Finite Group Databasebeta

← #13156582

Subgroups — #13156582

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723443C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63911C211yesmaximal, characteristic
6C2×C3×#3716C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591490—31nomaximal
9#10590877—31nomaximal
10#10590369—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20Dic3⋊(C3⋊Dic3)—22no—
21#88796—22no—
22#88794—22no—
23C3×#3716—22no—
24C2×#3754—31no—
25C2×#3744—31no—
26#64078—31no—
27C3×#940—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30C3×#874—31no—
31#63911D311yescharacteristic
32#63866—31no—
33#63866—31no—
34#63866—61no—
35C2×#3717—31no—
36C2×#3716—31no—
37C2×#3716—31no—
38C2×#3716—31no—
39C2×#3716—31no—
40C2×#3716—31no—
41C2×#3716—61no—
42C2×#3716—61no—
43C2×#3716D311yescharacteristic, complemented
44C2×#3715—31no—
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×#634—21no—
58C3×#634—21no—
59C2×C32×Dic3D412yes—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5175—91no—
66#5076—91no—
67#4693—91no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—121no—
77C2×#634C2×D311yescharacteristic
78C2×C3×#67—62no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#626—22no—
90C2×C32×C8—31no—
91C32×C42—31no—
92#3754—62no—
93C4⋊(C3⋊Dic3)—62no—
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95#3744—62no—
96C3×#214—62no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×#204—62no—
107Dic3⋊Dic3—64no—
108Dic3⋊Dic3—62no—
109Dic3⋊Dic3—62no—
110#3717—62no—
111#3716—22no—
112#3716—64no—
113#3716—62no—
114#3716—62no—
115#3716—62no—
116#3716—64no—
117#3716—62no—
118#3716—62no—
119#3715—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—42no—
123C2×C32×D3—41no—
124C2×C32×D3—42no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3⋊(C3⋊Dic3)—41no—
128C3×C3⋊Dic3—41no—
129C32×Dic3SD1614yescomplemented
130C2×C3×SD16—91no—
131C3×#127—91no—
132#944—91no—
133C2×#215—91no—
134#940—31no—
135C2×#214—31no—
136C2×#214—91no—
137C2×#214—91no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#204—31no—
142C2×#204—91no—
143C2×#204—91no—
144C2×#204—91no—
145C2×#204—91no—
146C2×#204—181no—
147#874—31no—
148#869—91no—
149#869—91no—
150#869—181no—
151C3×C4×C8—91no—
152C8×Dic3—31no—
153C4×C3⋊C8—31no—
154C23×C32—61no—
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—41no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—32no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—62no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—121no—
178C2×C3⋊Dic3—121no—
179C2×C3⋊Dic3—122no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182#634—21no—
183#634—21no—
184#634—61no—
185#634—61no—
186#634—62no—
187#634—62no—
188#634—61no—
189#634—61no—
190#634—61no—
191#634—61no—
192#634—121no—
193#634—121no—
194#634—121no—
195#634—121no—
196#634—122no—
197#634—122no—
198C3×#67—124no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3D1212yes—
204C2×C3×Dic3#6712yes—
205C3×D12—22no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—41no—
211C3×D12—62no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—121no—
217C3×D12—124no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—121no—
223C3×Dic6—121no—
224#626—44no—
225C32×C8—32no—
226C3⋊(C3⋊C8)D1212yescomplemented
227C3×C3⋊C8—22no—
228C3×C3⋊C8—32no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8D1212yescomplemented
232C3×C3⋊C8#6712yescomplemented
233#492—271noSylow
234C2×C33#12711yescharacteristic
235C2×C33C2×SD1612yes—
236C3×C3⋊D3—81no—
237C32×D3—81no—
238C2×C3×Q8—91no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—91no—
243C2×C3×D4—182no—
244C2×#67—62no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12—62no—
249C2×D12D3211yescharacteristic
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—61no—
253C2×Dic6—91no—
254C2×Dic6—91no—
255C2×Dic6—91no—
256C2×Dic6—181no—
257C2×Dic6—181no—
258C2×Dic6—181no—
259C2×Dic6—361no—
260C3×SD16—182no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—61no—
264C2×C3×C8—91no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268#215—182no—
269#214—62no—
270#214—182no—
271#214—182no—
272#214—182no—
273#214—184no—
274#214—182no—
275C4⋊Dic3—182no—
276C4⋊Dic3—182no—
277C4⋊Dic3—184no—
278Dic3⋊C4—182no—
279C4×Dic3D3211yescharacteristic
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8D3211yescharacteristic
284#204—62no—
285#204—182no—
286#204—182no—
287#204—184no—
288#204—182no—
289#204—182no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—121no—
295C22×C32—122no—
296C22×C32C2×D1211yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32C2×#6711yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—42no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—82no—
307C2×C3×D3—122no—
308C2×C3×D3—121no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—242no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—34no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4C2×D1211yescharacteristic
323C32×C4C2×D1211yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C3⋊Dic3—41no—
329C3⋊Dic3—122no—
330C3⋊Dic3—122no—
331C3⋊Dic3—122no—
332C3⋊Dic3—124no—
333C3⋊Dic3—121no—
334C3⋊Dic3—121no—
335C3⋊Dic3—121no—
336C3×Dic3—24no—
337C3×Dic3—122no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C3×Dic3#20414yescomplemented
341C3×Dic3#21414yescomplemented
342C2×SD16—91no—
343C2×SD16—271no—
344C2×SD16—271no—
345C2×SD16—271no—
346C4⋊Q8—271no—
347#127—91no—
348C4×C8—91no—
349C33#49211yescharacteristic, complemented, Sylow
350C23×C3—61no—
351C23×C3—61no—
352C23×C3—121no—
353C23×C3—181no—
354C22×D3—21no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—121no—
358C3×Q8—181no—
359C3×Q8—181no—
360C3×D4—62no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—121no—
366C3×D4—181no—
367C3×D4—181no—
368C3×D4—364no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—32no—
373C2×C3×C4—32no—
374C2×C3×C4—41no—
375C2×C3×C4—62no—
376C2×C3×C4—181no—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380#67—124no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—182no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—361no—
389C2×Dic3—361no—
390C2×Dic3—362no—
391C2×Dic3#62612yes—
392D12—21no—
393D12—21no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—121no—
400D12—124no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—121no—
406Dic6—121no—
407Dic6—182no—
408Dic6—182no—
409Dic6—181no—
410Dic6—181no—
411Dic6—181no—
412Dic6—181no—
413Dic6—181no—
414Dic6—181no—
415Dic6—361no—
416Dic6—361no—
417Dic6—361no—
418Dic6—361no—
419Dic6—362no—
420Dic6—362no—
421C3×C8—32no—
422C3×C8—32no—
423C3×C8—62no—
424C3×C8—92no—
425C3⋊C8—32no—
426C3⋊C8—32no—
427C3⋊C8—62no—
428C3⋊C8#62612yescomplemented
429C2×C32—22no—
430C2×C32—22no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—22no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—42no—
437C2×C32—241no—
438C2×C32#87411yescharacteristic
439C2×C32C2×#20412yes—
440C2×C32D3×SD1612yes—
441C2×C32C2×#21412yes—
442C2×C32#94011yescharacteristic
443C2×C32#94011yescharacteristic
444C3⋊D3—81no—
445C3×D3—82no—
446C3×D3—81no—
447C3×D3—81no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451C2×Q8—91no—
452C2×Q8—271no—
453C2×D4—31no—
454C2×D4—91no—
455C2×D4—182no—
456SD16—182no—
457SD16—542no—
458SD16—542no—
459SD16—542no—
460C2×C8—31no—
461C2×C8—91no—
462C4⋊C4—542no—
463C42—31no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—242no—
474C22×C3—362no—
475C22×C3—361no—
476C22×C3D3×D1211yescharacteristic
477C22×C3C2×#62611yescharacteristic
478C22×C3D3×#6711yescharacteristic
479C2×D3—42no—
480C2×D3—41no—
481C2×D3—122no—
482C2×D3—122no—
483C2×D3—121no—
484C2×D3—122no—
485C2×D3—121no—
486C2×D3—242no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—34no—
494C3×C4—34no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—64no—
498C3×C4—361no—
499C3×C4D3×D1211yescharacteristic
500C3×C4D3×D1211yescharacteristic
501C3×C4D3×D1211yescharacteristic
502C3×C4C2×#62611yescharacteristic
503C3×C4D3×#6711yescharacteristic
504C3×C4#381111yescharacteristic
505Dic3—122no—
506Dic3—121no—
507Dic3—121no—
508Dic3—362no—
509Dic3—362no—
510Dic3—362no—
511Dic3—364no—
512Dic3—361no—
513Dic3—361no—
514Dic3—361no—
515Dic3#371614yescomplemented
516C32—21no—
517C32—21no—
518C32—21no—
519C32—41no—
520C32#469311yescharacteristic, complemented
521C32#507611yescharacteristic, complemented
522C32#517511yescharacteristic, complemented
523C23—61no—
524C23—181no—
525Q8—181no—
526Q8—181no—
527Q8—541no—
528Q8—541no—
529D4—61no—
530D4—61no—
531D4—181no—
532D4—181no—
533D4—364no—
534C2×C4—32no—
535C2×C4—181no—
536C2×C4—541no—
537C2×C4D3311yescharacteristic
538C8—32no—
539C8—92no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—41no—
547C2×C3—42no—
548C2×C3—242no—
549C2×C3—241no—
550C2×C3—241no—
551C2×C3—721no—
552C2×C3D3×#20412yes—
553C2×C3C2×#371612yes—
554C2×C3#6391111yescharacteristic
555C2×C3#6391111yescharacteristic
556C2×C3D3×#21412yes—
557C2×C3#6397911yescharacteristic
558D3—81no—
559D3—242no—
560D3—241no—
561D3—241no—
562C22—121no—
563C22—122no—
564C22—361no—
565C22—362no—
566C22D3×#62611yescharacteristic
567C4—34no—
568C4—361no—
569C4—1081no—
570C4D3×#62611yescharacteristic
571C4#8895511yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3#1059036911yescharacteristic, complemented
577C3#1059087711yescharacteristic, complemented
578C3#1059149011yescharacteristic, complemented
579C2—241no—
580C2—721no—
581C2D3×#371612yes—
582C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88811C2semidirect
2. #11723443C2semidirect
3. C2×#88796C2semidirect
4. C2×#88794C2semidirect
6. C2×C3×#3716C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
43. C2×#3716D3semidirect
47. Dic3×C3⋊C8D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
129. C32×Dic3SD16semidirect
226. C3⋊(C3⋊C8)D12semidirect
231. C3×C3⋊C8D12semidirect
232. C3×C3⋊C8#67semidirect
340. C3×Dic3#204semidirect
341. C3×Dic3#214semidirect
349. C33#492semidirect
428. C3⋊C8#626semidirect
515. Dic3#3716semidirect
520. C32#4693semidirect
521. C32#5076semidirect
522. C32#5175semidirect
576. C3#10590369semidirect
577. C3#10590877semidirect
578. C3#10591490semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234432
C2×#887962
C2×#887942
C3×#639112
C2×C3×#37162
C3×Dic3×C3⋊C82
#105914903
#105908773
#105903693