Finite Group Databasebeta

← #10847307

Subgroups — #10847307

Order 768.

664 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79203C211yesmaximal, characteristic, complemented
2#78917C211yesmaximal, characteristic
3C2×#4761C211yesmaximal, characteristic
4#78544C211yesmaximal, characteristic, complemented
5C3×#1931C211yesmaximal, characteristic
6#70671C211yesmaximal, characteristic, complemented
7#70317C211yesmaximal, characteristic
8#70197C211yesmaximal, characteristic
9#70072C211yesmaximal, characteristic
10Dic3×#100C211yesmaximal, characteristic
11#69893C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68974C211yesmaximal, characteristic, complemented
14#68924C211yesmaximal, characteristic, complemented
15#68920C211yesmaximal, characteristic, complemented
16#25490—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18#5517C2211yescharacteristic
19C2×#888C2211yescharacteristic
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×#808C2211yescharacteristic
44#4876—21no—
45#4874—21no—
46#4856—21no—
47#4855—21no—
48#4766—21no—
49#4763—21no—
50#4761—21no—
51#4761—21no—
52#4761—21no—
53#4761—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#3165—31no—
69#3077—31no—
70C2×#476—31no—
71#2962—31no—
72#2090—31no—
73#2090—31no—
74#2063—31no—
75#2063—31no—
76#1997—31no—
77#1951—31no—
78#1951—31no—
79#1950—31no—
80#1931—31no—
81#1931D311yescharacteristic
82C4×#100—31no—
83C22×C3×D4C2311yescharacteristic
84C23×Dic3—21no—
85C22×Dic6—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#117C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×#217—21no—
93D4×Dic3—21no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97#926C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×#208C2311yescharacteristic
103C2×#208C2311yescharacteristic
104C2×#208C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106#890D411yescharacteristic
107#890D411yescharacteristic
108#888—21no—
109#888—21no—
110#888—21no—
111#888—21no—
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#808—21no—
158#808—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162C4⋊(C3⋊C8)—21no—
163C4⋊(C3⋊C8)—21no—
164#533—31no—
165C2×C4⋊Q8—31no—
166C2×C4×D4—31no—
167#501—61no—
168#501—61no—
169#500—61no—
170#498—61no—
171#478—61no—
172#478—61no—
173#476—61no—
174#476—61no—
175#476—61no—
176#476—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×#106—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—
233C2×Dic6—41no—
234C2×Dic6—41no—
235C2×Dic6—82no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16C2×Q812yes—
241C3×M16C2×Q812yes—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D411yescharacteristic
248C2×C3×C8C2×D411yescharacteristic
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C22⋊C4—21no—
254C3×C42—21no—
255#217—21no—
256#217—41no—
257#217—41no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D411yescharacteristic
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268Dic3⋊C4—21no—
269Dic3⋊C4—21no—
270Dic3⋊C4—41no—
271Dic3⋊C4—41no—
272C4×Dic3—21no—
273C4×Dic3C2×D411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208#4112yes—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D411yescharacteristic
289C2×C3⋊C8C2×D411yescharacteristic
290C22×Q8—61no—
291C22×D4C22×D311yescharacteristic
292C23×C4—61no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16—31no—
296C2×M16C22×D311yescharacteristic
297C2×M16C22×D311yescharacteristic
298C2×M16C22×D311yescharacteristic
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C4⋊Q8—31no—
302C4⋊Q8—31no—
303C4⋊Q8—61no—
304C4⋊Q8—61no—
305C4⋊Q8—61no—
306C4⋊Q8—61no—
307#125—31no—
308#125—31no—
309C22⋊Q8—61no—
310C22⋊Q8—61no—
311C4×Q8—61no—
312C4×Q8—61no—
313C4×D4—61no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317#117—31no—
318#117C22×D311yescharacteristic
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—61no—
324C2×C42—31no—
325#108—62no—
326#108—62no—
327#107—21no—
328#107—61no—
329#107—61no—
330#107—61no—
331#107—61no—
332#107—61no—
333#106—21no—
334#106—61no—
335#106—61no—
336#106—61no—
337#106—61no—
338#106—61no—
339#106—61no—
340#106—61no—
341#106—61no—
342#106—61no—
343C4⋊C8—31no—
344C4⋊C8—31no—
345C4⋊C8—31no—
346C4⋊C8—31no—
347C4⋊C8—61no—
348C4⋊C8—61no—
349C4⋊C8—61no—
350C4⋊C8—61no—
351C4⋊C8—61no—
352C4⋊C8—61no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—61no—
358#102—61no—
359#102—61no—
360#102—61no—
361#100—62no—
362#100Dic614yescomplemented
363#97—21no—
364#97—61no—
365#97—61no—
366#97—61no—
367C4×C8—21no—
368C4×C8—61no—
369C4×C8—61no—
370C4×C8—61no—
371C23×C3—22no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—81no—
375C23×C3—81no—
376C23×C3C2×#4111yescharacteristic
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—41no—
380C3×D4—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—81no—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×Q811yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—82no—
398C2×Dic3—82no—
399C2×Dic3—81no—
400C2×Dic3—81no—
401C2×Dic3#12111yescharacteristic
402C2×Dic3#12111yescharacteristic
403C2×Dic3#12111yescharacteristic
404C2×Dic3#12111yescharacteristic
405Dic6—42no—
406Dic6—81no—
407Dic6—82no—
408Dic6—82no—
409C3×C8—21no—
410C3×C8—21no—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—41no—
414C3⋊C8—21no—
415C3⋊C8—21no—
416C3⋊C8—22no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C24—21no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—121no—
423C2×Q8—121no—
424C2×Q8—242no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×Dic612yes—
431C22×C4—21no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—242no—
439C22×C4C23×D311yescharacteristic
440M16—21no—
441M16—21no—
442M16—21no—
443M16—21no—
444M16—32no—
445M16—61no—
446M16—61no—
447M16—61no—
448M16—61no—
449M16—61no—
450M16C2×Dic612yes—
451M16C2×Dic612yes—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—31no—
458C2×C8—31no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8D3×D411yescharacteristic
465C2×C8D3×D411yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—31no—
469C4⋊C4—31no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—121no—
481C4⋊C4—121no—
482C4⋊C4D3×D411yescharacteristic
483C4⋊C4D3×D411yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—61no—
486C22⋊C4—121no—
487C22⋊C4—121no—
488C42—21no—
489C42—31no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—61no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—42no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#53311yescharacteristic
503C22×C3#54111yescharacteristic
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—81no—
507C3×C4#54711yescharacteristic
508C3×C4#54811yescharacteristic
509C3×C4D4×Q811yescharacteristic
510C3×C4D4×Q811yescharacteristic
511Dic3—21no—
512Dic3—21no—
513Dic3—21no—
514Dic3—41no—
515Dic3—41no—
516Dic3—81no—
517Dic3—81no—
518Dic3#47112yes—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—81no—
523C23—81no—
524C23D3×#4111yescharacteristic
525Q8—122no—
526Q8—241no—
527Q8—242no—
528Q8—242no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—31no—
536C2×C4—31no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—41no—
540C2×C4—62no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—81no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—242no—
548C2×C4—242no—
549C2×C4—241no—
550C2×C4—241no—
551C2×C4C22×Dic611yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4D3×#4111yescharacteristic
555C2×C4#101011yescharacteristic
556C2×C4#101011yescharacteristic
557C8—21no—
558C8—21no—
559C8—22no—
560C8—22no—
561C8—41no—
562C8—61no—
563C8—61no—
564C8—62no—
565C8—121no—
566C8—121no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—82no—
570C2×C3#242911yescharacteristic
571C2×C3C2×#47111yescharacteristic
572C2×C3#306811yescharacteristic
573C22—21no—
574C22—21no—
575C22—42no—
576C22—82no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22#551711yescharacteristic
581C22D3×#12111yescharacteristic
582C22#570111yescharacteristic
583C4—32no—
584C4—41no—
585C4—41no—
586C4—61no—
587C4—61no—
588C4—61no—
589C4—81no—
590C4—121no—
591C4—121no—
592C4—241no—
593C4—241no—
594C4D4×Dic611yescharacteristic
595C4D4×Dic611yescharacteristic
596C4#565511yescharacteristic
597C4#567711yescharacteristic
598C3#2549011yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—82no—
602C2#7550011yescharacteristic
603C2D3×#47111yescharacteristic
604C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1931C2
socleC22×C3C2×#121
prime radicalC22×C3×D4C23
hypercenter#1931D3
hypocenterC3#25490
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. #79203C2semidirect
4. #78544C2semidirect
6. #70671C2semidirect
11. #69893C2semidirect
12. #68984C2semidirect
13. #68974C2semidirect
14. #68924C2semidirect
15. #68920C2semidirect
124. C3×#100Q8semidirect
362. #100Dic6semidirect
598. C3#25490semidirect

Maximal subgroups

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

subgroupindex
#792032
#789172
C2×#47612
#785442
C3×#19312
#706712
#703172
#701972
#700722
Dic3×#1002
#698932
#689842
#689742
#689242
#689202
#254903