Finite Group Databasebeta

← #10847348

Subgroups — #10847348

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79204C211yesmaximal, characteristic, complemented
2#78917C211yesmaximal, characteristic
3C2×#4763C211yesmaximal, characteristic
4#78544C211yesmaximal, characteristic, complemented
5C3×#1931C211yesmaximal, characteristic
6#70670C211yesmaximal, characteristic, complemented
7#70317C211yesmaximal, characteristic
8#70197C211yesmaximal, characteristic
9#70072C211yesmaximal, characteristic
10Dic3×#100C211yesmaximal, characteristic
11#69893C211yesmaximal, characteristic, complemented
12#68985C211yesmaximal, characteristic, complemented
13#68975C211yesmaximal, characteristic, complemented
14#68924C211yesmaximal, characteristic, complemented
15#68920C211yesmaximal, characteristic, complemented
16#25493—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18#5517C2211yescharacteristic
19C2×#890C2211yescharacteristic
20C3×#428C2211yescharacteristic
21C3×#418C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#100C2211yescharacteristic
24C3×#405C2211yescharacteristic
25C2×#833C2211yescharacteristic
26#5215C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#822C2211yescharacteristic
29#5122C2211yescharacteristic
30#5121C2211yescharacteristic
31#5121C2211yescharacteristic
32#5118C2211yescharacteristic
33Dic3×M16C2211yescharacteristic
34Dic3×M16C2211yescharacteristic
35C2×#819C2211yescharacteristic
36C2×#817C2211yescharacteristic
37#5102C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4999C2211yescharacteristic
40#4995C2211yescharacteristic
41#4995C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4878—21no—
45#4876—21no—
46#4858—21no—
47#4856—21no—
48#4766—21no—
49#4763—21no—
50#4763—21no—
51#4763—21no—
52#4763—21no—
53#4763—21no—
54#4761—21no—
55#4759—21no—
56#4701—21no—
57#4701—21no—
58#4679—21no—
59C8×Dic6—21no—
60C3×#343C2211yescharacteristic
61C3×#341C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4530C2211yescharacteristic
68#3166—31no—
69#3077—31no—
70C2×#478—31no—
71#2962—31no—
72#2090—31no—
73#2089—31no—
74#2063—31no—
75#2063—31no—
76#1998—31no—
77#1952—31no—
78#1951—31no—
79#1950—31no—
80#1931—31no—
81#1931D311yescharacteristic
82C4×#100—31no—
83C22×C3×D4C2311yescharacteristic
84C23×Dic3—21no—
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×#217—21no—
92D4×Dic3—21no—
93D4×Dic3—21no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96#926C2311yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×#208C2311yescharacteristic
103C2×#208C2311yescharacteristic
104C2×#208C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106#890—21no—
107#890—21no—
108#890—21no—
109#890—21no—
110#890D411yescharacteristic
111#890D411yescharacteristic
112#888D411yescharacteristic
113#888D411yescharacteristic
114#878—21no—
115#878—21no—
116C4×Dic6—21no—
117C4×Dic6—21no—
118C3×#107—21no—
119C3×#106—21no—
120C3×#102—21no—
121C3×#102—21no—
122C3×#102—21no—
123C3×#102—21no—
124C3×#100Q814yescomplemented
125C3×#97—21no—
126C3×C4×C8—21no—
127#833—22no—
128#832—21no—
129#832—21no—
130#832—21no—
131#832—21no—
132#822—22no—
133#819—22no—
134#818—21no—
135#818—21no—
136#817—21no—
137#817—21no—
138#817—21no—
139#817—21no—
140#815—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8D411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149Dic3⋊C8D411yescharacteristic
150Dic3⋊C8D411yescharacteristic
151C8×Dic3—21no—
152C8×Dic3—21no—
153C8×Dic3—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#807—21no—
158#807—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162C4⋊(C3⋊C8)—21no—
163C4⋊(C3⋊C8)—21no—
164#533—31no—
165C2×#125—31no—
166C2×C4×D4—31no—
167#501—61no—
168#501—61no—
169#499—61no—
170#499—61no—
171#478—61no—
172#478—61no—
173#478—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#474—61no—
179#446—61no—
180#446—61no—
181#446—61no—
182C8×Q8—61no—
183C2×#108—31no—
184C2×#108—31no—
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428C2×D311yescharacteristic
191C2×#107—31no—
192C2×#106—31no—
193#424—31no—
194#424—31no—
195#423—31no—
196C2×C4⋊C8—31no—
197#418—31no—
198#418C2×D311yescharacteristic
199C2×#102—31no—
200C2×#102C2×D311yescharacteristic
201C2×#100—31no—
202C2×#100C2×D311yescharacteristic
203#405—31no—
204#405C2×D311yescharacteristic
205C4×M16—31no—
206C4×M16—31no—
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#341C2×D311yescharacteristic
215C24×C3—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4C2×Q812yes—
222C22×C3×C4—21no—
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—41no—
227C22×Dic3—41no—
228C22×Dic3—82no—
229C22×Dic3#4111yescharacteristic
230C22×Dic3#4111yescharacteristic
231C2×Dic6—21no—
232C2×Dic6—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C3×M16C2×Q812yes—
238C3×M16C2×Q812yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C22⋊C4—21no—
251C3×C42—21no—
252#217—21no—
253#217—41no—
254#217—41no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—41no—
262C4⋊Dic3—41no—
263C4⋊Dic3C2×D411yescharacteristic
264C4⋊Dic3C2×D411yescharacteristic
265Dic3⋊C4—21no—
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268Dic3⋊C4—21no—
269Dic3⋊C4—41no—
270Dic3⋊C4—41no—
271C4×Dic3—21no—
272C4×Dic3C2×D411yescharacteristic
273C4×Dic3C2×D411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276#208—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208#4112yes—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D411yescharacteristic
288C2×C3⋊C8C2×D411yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—61no—
291C2×M16—31no—
292C2×M16—31no—
293C2×M16—31no—
294C2×M16C22×D311yescharacteristic
295C2×M16C22×D311yescharacteristic
296C2×M16C22×D311yescharacteristic
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—31no—
300C4⋊Q8—31no—
301#125—31no—
302#125—31no—
303#125—61no—
304#125—61no—
305#125—61no—
306#125—61no—
307C22⋊Q8—61no—
308C22⋊Q8—61no—
309C4×Q8—61no—
310C4×Q8—61no—
311C4×D4—61no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4—61no—
315#117—31no—
316#117C22×D311yescharacteristic
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—61no—
323C2×C42—31no—
324#108—62no—
325#108—62no—
326#107—21no—
327#107—61no—
328#107—61no—
329#107—61no—
330#107—61no—
331#107—61no—
332#107—61no—
333#107—61no—
334#106—21no—
335#106—61no—
336#106—61no—
337#106—61no—
338#106—61no—
339#106—61no—
340#106—61no—
341#106—61no—
342C4⋊C8—31no—
343C4⋊C8—31no—
344C4⋊C8—31no—
345C4⋊C8—31no—
346C4⋊C8—61no—
347C4⋊C8—61no—
348C4⋊C8—61no—
349C4⋊C8—61no—
350C4⋊C8—61no—
351C4⋊C8—61no—
352#102—21no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—61no—
357#102—61no—
358#102—61no—
359#102—61no—
360#100—62no—
361#100Dic614yescomplemented
362#97—21no—
363#97—61no—
364#97—61no—
365#97—61no—
366C4×C8—21no—
367C4×C8—61no—
368C4×C8—61no—
369C4×C8—61no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—81no—
374C23×C3—81no—
375C23×C3C2×#4111yescharacteristic
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—81no—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×Q811yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—82no—
397C2×Dic3—82no—
398C2×Dic3—81no—
399C2×Dic3—81no—
400C2×Dic3#12111yescharacteristic
401C2×Dic3#12111yescharacteristic
402C2×Dic3#12111yescharacteristic
403C2×Dic3#12111yescharacteristic
404Dic6—42no—
405Dic6—81no—
406C3×C8—21no—
407C3×C8—21no—
408C3×C8—22no—
409C3×C8—22no—
410C3×C8—41no—
411C3⋊C8—21no—
412C3⋊C8—21no—
413C3⋊C8—22no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—21no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×Dic612yes—
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—121no—
431C22×C4—121no—
432C22×C4—242no—
433C22×C4C23×D311yescharacteristic
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—32no—
439M16—61no—
440M16—61no—
441M16—61no—
442M16—61no—
443M16—61no—
444M16C2×Dic612yes—
445M16C2×Dic612yes—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—31no—
452C2×C8—31no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8D3×D411yescharacteristic
459C2×C8D3×D411yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—31no—
463C4⋊C4—31no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—121no—
475C4⋊C4—121no—
476C4⋊C4—121no—
477C4⋊C4—121no—
478C4⋊C4D3×D411yescharacteristic
479C4⋊C4D3×D411yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—61no—
482C22⋊C4—121no—
483C22⋊C4—121no—
484C42—21no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—61no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—42no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3C2×#12111yescharacteristic
498C22×C3#53311yescharacteristic
499C22×C3#54111yescharacteristic
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—81no—
503C3×C4#54711yescharacteristic
504C3×C4#54811yescharacteristic
505C3×C4D4×Q811yescharacteristic
506C3×C4D4×Q811yescharacteristic
507Dic3—21no—
508Dic3—21no—
509Dic3—21no—
510Dic3—21no—
511Dic3—41no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—81no—
519C23—81no—
520C23D3×#4111yescharacteristic
521Q8—122no—
522Q8—241no—
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—81no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—242no—
542C2×C4—242no—
543C2×C4—241no—
544C2×C4—241no—
545C2×C4C22×Dic611yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4#101011yescharacteristic
550C2×C4#101011yescharacteristic
551C8—21no—
552C8—21no—
553C8—22no—
554C8—22no—
555C8—41no—
556C8—61no—
557C8—61no—
558C8—62no—
559C8—121no—
560C8—121no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—82no—
564C2×C3#242911yescharacteristic
565C2×C3#306711yescharacteristic
566C2×C3#306811yescharacteristic
567C22—21no—
568C22—21no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—81no—
573C22—81no—
574C22#551711yescharacteristic
575C22D3×#12111yescharacteristic
576C22#570111yescharacteristic
577C4—41no—
578C4—41no—
579C4—61no—
580C4—61no—
581C4—61no—
582C4—61no—
583C4—81no—
584C4—121no—
585C4—121no—
586C4—241no—
587C4—241no—
588C4D4×Dic611yescharacteristic
589C4D4×Dic611yescharacteristic
590C4#565511yescharacteristic
591C4#567711yescharacteristic
592C3#2549311yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—82no—
596C2#7550011yescharacteristic
597C2#8092911yescharacteristic
598C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1931C2
socleC22×C3C2×#121
prime radicalC22×C3×D4C23
hypercenter#1931D3
hypocenterC3#25493
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1931

Complements

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

NH ≅ G/Ntype
1. #79204C2semidirect
4. #78544C2semidirect
6. #70670C2semidirect
11. #69893C2semidirect
12. #68985C2semidirect
13. #68975C2semidirect
14. #68924C2semidirect
15. #68920C2semidirect
124. C3×#100Q8semidirect
361. #100Dic6semidirect
592. C3#25493semidirect

Maximal subgroups

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

subgroupindex
#792042
#789172
C2×#47632
#785442
C3×#19312
#706702
#703172
#701972
#700722
Dic3×#1002
#698932
#689852
#689752
#689242
#689202
#254933