Finite Group Databasebeta

← #13161977

Subgroups — #13161977

Order 1728.

1046 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 103 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×#64361C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3#11723661C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5#11722077C212yesmaximal, complemented
6#10591511—31nomaximal
7#10591510—31nomaximal
8#10591509—31nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—22no—
11C2×C3×#638—22no—
12C3×#3830—22no—
13C3×D4×C3⋊D3—22no—
14C3×#3828C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
17C3×#3748C2212yescomplemented
18#88811—22no—
19#88810—22no—
20#64361D311yescharacteristic, complemented
21C3×#1004—31no—
22C3×#1004—31no—
23C3×#1004—61no—
24#64149D311yescharacteristic, complemented
25C3×#940—31no—
26C3×#940—31no—
27C3×#940—61no—
28#63830—32no—
29#63825—32no—
30#63823—32no—
31#63660D311yescharacteristic, complemented
32C3×#833—31no—
33C3×#833—31no—
34C3×#833—61no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C23×C33D412yes—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3—21no—
41C33×D4—22no—
42C2×C33×C4C2311yescharacteristic
43C3×#638—24no—
44C3×#638—22no—
45C3×#638—44no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—22no—
49C3×C4⋊(C3⋊D3)D412yescomplemented
50C3×C4×C3⋊D3—22no—
51C3×#634D412yescomplemented
52C3×C3⋊(C3⋊C8)—22no—
53#5241—91no—
54#5077—91no—
55#4747—91no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—41no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C2×D311yescharacteristic
63C2×#638—22no—
64C2×#638—22no—
65#3830—22no—
66D4×C3⋊D3—22no—
67#3828C2×D312yes—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
69C2×C3×#67—64no—
70C2×C3×#67—62no—
71C2×C3×#67—62no—
72C2×C3×#67—64no—
73C2×C3×#67—62no—
74C2×C3×#67—62no—
75C3×#237—62no—
76C3×#237—62no—
77C3×#237—64no—
78C3×D3×D4—62no—
79C3×D3×D4—62no—
80C3×D3×D4—64no—
81C3×#235—32no—
82C3×#235—32no—
83C3×#235—62no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C4×C3⋊Dic3C2×D311yescharacteristic, complemented
88#3748C2×D312yescomplemented
89C3×C4×Dic3—31no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—61no—
92C3×#208—32no—
93C3×#208—32no—
94C3×#208—62no—
95#3717—62no—
96#3716—62no—
97#3715—62no—
98#3714—62no—
99#3714—62no—
100#3713—62no—
101C22×C33—22no—
102C22×C33—42no—
103C22×C33C2×D411yescharacteristic
104C2×C3×C3⋊D3—22no—
105C2×C3×C3⋊D3—42no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—82no—
108C33×C4C2×D412yes—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—22no—
111C3×H(5,𝔽2)—91no—
112#1004—31no—
113#1004—31no—
114#1004—61no—
115C3×#127—91no—
116#949—92no—
117#940—31no—
118#940—31no—
119#940—61no—
120#932—92no—
121#908—92no—
122C3×#100—91no—
123#837—92no—
124#833—31no—
125#833—31no—
126#833—61no—
127#825—92no—
128#805—92no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32—42no—
133C23×C32D1212yes—
134C23×C32#6712yes—
135C23×C32#6712yes—
136C22×C3⋊D3—22no—
137C22×C3⋊D3—21no—
138C22×C3×D3—64no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C32×D4—22no—
145C32×D4—24no—
146C32×D4—22no—
147C32×D4—22no—
148C32×D4—42no—
149C32×D4—42no—
150C32×D4—44no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158#638—24no—
159#638—22no—
160#638—44no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3—21no—
163C4⋊(C3⋊D3)—22no—
164C4⋊(C3⋊D3)#6712yescomplemented
165C4×C3⋊D3—22no—
166#634#6712yescomplemented
167C3×#67—64no—
168C3×#67—68no—
169C3×#67—64no—
170C3×#67—64no—
171C3×#67—62no—
172C3×#67—62no—
173C3×#67—124no—
174C3×#67—124no—
175C3×#67—128no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—64no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C3×D12—32no—
183C3×D12—32no—
184C3×D12—62no—
185C3×D12—62no—
186C3×D12—62no—
187C3×D12—122no—
188C3×C4×D3—64no—
189C3×C4×D3—62no—
190C3×C4×D3—62no—
191C3×Dic6—32no—
192C3×Dic6—32no—
193C3×Dic6—62no—
194C3⋊(C3⋊C8)—22no—
195C3×C3⋊C8—62no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—122no—
198#453—271noSylow
199C2×C33—21no—
200C2×C33—42no—
201C2×C33#12011yescharacteristic
202C3×C3⋊D3—42no—
203C3×C3⋊D3—81no—
204C3×#41—92no—
205C3×#41—182no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—41no—
210C2×C3×D4—91no—
211C2×C3×D4—182no—
212C2×C3×D4—182no—
213C2×C3×D4—182no—
214C2×C3×D4D3211yescharacteristic
215C2×C3×D4D3211yescharacteristic
216C2×C3×D4D3211yescharacteristic
217C2×#67—64no—
218C2×#67—62no—
219C2×#67—62no—
220C2×#67—64no—
221C2×#67—62no—
222C2×#67—62no—
223#237—62no—
224#237—62no—
225#237—64no—
226D3×D4—62no—
227D3×D4—62no—
228D3×D4—64no—
229#235—32no—
230#235—32no—
231#235—62no—
232C2×D12—31no—
233C2×D12—31no—
234C2×D12—61no—
235C3×M16—92no—
236C3×C42—91no—
237#215—182no—
238#214—182no—
239#213—182no—
240#213—182no—
241C4×Dic3—31no—
242C4×Dic3—31no—
243C4×Dic3—61no—
244#208—32no—
245#208—32no—
246#208—62no—
247D24—182no—
248#204—182no—
249C22×C32—22no—
250C22×C32—24no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—44no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—42no—
263C22×C32—44no—
264C22×C32—82no—
265C22×C32—82no—
266C22×C32—84no—
267C22×C32C2×D1211yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C22×C32C2×#6711yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3—42no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—82no—
274C2×C3×D3—64no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—122no—
278C2×C3×D3—124no—
279C2×C3×D3—122no—
280C2×C3×D3—122no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—244no—
284C2×C3×D3—242no—
285C2×C3×D3—242no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—42no—
290C32×C4D3×D412yes—
291C32×C4D3×D412yes—
292C32×C4C2×#6712yes—
293C3⋊Dic3—22no—
294C3⋊Dic3—22no—
295C3×Dic3—64no—
296C3×Dic3—64no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301H(5,𝔽2)—91no—
302AGL(1,ℤ/8)—272no—
303#127—91no—
304#104—272no—
305#100—91no—
306C33#45311yescharacteristic, complemented, Sylow
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—42no—
311C23×C3—182no—
312C23×C3—181no—
313C23×C3#62512yes—
314C23×C3#62612yes—
315C23×C3#62612yes—
316C22×D3—64no—
317C22×D3—62no—
318C22×D3—62no—
319C22×D3—62no—
320C22×D3—61no—
321C22×D3—61no—
322C3×Q8—92no—
323C3×D4—24no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—44no—
330C3×D4—92no—
331C3×D4—184no—
332C3×D4—182no—
333C3×D4—182no—
334C3×D4—364no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—182no—
340C2×C3×C4—181no—
341C2×C3×C4—182no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—64no—
346#67—68no—
347#67—64no—
348#67—64no—
349#67—62no—
350#67—62no—
351#67—124no—
352#67—124no—
353#67—128no—
354C2×Dic3—62no—
355C2×Dic3—64no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360D12—32no—
361D12—32no—
362D12—62no—
363D12—62no—
364D12—62no—
365D12—122no—
366C4×D3—64no—
367C4×D3—62no—
368C4×D3—62no—
369Dic6—32no—
370Dic6—32no—
371Dic6—62no—
372C3×C8—182no—
373C3⋊C8—62no—
374C3⋊C8—62no—
375C3⋊C8—122no—
376C2×C32—21no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32—44no—
389C2×C32—42no—
390C2×C32—42no—
391C2×C32—82no—
392C2×C32—82no—
393C2×C32—84no—
394C2×C32#88211yescharacteristic
395C2×C32#93711yescharacteristic
396C2×C32#95311yescharacteristic
397C3⋊D3—42no—
398C3⋊D3—81no—
399C3×D3—124no—
400C3×D3—122no—
401C3×D3—122no—
402C3×D3—242no—
403C3×D3—241no—
404C3×D3—241no—
405#41—92no—
406#41—182no—
407C2×D4—91no—
408C2×D4—182no—
409C2×D4—182no—
410C2×D4—182no—
411C2×D4#111311yescharacteristic
412SD16—542no—
413D8—542no—
414M16—92no—
415C42—91no—
416C22×C3—24no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—44no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—44no—
430C22×C3—41no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—84no—
434C22×C3—182no—
435C22×C3—362no—
436C22×C3—361no—
437C22×C3—722no—
438C22×C3C2×#62511yescharacteristic
439C22×C3C2×#62611yescharacteristic
440C22×C3C2×#62611yescharacteristic
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—122no—
445C2×D3—124no—
446C2×D3—122no—
447C2×D3—122no—
448C2×D3—121no—
449C2×D3—121no—
450C2×D3—244no—
451C2×D3—242no—
452C2×D3—242no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—182no—
458C3×C4—182no—
459C3×C4D3×#6712yes—
460C3×C4D3×#6712yes—
461C3×C4#381112yes—
462Dic3—64no—
463Dic3—64no—
464Dic3—62no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468C32—21no—
469C32—21no—
470C32—21no—
471C32—41no—
472C32#474711yescharacteristic, complemented
473C32#507711yescharacteristic, complemented
474C32#524111yescharacteristic, complemented
475C23—182no—
476C23—181no—
477C23#629512yes—
478Q8—92no—
479D4—22no—
480D4—92no—
481D4—184no—
482D4—182no—
483D4—182no—
484D4—364no—
485C2×C4—182no—
486C2×C4—181no—
487C2×C4—182no—
488C2×C4C2×#111311yescharacteristic
489C8—182no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—42no—
500C2×C3—44no—
501C2×C3—42no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3—41no—
505C2×C3—82no—
506C2×C3—82no—
507C2×C3—84no—
508C2×C3—362no—
509C2×C3—721no—
510C2×C3#6397611yescharacteristic
511C2×C3#6397711yescharacteristic
512C2×C3#6398111yescharacteristic
513D3—124no—
514D3—122no—
515D3—122no—
516D3—242no—
517D3—241no—
518D3—241no—
519C22—22no—
520C22—42no—
521C22—182no—
522C22—362no—
523C22—361no—
524C22—722no—
525C22C2×#629511yescharacteristic
526C4—182no—
527C4—182no—
528C4#8904912yes—
529C3—21no—
530C3—21no—
531C3—21no—
532C3—41no—
533C3#1059150911yescharacteristic, complemented
534C3#1059151011yescharacteristic, complemented
535C3#1059151111yescharacteristic, complemented
536C2—21no—
537C2—42no—
538C2—362no—
539C2—721no—
540C2#1172372711yescharacteristic
subgroupG/H
centerC2#11723727
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×D4#1113
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. C3×#64149C2semidirect
3. #11723661C2semidirect
4. C3×#63660C2semidirect
5. #11722077C2semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
16. C3×C4×C3⋊Dic3C22semidirect
17. C3×#3748C22semidirect
20. #64361D3semidirect
24. #64149D3semidirect
31. #63660D3semidirect
49. C3×C4⋊(C3⋊D3)D4semidirect
51. C3×#634D4semidirect
68. C2×C4⋊(C3⋊D3)C2×D3semidirect
87. C4×C3⋊Dic3C2×D3semidirect
88. #3748C2×D3semidirect
164. C4⋊(C3⋊D3)#67semidirect
166. #634#67semidirect
306. C33#453semidirect
472. C32#4747semidirect
473. C32#5077semidirect
474. C32#5241semidirect
533. C3#10591509semidirect
534. C3#10591510semidirect
535. C3#10591511semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
C3×#641492
#117236612
C3×#636602
#117220772
#105915113
#105915103
#105915093