Finite Group Databasebeta

← #13165529

Subgroups — #13165529

Order 1728.

814 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 58 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×D3×#625C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3C2×#88935C211yesmaximal, characteristic, complemented
4#11722955C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593519—31nomaximal
9C22×C3×D32—21no—
10C2×#6295C2211yescharacteristic
11C2×#6295C2211yescharacteristic, complemented
12C2×#6290—21no—
13C2×D3×C3⋊Dic3C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#625C2211yescharacteristic
16C2×C3⋊#622C2211yescharacteristic, complemented
17C2×C3×#622C2211yescharacteristic, complemented
18D3×#625—22no—
19D3×#625—22no—
20#88941—22no—
21#88935—22no—
22#88803—21no—
23C3×#3724C2211yescharacteristic, complemented
24C2×D3×#67—61no—
25C22×#625—31no—
26C2×#3774—31no—
27C2×#3774—31no—
28#63980—31no—
29#63886—61no—
30C22⋊#622—31no—
31#63739—31no—
32#63739D311yescharacteristic, complemented
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—21no—
35C22×C32×D3—21no—
36C22×C32×D3D411yescharacteristic
37C2×C3×D32—24no—
38C2×C3×D32—42no—
39C2×C3×D32—42no—
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3C2311yescharacteristic
42C2×C32×Dic3—21no—
43#6295—21no—
44#6295—21no—
45#6295—21no—
46#6295D412yescomplemented
47#6290—24no—
48D3×C3⋊Dic3—21no—
49D3×C3⋊Dic3—21no—
50C3×#626—42no—
51C3×#625—21no—
52C3×#625D412yescomplemented
53C3⋊#622—21no—
54C3×#622—21no—
55#4773—91nomaximal
56C23×C3×D3—61no—
57C23×C3×D3—61no—
58C22×D32—21no—
59C2×#638—61no—
60C22×C3⋊Dic3—31no—
61C2×C3×#67—61no—
62C2×C3×#67—61no—
63C2×C3×D12—61no—
64D3×#67—64no—
65D3×#67—64no—
66C2×#626—21no—
67C2×#626—61no—
68C2×#626—61no—
69C2×#625—31no—
70C2×#625—31no—
71C2×#625—62no—
72C2×#625—61no—
73C2×#625—61no—
74C2×#625—62no—
75C2×#625—121no—
76C2×#625C2×D311yescharacteristic
77C2×D3×Dic3—61no—
78C2×D3×Dic3—61no—
79C2×#622—31no—
80C2×#622C2×D311yescharacteristic, complemented
81#3774—62no—
82#3774—62no—
83#3752—61no—
84C3×Dic3⋊C4—61no—
85#3724C2×D311yescharacteristic, complemented
86(C2×D3)⋊Dic3—61no—
87(C2×D3)⋊Dic3—61no—
88(C2×D3)⋊Dic3—121no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C32×D3—21no—
97C2×C32×D3—21no—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C2×C32×D3D812yescomplemented
102C3×D32—48no—
103C3⋊(C3⋊Dic3)—21no—
104C3×C3⋊Dic3C2×D411yescharacteristic
105C3×C3⋊Dic3C2×D411yescharacteristic
106C32×Dic3—41no—
107C22×#67—181no—
108C2×D3×D4—91no—
109#939—181no—
110C2×#213—91no—
111C2×D24—91no—
112#895—91no—
113C3×#102—91no—
114(C2×D3)⋊C8—91no—
115#809—91no—
116C23×C32—31no—
117C23×C32—61no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—21no—
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—41no—
125C22×C3×D3—64no—
126C22×C3×D3—64no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—121no—
132C22×C3×D3—121no—
133C22×C3×D3—122no—
134C22×C3×D3—122no—
135C22×C3×D3—122no—
136C22×C3×D3—122no—
137C2×D32—24no—
138C2×D32—42no—
139C2×D32—42no—
140C2×C32×C4—61no—
141#638—64no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—121no—
148C2×C3⋊Dic3—121no—
149C2×C3⋊Dic3C22×D311yescharacteristic
150C3×#67—62no—
151C3×#67—62no—
152C3×#67—122no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C3×D12—122no—
159#626—42no—
160#626—62no—
161#626—62no—
162#626—121no—
163#626—121no—
164#625—21no—
165#625—32no—
166#625—64no—
167#625—64no—
168#625—61no—
169#625—61no—
170#625—61no—
171#625—124no—
172#625D1212yescomplemented
173D3×Dic3—62no—
174D3×Dic3—62no—
175D3×Dic3—121no—
176D3×Dic3—121no—
177#622—21no—
178#622—61no—
179#447—271noSylow
180C2×C33#12011yescharacteristic
181C2×C33C2×D811yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C3×C3⋊D3—42no—
184C3×C3⋊D3—81no—
185C32×D3—22no—
186C32×D3—42no—
187C24×C3—181no—
188C23×D3—61no—
189C23×D3—61no—
190C2×C3×D4—91no—
191C2×C3×D4—181no—
192C2×#67—61no—
193C2×#67—61no—
194C2×#67—181no—
195C2×#67—181no—
196C2×#67—181no—
197C2×#67—184no—
198C2×#67—184no—
199C2×#67—181no—
200C2×#67—181no—
201C2×#67—361no—
202C22×Dic3—181no—
203C22×Dic3—181no—
204D3×D4—182no—
205D3×D4—182no—
206C2×D12—61no—
207C2×D12—91no—
208C2×D12—91no—
209C2×C4×D3—91no—
210C2×C3×C8—91no—
211C3×C4⋊C4—91no—
212#217—181no—
213#213—182no—
214(C2×D3)⋊C4—181no—
215(C2×D3)⋊C4—181no—
216(C2×D3)⋊C4—361no—
217Dic3⋊C4—61no—
218C2×C3⋊C8—91no—
219D24—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—32no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—62no—
226C22×C32—61no—
227C22×C32—61no—
228C22×C32—121no—
229C22×C32—121no—
230C22×C32D3×D411yescharacteristic
231C2×C3⋊D3—22no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3×D3—22no—
238C2×C3×D3—22no—
239C2×C3×D3—22no—
240C2×C3×D3—22no—
241C2×C3×D3—22no—
242C2×C3×D3—22no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—42no—
246C2×C3×D3—41no—
247C2×C3×D3—41no—
248C2×C3×D3—41no—
249C2×C3×D3—41no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—81no—
253C2×C3×D3—81no—
254C2×C3×D3—122no—
255C2×C3×D3—128no—
256C2×C3×D3—128no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—122no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—241no—
269C2×C3×D3—241no—
270C2×C3×D3—241no—
271C2×C3×D3—241no—
272D32—48no—
273C32×C4—121no—
274C3⋊Dic3—62no—
275C3⋊Dic3—61no—
276C3⋊Dic3—121no—
277C3⋊Dic3—122no—
278C3⋊Dic3—241no—
279C3⋊Dic3C2×D1211yescharacteristic
280C3⋊Dic3D3×D411yescharacteristic
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C3×Dic3—81no—
288C22×D4—271no—
289C2×D8—271no—
290C2×D8—271no—
291#121—271no—
292#102—91no—
293#102—271no—
294C22⋊C8—271no—
295C33#44711yescharacteristic, complemented, Sylow
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—61no—
299C23×C3—61no—
300C23×C3—121no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—184no—
304C23×C3—362no—
305C23×C3—362no—
306C22×D3—21no—
307C22×D3—64no—
308C22×D3—64no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—121no—
314C22×D3—121no—
315C22×D3—122no—
316C22×D3—122no—
317C22×D3—122no—
318C22×D3—122no—
319C22×D3GO+(4,𝔽2)11yescharacteristic
320C3×D4—92no—
321C3×D4—181no—
322C3×D4—362no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—91no—
326C2×C3×C4—121no—
327#67—62no—
328#67—62no—
329#67—122no—
330#67—182no—
331#67—182no—
332#67—184no—
333#67—184no—
334#67—184no—
335#67—361no—
336#67—361no—
337#67—364no—
338C2×Dic3—21no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—91no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—182no—
345C2×Dic3—182no—
346C2×Dic3—361no—
347C2×Dic3—361no—
348C2×Dic3—361no—
349C2×Dic3—361no—
350D12—92no—
351D12—122no—
352D12—181no—
353D12—181no—
354D12—181no—
355C4×D3—181no—
356C4×D3—181no—
357C3×C8—181no—
358C3⋊C8—181no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—62no—
372C2×C32—122no—
373C2×C32#88211yescharacteristic
374C2×C32D3×D811yescharacteristic
375C2×C32#91411yescharacteristic
376C3⋊D3—42no—
377C3⋊D3—81no—
378C3×D3—24no—
379C3×D3—42no—
380C3×D3—42no—
381C3×D3—42no—
382C3×D3—82no—
383C3×D3—122no—
384C3×D3—122no—
385C3×D3—242no—
386C3×D3—241no—
387C3×D3—242no—
388C3×D3—241no—
389C24—181no—
390C2×D4—91no—
391C2×D4—181no—
392C2×D4—271no—
393C2×D4—271no—
394C2×D4—541no—
395C2×D4—542no—
396C2×D4—542no—
397C22×C4—271no—
398D8—542no—
399D8—542no—
400C2×C8—91no—
401C2×C8—271no—
402C4⋊C4—91no—
403C22⋊C4—541no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—122no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3—182no—
422C22×C3—241no—
423C22×C3—241no—
424C22×C3—361no—
425C22×C3—361no—
426C22×C3—361no—
427C22×C3—361no—
428C22×C3—361no—
429C22×C3—368no—
430C22×C3C2×GO+(4,𝔽2)11yescharacteristic
431C2×D3—21no—
432C2×D3—21no—
433C2×D3—22no—
434C2×D3—41no—
435C2×D3—41no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—122no—
439C2×D3—128no—
440C2×D3—128no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—122no—
449C2×D3—121no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—241no—
453C2×D3—241no—
454C2×D3—241no—
455C2×D3—241no—
456C2×D3#377412yescomplemented
457C3×C4—91no—
458C3×C4—91no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—241no—
462Dic3—41no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466Dic3—61no—
467Dic3—182no—
468Dic3—181no—
469Dic3—361no—
470Dic3—362no—
471Dic3—721no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32—41no—
476C32#477311yescharacteristic, complemented
477C23—31no—
478C23—61no—
479C23—184no—
480C23—181no—
481C23—181no—
482C23—362no—
483C23—362no—
484D4—92no—
485D4—181no—
486D4—272no—
487D4—362no—
488D4—541no—
489D4—541no—
490D4—541no—
491D4—544no—
492C2×C4—61no—
493C2×C4—91no—
494C2×C4—271no—
495C2×C4—541no—
496C2×C4—541no—
497C8—181no—
498C8—541no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—64no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—122no—
515C2×C3—242no—
516C2×C3—362no—
517C2×C3—721no—
518C2×C3C2×#377411yescharacteristic
519C2×C3#6423611yescharacteristic
520C2×C3#6424111yescharacteristic
521D3—22no—
522D3—42no—
523D3—122no—
524D3—122no—
525D3—242no—
526D3—241no—
527D3—242no—
528D3—241no—
529C22—32no—
530C22—61no—
531C22—61no—
532C22—62no—
533C22—121no—
534C22—121no—
535C22—182no—
536C22—368no—
537C22—361no—
538C22—361no—
539C22—361no—
540C22—361no—
541C22—361no—
542C22D3×GO+(4,𝔽2)11yescharacteristic
543C4—91no—
544C4—91no—
545C4—121no—
546C4—541no—
547C3—21no—
548C3—21no—
549C3—41no—
550C3—41no—
551C3#1059351911yescharacteristic, complemented
552C2—62no—
553C2—122no—
554C2—362no—
555C2—721no—
556C2#1172411311yescharacteristic
557C2D3×#377411yescharacteristic
558C2#1172413911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#447
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×D3×#625C2semidirect
2. C2×#88941C2semidirect
3. C2×#88935C2semidirect
4. #11722955C2semidirect
5. #11722772C2semidirect
6. C3×#63739C2semidirect
7. (C2×D3)⋊#622C2semidirect
11. C2×#6295C22semidirect
16. C2×C3⋊#622C22semidirect
17. C2×C3×#622C22semidirect
23. C3×#3724C22semidirect
32. #63739D3semidirect
46. #6295D4semidirect
52. C3×#625D4semidirect
80. C2×#622C2×D3semidirect
85. #3724C2×D3semidirect
101. C2×C32×D3D8semidirect
172. #625D12semidirect
295. C33#447semidirect
456. C2×D3#3774semidirect
476. C32#4773semidirect
551. C3#10593519semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6252
C2×#889412
C2×#889352
#117229552
#117227722
C3×#637392
(C2×D3)⋊#6222
#105935193
#47739