3×#67—62no—
84C2×#627C2×D312yescomplemented
85C2×#624—21no—
86#3805—32no—
87#3804—62no—
88#3802—31no—
89D3×D12—61no—
90C4×D32—62no—
91C4×D32—61no—
92#3799—61no—
93#3798—32no—
94#3798—62no—
95#3797—21no—
96#3797—21no—
97#3797—31no—
98#3797C2×D312yes—
99#3797C2×D312yescomplemented
100#3796—61no—
101#3795—61no—
102D3×Dic6—61no—
103#3792—32no—
104#3788—32no—
105#3787—32no—
106#3776—38no—
107#3775—38no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C3×C3⋊D3C2×D411yescharacteristic
113#1113—42no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3C2×D411yescharacteristic
118C3×C3⋊Dic3C2411yescharacteristic
119C32×Dic3—24no—
120C22×C3×Q8—91no—
121D3×#41—181no—
122C2×D3×Q8—61no—
123C2×#235—91no—
124C2×#216—92no—
125#942—94no—
126#942—94no—
127C2×#215—92no—
128C2×#208—91no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C2×D32—62no—
133C2×D32—61no—
134C32×Q8—62no—
135C32×Q8—62no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4—41no—
140C2×C32×C4—62no—
141C2×C32×C4#6711yescharacteristic
142#638—22no—
143C2×C3⋊Dic3C22×D311yescharacteristic
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—22no—
146C4×C3⋊D3#6711yescharacteristic
147C4×C3⋊D3#6711yescharacteristic
148C4×C3⋊D3C22×D311yescharacteristic
149C4×C3⋊D3C22×D311yescharacteristic
150#634—21no—
151C3×#67—62no—
152C3×#67—62no—
153C3×#67—122no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—121no—
162C3×C4×D3—61no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×C4×D3—64no—
171C3×C4×D3—64no—
172C3×C4×D3—62no—
173C3×C4×D3—62no—
174C3×C4×D3—122no—
175C3×Dic6—22no—
176C3×Dic6—22no—
177C3×Dic6—22no—
178C3×Dic6—22no—
179C3×Dic6—42no—
180C3×Dic6—42no—
181C3×Dic6—64no—
182C3×Dic6—64no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C3×Dic6—121no—
186#627—22no—
187#627—32no—
188#627—62no—
189#627C22×D314yes—
190#626—62no—
191#626—62no—
192#626—62no—
193#625—32no—
194#625—62no—
195#624—22no—
196#624—22no—
197#624—62no—
198#624—61no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201D3×Dic3—62no—
202#622—34no—
203C2×#137—271noSylow
204C2×C33—21no—
205C2×C33C22×D411yescharacteristic
206C3×C3⋊D3—21no—
207C3×C3⋊D3—42no—
208C3×C3⋊D3#13712yescomplemented
209C3×#41—181no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—92no—
213C2×C3×Q8—92no—
214C2×C3×Q8—92no—
215C2×C3×Q8—121no—
216C2×C3×Q8—181no—
217C2×C3×Q8—181no—
218C22×C3×C4—91no—
219C22×C3×C4—181no—
220C2×#67—181no—
221#239—181no—
222D3×Q8—62no—
223D3×Q8—62no—
224D3×Q8—62no—
225D3×Q8—62no—
226D3×Q8—181no—
227#237—182no—
228#237—181no—
229D3×D4—181no—
230D3×D4—182no—
231#235—61no—
232#235—61no—
233#235—92no—
234#235—92no—
235#235—121no—
236#235—182no—
237#235—181no—
238C2×D12—91no—
239C2×C4×D3—61no—
240C2×C4×D3—61no—
241C2×C4×D3—62no—
242C2×C4×D3—181no—
243C2×C4×D3—182no—
244C2×Dic6—21no—
245C2×Dic6—62no—
246C2×Dic6—91no—
247#216—98no—
248#215—98no—
249#208—92no—
250#208—92no—
251C2×C3⋊C8—92no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32C2×#6711yescharacteristic
257C2×C3⋊D3—22no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3C2×#6711yescharacteristic
260C2×C3⋊D3C2×#6711yescharacteristic
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—64no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270D32—124no—
271D32—122no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—64no—
281C32×C4C2×#6711yescharacteristic
282C32×C4C2×#6711yescharacteristic
283C3⋊Dic3—22no—
284C3⋊Dic3C2×#6711yescharacteristic
285C3⋊Dic3C23×D311yescharacteristic
286C3×Dic3—24no—
287C3×Dic3—24no—
288C3×Dic3—44no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—122no—
296C2×#41—271no—
297C22×Q8—91no—
298#137—274no—
299#137—274no—
300C2×Q16—272no—
301C2×SD16—272no—
302C2×M16—271no—
303C33C2×#13711yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C22×D3—61no—
306C22×D3—61no—
307C22×D3—181no—
308C22×D3—182no—
309C3×Q8—62no—
310C3×Q8—62no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—94no—
314C3×Q8—122no—
315C3×Q8—122no—
316C3×Q8—181no—
317C3×Q8—182no—
318C3×D4—181no—
319C3×D4—182no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—122no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—182no—
335C2×C3×C4GO+(4,𝔽2)11yescharacteristic
336#67—62no—
337#67—62no—
338#67—122no—
339#67—182no—
340#67—182no—
341#67—182no—
342C2×Dic3—22no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—181no—
346C2×Dic3—182no—
347D12—61no—
348D12—61no—
349D12—92no—
350D12—121no—
351D12—182no—
352C4×D3—61no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—64no—
361C4×D3—64no—
362C4×D3—62no—
363C4×D3—62no—
364C4×D3—122no—
365C4×D3—182no—
366C4×D3—182no—
367C4×D3—182no—
368Dic6—22no—
369Dic6—22no—
370Dic6—64no—
371Dic6—64no—
372Dic6—61no—
373Dic6—61no—
374Dic6—92no—
375Dic6—121no—
376Dic6—182no—
377C3⋊C8—94no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—42no—
386C2×C32—81no—
387C2×C32C22×#6711yescharacteristic
388C3⋊D3—21no—
389C3⋊D3—42no—
390C3⋊D3#94212yescomplemented
391C3×D3—62no—
392C3×D3—62no—
393C3×D3—62no—
394C3×D3—121no—
395C3×D3—122no—
396C3×D3—122no—
397C3×D3—242no—
398#41—181no—
399#41—272no—
400#41—272no—
401#41—541no—
402C2×Q8—61no—
403C2×Q8—92no—
404C2×Q8—92no—
405C2×Q8—92no—
406C2×Q8—181no—
407C2×Q8—181no—
408C2×Q8—271no—
409C2×D4—271no—
410C2×D4—541no—
411C22×C4—91no—
412C22×C4—181no—
413C22×C4—541no—
414Q16—278no—
415SD16—278no—
416M16—272no—
417M16—272no—
418C2×C8—272no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—91no—
424C22×C3—91no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3C2×GO+(4,𝔽2)11yescharacteristic
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—64no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—122no—
436C2×D3—122no—
437C2×D3—364no—
438C2×D3—362no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—64no—
448C3×C4—64no—
449C3×C4—91no—
450C3×C4—91no—
451C3×C4—124no—
452C3×C4—182no—
453C3×C4C2×GO+(4,𝔽2)11yescharacteristic
454C3×C4C2×GO+(4,𝔽2)11yescharacteristic
455Dic3—24no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—61no—
460Dic3—62no—
461Dic3—62no—
462Dic3—122no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32—41no—
467C32C2×#94211yescharacteristic, complemented
468C23—91no—
469C23—541no—
470Q8—62no—
471Q8—62no—
472Q8—94no—
473Q8—182no—
474Q8—181no—
475Q8—272no—
476D4—182no—
477D4—181no—
478D4—272no—
479D4—542no—
480C2×C4—62no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—182no—
487C2×C4—182no—
488C2×C4—182no—
489C2×C4—542no—
490C2×C4—541no—
491C2×C4D32⋊D311yescharacteristic
492C8—274no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—81no—
502C2×C3—92no—
503C2×C3—181no—
504C2×C3—362no—
505C2×C3C22×GO+(4,𝔽2)11yescharacteristic
506D3—62no—
507D3—62no—
508D3—62no—
509D3—121no—
510D3—122no—
511D3—122no—
512D3—242no—
513C22—91no—
514C22—91no—
515C22—182no—
516C22—182no—
517C22—1082no—
518C22C2×D32⋊D311yescharacteristic
519C4—64no—
520C4—91no—
521C4—91no—
522C4—182no—
523C4C2×D32⋊D311yescharacteristic
524C4C2×D32⋊D311yescharacteristic
525C3—21no—
526C3—21no—
527C3—41no—
528C3—41no—
529C3#1059650511yescharacteristic, complemented
530C2—21no—
531C2—92no—
532C2—181no—
533C2—362no—
534C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FittingC2×C33×C4D4
socleC2×C33C22×D4
prime radical#11724618C2
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#137
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. #11724511C2semidirect
4. #11724326C2semidirect
5. #11724311C2semidirect
6. #11724308C2semidirect
7. #11724307C2semidirect
19. C2×C3×#627C22semidirect
24. C3×#3797C22semidirect
25. #88993C22semidirect
26. #88989C22semidirect
28. #88938C22semidirect
29. #88936C22semidirect
33. C2×#3797D3semidirect
84. C2×#627C2×D3semidirect
99. #3797C2×D3semidirect
208. C3×C3⋊D3#137semidirect
303. C33C2×#137semidirect
390. C3⋊D3#942semidirect
467. C32C2×#942semidirect
529. C3#10596505semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×#37972
#117245112
#117243262
#117243112
#117243082
#117243072
#105965053
C2×#9429