Finite Group Databasebeta

← #10898393

Subgroups — #10898393

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76560C211yesmaximal, characteristic, complemented
2#75969C211yesmaximal, characteristic
3C3×#1935C211yesmaximal, characteristic, complemented
4#70280C211yesmaximal, characteristic
5#70276C211yesmaximal, characteristic
6#70261C211yesmaximal, characteristic, complemented
7#70241C211yesmaximal, characteristic, complemented
8#69832C211yesmaximal, characteristic, complemented
9#69818C211yesmaximal, characteristic
10#69676C211yesmaximal, characteristic
11#69618C211yesmaximal, characteristic, complemented
12#68900C211yesmaximal, characteristic
13#68895C211yesmaximal, characteristic
14#68817C211yesmaximal, characteristic, complemented
15#68809C211yesmaximal, characteristic, complemented
16#29440—31nomaximal, Sylow
17C22×#215C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×C3×#103C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C3×#391C2211yescharacteristic
23C3×#389C2211yescharacteristic
24#5229C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4990—21no—
32#4988C2211yescharacteristic
33#4987—21no—
34#4984C2211yescharacteristic
35#4984C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4939C2211yescharacteristic
43#4932C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4670—21no—
47#4664—21no—
48C3⋊#401—21no—
49#4660—21no—
50#4659—21no—
51#4658—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2645—31no—
61#2319—31no—
62#2088—31no—
63#2066—31no—
64#2065—31no—
65#2055—31no—
66#1968—31no—
67#1941—31no—
68#1936—31no—
69#1935D311yescharacteristic, complemented
70#1927—31no—
71#1919—31no—
72#1877—31no—
73#1864—31no—
74#1768—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#215—22no—
86C2×#215—22no—
87C2×#215—41no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105#893—22no—
106#893—22no—
107C3×#103—22no—
108C3×#103—41no—
109C3×#97—41no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113#835—22no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95D411yescharacteristic
124C3⋊#95D411yescharacteristic
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#821—22no—
128#821—41no—
129Dic3⋊C8—22no—
130Dic3⋊C8—22no—
131#809—22no—
132#808—22no—
133#808—22no—
134#808—41no—
135C4×C3⋊C8—41no—
136C22×SD16—31no—
137C2×C4×D4—31no—
138C2×#106—31no—
139C2×#106—31no—
140C2×#106—31no—
141C2×C4⋊C8—31no—
142C2×C4⋊C8—31no—
143C2×#103—31no—
144C2×#103C2×D311yescharacteristic
145C2×#103C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#97C2×D311yescharacteristic
150C2×C4×C8—31no—
151#401—61no—
152#400—61no—
153#400—61no—
154#400—61no—
155#391—31no—
156#391—31no—
157#391C2×D311yescharacteristic
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389C2×D311yescharacteristic
165#388—61no—
166#388—61no—
167#381—61no—
168#380—61no—
169C4×C4⋊C4—31no—
170#376—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12#4112yes—
206C2×C4×D3—24no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—81no—
221C3×C4⋊C4#4112yes—
222C3×C42—21no—
223C3×C42—21no—
224#215—44no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—81no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—81no—
237C4⋊Dic3—81no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8#4112yes—
250C2×C3⋊C8#4112yes—
251C22×Q8C22×D311yescharacteristic
252C22×D4—31no—
253C23×C4—61no—
254C2×SD16—62no—
255C2×SD16—62no—
256C2×SD16—121no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261C4×D4—62no—
262C4×D4—62no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283#106—62no—
284#106—62no—
285#106—121no—
286C4⋊C8—62no—
287C4⋊C8—62no—
288#103—22no—
289#103—41no—
290#103—62no—
291#102—62no—
292#102—62no—
293#102—121no—
294#97—41no—
295C4×C8—121no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—31no—
300#95—31no—
301#95—31no—
302#95—31no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C3×Q8—41no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—41no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369D12—22no—
370D12—22no—
371D12—41no—
372D12—41no—
373C4×D3—84no—
374C3×C8—41no—
375C3×C8—41no—
376C3⋊C8—22no—
377C3⋊C8—22no—
378C24—61no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—32no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—64no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—241no—
404C22×C4—241no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—32no—
416C2×C8—32no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—81no—
433C4⋊C4—241no—
434C4⋊C4—241no—
435C4⋊C4—241no—
436C4⋊C4—241no—
437C4⋊C4#23512yes—
438C22⋊C4—64no—
439C22⋊C4—241no—
440C22⋊C4—241no—
441C22⋊C4—241no—
442C22⋊C4—241no—
443C42—21no—
444C42—21no—
445C42—61no—
446C42—61no—
447C42—61no—
448C42—61no—
449C42—241no—
450C22×C3#52511yescharacteristic
451C22×C3#53211yescharacteristic
452C22×C3#53711yescharacteristic
453C22×C3#57711yescharacteristic
454C22×C3#57711yescharacteristic
455C22×C3#57711yescharacteristic
456C22×C3#57711yescharacteristic
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C2×D3—81no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4#55111yescharacteristic
471C3×C4#55211yescharacteristic
472C3×C4#55311yescharacteristic
473C3×C4#55611yescharacteristic
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480Dic3—81no—
481C23—62no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23C2×D3×D411yescharacteristic
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493D4—62no—
494D4—62no—
495D4—121no—
496D4—121no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—244no—
537C2×C4C2×#23511yescharacteristic
538C2×C4#100411yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C2×C4D3×#4111yescharacteristic
542C2×C4#101011yescharacteristic
543C8—41no—
544C8—41no—
545C8—62no—
546C8—62no—
547C2×C3#260111yescharacteristic
548C2×C3#312311yescharacteristic
549C2×C3#312511yescharacteristic
550C2×C3#317311yescharacteristic
551C2×C3#317411yescharacteristic
552C2×C3#326011yescharacteristic
553C2×C3#326311yescharacteristic
554D3—82no—
555C22—122no—
556C22—122no—
557C22—122no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22—241no—
562C22#554711yescharacteristic
563C22#559711yescharacteristic
564C22#566811yescharacteristic
565C22#577511yescharacteristic
566C22#577811yescharacteristic
567C22#577811yescharacteristic
568C22#577811yescharacteristic
569C4—41no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—121no—
576C4—121no—
577C4—121no—
578C4—121no—
579C4—241no—
580C4—241no—
581C4—241no—
582C4#553311yescharacteristic
583C4#555811yescharacteristic
584C4#565911yescharacteristic
585C4#570811yescharacteristic
586C3#2944011yescharacteristic, complemented, Sylow
587C2—242no—
588C2#7564411yescharacteristic
589C2#7978711yescharacteristic
590C2#7985011yescharacteristic
591C2#8079811yescharacteristic
592C2#8089911yescharacteristic
593C2#8166611yescharacteristic
594C2#8172911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1935C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1935D3
hypocenterC3#29440
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
1. #76560C2semidirect
3. C3×#1935C2semidirect
6. #70261C2semidirect
7. #70241C2semidirect
8. #69832C2semidirect
11. #69618C2semidirect
14. #68817C2semidirect
15. #68809C2semidirect
69. #1935D3semidirect
586. C3#29440semidirect

Maximal subgroups

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

subgroupindex
#765602
#759692
C3×#19352
#702802
#702762
#702612
#702412
#698322
#698182
#696762
#696182
#689002
#688952
#688172
#688092
#294403