Finite Group Databasebeta

← #10780315

Subgroups — #10780315

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3251C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#83129C211yesmaximal, characteristic, complemented
4#83123C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71123C211yesmaximal, characteristic, complemented
7#71119C211yesmaximal, characteristic, complemented
8#71107C211yesmaximal, characteristic, complemented
9#71103C211yesmaximal, characteristic, complemented
10#71026C211yesmaximal, characteristic, complemented
11#71022C211yesmaximal, characteristic, complemented
12#71010C211yesmaximal, characteristic, complemented
13#71006C211yesmaximal, characteristic, complemented
14#70507C211yesmaximal, characteristic, complemented
15#70492C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C3×#529C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#478—21no—
21C3×#475C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#469—21no—
24C3×#468—21no—
25C3×#463—21no—
26C3×#460C2211yescharacteristic
27C3×#459C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5232—21no—
30#5232—21no—
31#5079C2211yescharacteristic
32#5079C2211yescharacteristic
33#5070C2211yescharacteristic
34#5070C2211yescharacteristic
35#5064—21no—
36#5055—21no—
37#5055—21no—
38#5055—21no—
39#5043—21no—
40#5033—21no—
41#5024C2211yescharacteristic
42Q8×C3⋊C8C2211yescharacteristic
43#5021C2211yescharacteristic
44#5021C2211yescharacteristic
45#5012—21no—
46#5012C2211yescharacteristic
47#5011—21no—
48D4×C3⋊C8C2211yescharacteristic
49#5008C2211yescharacteristic
50#5008C2211yescharacteristic
51#5000—21no—
52#4997—21no—
53#4926C2211yescharacteristic
54#4925C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4481C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3251D311yescharacteristic
69#3238—31no—
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698—31no—
75#1697—31no—
76#1684—31no—
77#1683—31no—
78#1678—31no—
79#1677—31no—
80#1569—31no—
81#1553—31no—
82#1548—31no—
83C2×C3×#41—21no—
84C22×Dic6—21no—
85C2×C3×SD16—21no—
86C2×C3×D8—21no—
87C2×C3×M16—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#126—21no—
91C3×#125D411yescharacteristic
92C3×#124—21no—
93C3×#124D411yescharacteristic
94C3×#123—21no—
95C3×C22⋊Q8D411yescharacteristic
96C3×#121—21no—
97C3×#121—21no—
98C3×#121D411yescharacteristic
99C3×C4×Q8C2311yescharacteristic
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4C2311yescharacteristic
104C3×#117—21no—
105C2×C3×C42C2311yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×#208—21no—
109C2×#208D411yescharacteristic
110C2×#208D411yescharacteristic
111C22×C3⋊C8—21no—
112#869—21no—
113#869—21no—
114#869—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×#106—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×#102—21no—
126#835—21no—
127#835—21no—
128#835—21no—
129#835—21no—
130#835—21no—
131#835—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#830—21no—
139#830—21no—
140#830—21no—
141#830—21no—
142#808—21no—
143#808—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532C2×D311yescharacteristic
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#483—61no—
172#480—61no—
173#478—21no—
174#477—31no—
175#477—31no—
176#476—31no—
177#476—31no—
178#475C2×D311yescharacteristic
179#474C2×D311yescharacteristic
180#469—21no—
181#468—21no—
182#463—21no—
183#460C2×D311yescharacteristic
184#459C2×D311yescharacteristic
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423C2×D311yescharacteristic
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—42no—
212C3×#41—42no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—41no—
220C2×C3×D4C2×D411yescharacteristic
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4111yescharacteristic
226C22×Dic3—41no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C2×Dic6—82no—
231C3×SD16—81no—
232C3×D8—81no—
233C3×M16—81no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C42—21no—
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42#4111yescharacteristic
255C3×C42C2411yescharacteristic
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—41no—
261C4⋊Dic3—41no—
262#208—41no—
263#208—41no—
264#208—42no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—42no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×#41—21no—
275C22×Q8—61no—
276C2×SD16—21no—
277C2×D8—21no—
278C2×M16—21no—
279C2×M16—31no—
280C2×M16—31no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8C22×D311yescharacteristic
289#127C22×D311yescharacteristic
290#126—21no—
291#125#6711yescharacteristic
292#124—21no—
293#124#6711yescharacteristic
294#123—21no—
295C22⋊Q8#6711yescharacteristic
296#121—21no—
297#121—21no—
298#121#6711yescharacteristic
299C4×Q8C22×D311yescharacteristic
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C22×D311yescharacteristic
304#117—21no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42C22×D311yescharacteristic
308#107—21no—
309#107—61no—
310#107—61no—
311#106—21no—
312#106—61no—
313#106—61no—
314C4⋊C8—21no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—21no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—21no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—31no—
344#97—31no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8—61no—
351C23×C3—21no—
352C23×C3—41no—
353C23×C3#12111yescharacteristic
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—81no—
365C3×D4—81no—
366C3×D4—82no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4#12111yescharacteristic
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—82no—
388Dic6—82no—
389Dic6—81no—
390Dic6—81no—
391C3×C8—41no—
392C3×C8—41no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—121no—
407C2×Q8—242no—
408C2×Q8C2×#6711yescharacteristic
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—41no—
414C2×D4C2×#6711yescharacteristic
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—121no—
418C22×C4D3×D411yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421SD16—81no—
422D8—81no—
423M16—81no—
424M16—121no—
425M16—121no—
426M16—122no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4C2×#6711yescharacteristic
451C4⋊C4C2×#6711yescharacteristic
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C42—21no—
460C42D3×D411yescharacteristic
461C42#23711yescharacteristic
462C42C2×#6711yescharacteristic
463C42C23×D311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3C2×#12111yescharacteristic
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4#57511yescharacteristic
483C3×C4#57511yescharacteristic
484C3×C4#57711yescharacteristic
485C3×C4#57811yescharacteristic
486Dic3—81no—
487Dic3—81no—
488C23—21no—
489C23—41no—
490C23#93911yescharacteristic
491Q8—21no—
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—241no—
496Q8—241no—
497Q8—242no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—81no—
505D4—81no—
506D4—82no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—82no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—242no—
522C2×C4#93911yescharacteristic
523C2×C4#93911yescharacteristic
524C2×C4#93911yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×#23711yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—41no—
529C8—41no—
530C8—61no—
531C8—61no—
532C8—61no—
533C8—61no—
534C8—121no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—41no—
539C2×C3—81no—
540C2×C3—161no—
541C2×C3#304011yescharacteristic
542C2×C3#306311yescharacteristic
543C2×C3#306811yescharacteristic
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#93911yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—241no—
563C4—241no—
564C4#577311yescharacteristic
565C4#577311yescharacteristic
566C4#583611yescharacteristic
567C4#583811yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—41no—
570C2—81no—
571C2—161no—
572C2#8342311yescharacteristic
573C2#8393311yescharacteristic
574C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3251C2
socleC22×C3C2×#121
hypercenter#3251D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3251

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #83129C2semidirect
4. #83123C2semidirect
5. #71808C2semidirect
6. #71123C2semidirect
7. #71119C2semidirect
8. #71107C2semidirect
9. #71103C2semidirect
10. #71026C2semidirect
11. #71022C2semidirect
12. #71010C2semidirect
13. #71006C2semidirect
14. #70507C2semidirect
15. #70492C2semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
C3×#32512
#832832
#831292
#831232
#718082
#711232
#711192
#711072
#711032
#710262
#710222
#710102
#710062
#705072
#704922
#193823