Finite Group Databasebeta

← #13161538

Subgroups — #13161538

Order 1728.

1060 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 91 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#11724633C211yesmaximal, characteristic, complemented
2C3×#64091C211yesmaximal, characteristic, complemented
3#11723727C212yesmaximal, complemented
4C3×#63633C212yesmaximal, complemented
5#11722083C211yesmaximal, characteristic, complemented
6#10590410—31nomaximal
7#10590407—32nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic, complemented
9C2×C3×#638C2211yescharacteristic
10C2×C3×#638C2212yes—
11C2×C3×C4⋊(C3⋊D3)—21no—
12#89049—22no—
13#89049—21no—
14C2×#6295—22no—
15#89046—22no—
16#89046—21no—
17C33×C22⋊C4C2211yescharacteristic, complemented
18C3×#3757C2212yescomplemented
19C3×#3752—21no—
20#88814—22no—
21#64330—32no—
22#64330—31no—
23#64091D311yescharacteristic, complemented
24C3×#882—32no—
25C3×#882—61no—
26#63981—32no—
27#63977—32no—
28#63976—32no—
29#63633D312yescomplemented
30C3×#806—32no—
31C3×#806—32no—
32C3×#806—62no—
33#63581—31no—
34#63580—32no—
35C23×C33C2311yescharacteristic
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3—22no—
38C22×C3×C3⋊D3—42no—
39C22×C3×C3⋊D3D412yescomplemented
40C22×C3×C3⋊D3D411yescharacteristic, complemented
41C2×#1113—21no—
42C2×C33×C4—21no—
43C3×#638—22no—
44C3×#638—22no—
45C3×#638—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3D412yescomplemented
48C2×C3×C3⋊Dic3D411yescharacteristic, complemented
49C3×C4⋊(C3⋊D3)—42no—
50#6296—21no—
51#6295—21no—
52#6295—22no—
53#6295—44no—
54#6294—22no—
55#6294—21no—
56#5032—91no—
57#4741—92no—
58C23×C3⋊D3C2×D311yescharacteristic, complemented
59C23×C3×D3—32no—
60C23×C3×D3—61no—
61C2×#638C2×D312yes—
62C2×#638C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)—21no—
64C2×C3×#67—32no—
65C2×C3×#67—32no—
66C2×C3×#67—32no—
67C2×C3×#67—61no—
68C2×C3×#67—62no—
69C2×C3×D12—62no—
70C2×C3×D12—62no—
71#3811—62no—
72#3811—61no—
73D3×#67—62no—
74D3×#67—62no—
75D3×#67—62no—
76C2×#626—62no—
77C2×#626—62no—
78C2×#625—62no—
79#3805—62no—
80#3805—61no—
81#3804—62no—
82#3804—62no—
83#3804—62no—
84C32×C22⋊C4—21no—
85C32×C22⋊C4—22no—
86C32×C22⋊C4—41no—
87C32×C22⋊C4C2×D312yescomplemented
88C32×C22⋊C4C2×D311yescharacteristic, complemented
89#3757C2×D312yescomplemented
90#3752—21no—
91C3×#217—32no—
92C3×#217—32no—
93C3×#217—62no—
94C3×(C2×D3)⋊C4—62no—
95C3×(C2×D3)⋊C4—62no—
96#3722—62no—
97(C2×D3)⋊Dic3—62no—
98(C2×D3)⋊Dic3—62no—
99C22×C33—41no—
100C22×C33C2×D412yes—
101C22×C33C2×D411yescharacteristic
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—42no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—42no—
111#1113—81no—
112C33×C4—41no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—22no—
115C3×C3⋊Dic3—42no—
116#1004—92no—
117#1004—91no—
118C3×#120—91no—
119#953—92no—
120#937—92no—
121#882—32no—
122#882—61no—
123#882—92no—
124C3×#99—92no—
125#834—91no—
126#806—32no—
127#806—32no—
128#806—62no—
129#806—92no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—41no—
133C23×C32C22×D312yes—
134C23×C32C22×D311yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3D1212yescomplemented
139C22×C3⋊D3#6711yescharacteristic, complemented
140C22×C3×D3—32no—
141C22×C3×D3—32no—
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—122no—
150C22×C3×D3—124no—
151C22×C3×D3—124no—
152C2×D32—62no—
153C2×D32—61no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—42no—
158C2×C32×C4—42no—
159#638—22no—
160#638—22no—
161#638—22no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3D1212yescomplemented
164C2×C3⋊Dic3#6711yescharacteristic, complemented
165C4⋊(C3⋊D3)—42no—
166C3×#67—62no—
167C3×#67—62no—
168C3×#67—62no—
169C3×#67—62no—
170C3×#67—62no—
171C3×#67—62no—
172C3×#67—122no—
173C3×#67—122no—
174C3×#67—122no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—122no—
183C3×D12—124no—
184C3×D12—124no—
185#627—62no—
186#627—61no—
187#626—62no—
188#626—62no—
189#626—62no—
190#626—124no—
191#626—124no—
192#625—62no—
193#625—61no—
194#625—124no—
195#624—62no—
196#624—61no—
197D3×Dic3—62no—
198D3×Dic3—62no—
199D3×Dic3—62no—
200B(4,𝔽2)—271noSylow
201C2×C33—21no—
202C2×C33—22no—
203C2×C33#12011yescharacteristic
204C3×C3⋊D3—42no—
205C3×C3⋊D3—41no—
206C3×C3⋊D3—42no—
207C24×C3—91no—
208C23×D3—32no—
209C23×D3—61no—
210C2×C3×D4—91no—
211C2×C3×D4—92no—
212C2×C3×D4—181no—
213C2×#67—32no—
214C2×#67—32no—
215C2×#67—32no—
216C2×#67—62no—
217C2×#67—61no—
218C2×#67—182no—
219C2×#67—182no—
220C2×#67—182no—
221C2×#67—181no—
222#237—182no—
223#237—182no—
224#237—182no—
225D3×D4—182no—
226D3×D4—182no—
227D3×D4—182no—
228#235—182no—
229#235—181no—
230C2×D12—62no—
231C2×D12—62no—
232C2×D12—182no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—181no—
238C3×C22⋊C4D3211yescharacteristic, complemented
239C3×C22⋊C4D3212yescomplemented
240#217—32no—
241#217—32no—
242#217—62no—
243#217—182no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—62no—
246(C2×D3)⋊C4—182no—
247(C2×D3)⋊C4—182no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—42no—
257C22×C32—42no—
258C22×C32—82no—
259C22×C32—82no—
260C22×C32C2×D1212yes—
261C22×C32D3×D412yes—
262C22×C32D3×D412yes—
263C22×C32D3×D412yes—
264C22×C32C2×#6711yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—24no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—42no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—42no—
274C2×C3×D3—64no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—124no—
282C2×C3×D3—122no—
283C2×C3×D3—121no—
284C2×C3×D3—122no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—124no—
290C2×C3×D3—122no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—242no—
296D32—242no—
297D32—241no—
298C32×C4—41no—
299C32×C4—42no—
300C32×C4—42no—
301C32×C4—82no—
302C32×C4—82no—
303C3⋊Dic3—22no—
304C3⋊Dic3—21no—
305C3⋊Dic3—42no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—122no—
312C3×Dic3—122no—
313C3×Dic3—242no—
314H(5,𝔽2)—271no—
315#120—91no—
316#120—272no—
317#99—92no—
318#99—271no—
319C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—41no—
323C23×C3—92no—
324C23×C3—91no—
325C23×C3—182no—
326C23×C3—181no—
327C23×C3—362no—
328C23×C3C2×D3211yescharacteristic
329C23×C3C2×D3212yes—
330C22×D3—32no—
331C22×D3—32no—
332C22×D3—32no—
333C22×D3—61no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—62no—
338C22×D3—62no—
339C22×D3—122no—
340C22×D3—124no—
341C22×D3—124no—
342C22×D3—182no—
343C22×D3—181no—
344C3×D4—182no—
345C3×D4—182no—
346C3×D4—182no—
347C3×D4—362no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—92no—
354C2×C3×C4—91no—
355C2×C3×C4—182no—
356#67—62no—
357#67—62no—
358#67—62no—
359#67—62no—
360#67—62no—
361#67—62no—
362#67—122no—
363#67—122no—
364#67—122no—
365#67—182no—
366#67—182no—
367#67—182no—
368#67—364no—
369#67—364no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—62no—
374C2×Dic3—61no—
375C2×Dic3—62no—
376C2×Dic3—62no—
377C2×Dic3—122no—
378C2×Dic3—182no—
379C2×Dic3—181no—
380D12—124no—
381D12—124no—
382D12—182no—
383D12—181no—
384D12—364no—
385C4×D3—182no—
386C4×D3—182no—
387C4×D3—182no—
388Dic6—182no—
389Dic6—181no—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—41no—
399C2×C32—42no—
400C2×C32—42no—
401C2×C32—42no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—82no—
405C2×C32#88212yes—
406C2×C32#93711yescharacteristic
407C3⋊D3—42no—
408C3⋊D3—42no—
409C3⋊D3—41no—
410C3×D3—124no—
411C3×D3—122no—
412C3×D3—122no—
413C3×D3—124no—
414C3×D3—122no—
415C3×D3—241no—
416C3×D3—242no—
417C24—91no—
418#41—542no—
419#41—541no—
420C2×D4—92no—
421C2×D4—91no—
422C2×D4—181no—
423C2×D4—542no—
424C2×D4—542no—
425C2×D4—541no—
426C22⋊C4—92no—
427C22⋊C4—181no—
428C22⋊C4—542no—
429C22⋊C4#111311yescharacteristic, complemented
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—41no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—184no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—181no—
446C22×C3—362no—
447C22×C3—361no—
448C22×C3—362no—
449C22×C3—361no—
450C22×C3—361no—
451C22×C3D3×D1212yes—
452C22×C3D3×D1212yes—
453C22×C3#380212yes—
454C22×C3D3×#6712yes—
455C22×C3#381111yescharacteristic
456C2×D3—64no—
457C2×D3—64no—
458C2×D3—64no—
459C2×D3—64no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—124no—
464C2×D3—121no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—122no—
471C2×D3—124no—
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—122no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—242no—
478C2×D3—722no—
479C2×D3—721no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—41no—
483C3×C4—82no—
484C3×C4—82no—
485C3×C4—182no—
486C3×C4—181no—
487C3×C4—362no—
488Dic3—62no—
489Dic3—62no—
490Dic3—62no—
491Dic3—121no—
492Dic3—122no—
493Dic3—122no—
494Dic3—122no—
495Dic3—242no—
496C32—22no—
497C32—21no—
498C32—41no—
499C32#474112yescomplemented
500C32#503211yescharacteristic, complemented
501C23—92no—
502C23—91no—
503C23—182no—
504C23—181no—
505C23—362no—
506C23—541no—
507C23C2×#111311yescharacteristic
508Q8—541no—
509D4—182no—
510D4—182no—
511D4—182no—
512D4—362no—
513D4—542no—
514D4—541no—
515D4—1084no—
516C2×C4—21no—
517C2×C4—92no—
518C2×C4—91no—
519C2×C4—182no—
520C2×C4—541no—
521C2×C4—542no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—22no—
530C2×C3—42no—
531C2×C3—42no—
532C2×C3—42no—
533C2×C3—42no—
534C2×C3—42no—
535C2×C3—41no—
536C2×C3—82no—
537C2×C3—362no—
538C2×C3—361no—
539C2×C3—362no—
540C2×C3#6392212yes—
541C2×C3#6392311yescharacteristic
542D3—124no—
543D3—124no—
544D3—122no—
545D3—122no—
546D3—122no—
547D3—241no—
548D3—242no—
549C22—41no—
550C22—182no—
551C22—184no—
552C22—181no—
553C22—182no—
554C22—361no—
555C22—362no—
556C22—361no—
557C22—362no—
558C22—361no—
559C22—2161no—
560C22C4⋊#111312yes—
561C22#8904911yescharacteristic
562C4—41no—
563C4—182no—
564C4—181no—
565C4—362no—
566C3—21no—
567C3—22no—
568C3—41no—
569C3#1059040712yescomplemented
570C3#1059041011yescharacteristic, complemented
571C2—21no—
572C2—22no—
573C2—362no—
574C2—361no—
575C2—362no—
576C2#1172370111yescharacteristic
subgroupG/H
centerC2#11723701
commutator, cosocleC23×C33C23
FrattiniC23C2×#1113
FittingC33×C22⋊C4C22
socleC2×C33#120
hypercenterC22⋊C4#1113
hypocenterC33B(4,𝔽2)
LCSG ▷ C23×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11724633C2semidirect
2. C3×#64091C2semidirect
3. #11723727C2semidirect
4. C3×#63633C2semidirect
5. #11722083C2semidirect
8. C23×C3×C3⋊D3C22semidirect
17. C33×C22⋊C4C22semidirect
18. C3×#3757C22semidirect
23. #64091D3semidirect
29. #63633D3semidirect
39. C22×C3×C3⋊D3D4semidirect
40. C22×C3×C3⋊D3D4semidirect
47. C2×C3×C3⋊Dic3D4semidirect
48. C2×C3×C3⋊Dic3D4semidirect
58. C23×C3⋊D3C2×D3semidirect
87. C32×C22⋊C4C2×D3semidirect
88. C32×C22⋊C4C2×D3semidirect
89. #3757C2×D3semidirect
138. C22×C3⋊D3D12semidirect
139. C22×C3⋊D3#67semidirect
163. C2×C3⋊Dic3D12semidirect
164. C2×C3⋊Dic3#67semidirect
238. C3×C22⋊C4D32semidirect
239. C3×C22⋊C4D32semidirect
319. C33B(4,𝔽2)semidirect
429. C22⋊C4#1113semidirect
499. C32#4741semidirect
500. C32#5032semidirect
569. C3#10590407semidirect
570. C3#10590410semidirect

Maximal subgroups

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

subgroupindex
#117246332
C3×#640912
#117237272
C3×#636332
#117220832
#105904103
#105904073