Finite Group Databasebeta

← #10858871

Subgroups — #10858871

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73494C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3C3×#1850C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5#71969C211yesmaximal, characteristic
6#70489C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70462C211yesmaximal, characteristic, complemented
9#70461C211yesmaximal, characteristic, complemented
10#69931C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#68690C211yesmaximal, characteristic, complemented
13#68684C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#26417—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4949—21no—
35#4948—21no—
36#4945—21no—
37#4944—21no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4654—21no—
45#4652—21no—
46#4649—21no—
47#4647—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2441—31no—
61#2362—31no—
62#2100—31no—
63#2086—31no—
64#2068—31no—
65#2026—31no—
66#1989—31no—
67#1919—31no—
68#1892—31no—
69#1873—31no—
70#1865—31no—
71#1854—31no—
72#1850D311yescharacteristic
73#1803—31no—
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8—22no—
78C22×C3×C8—41no—
79C22×C3×C8—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92#926—22no—
93#926—22no—
94#926—41no—
95#926—41no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924Q814yes—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#870—22no—
119#870—41no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×C4⋊C8—41no—
123C3×C4⋊C8—41no—
124C3×C22⋊C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#818—41no—
135#817—22no—
136#816—22no—
137#816—41no—
138#816—41no—
139#816—41no—
140C23×C8C2×D311yescharacteristic
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#399—61no—
156#396—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389—61no—
162#388—61no—
163#388—61no—
164#387—61no—
165#387—61no—
166#384—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×Q812yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6C2×Q812yes—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—44no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—41no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229#217—24no—
230#217—42no—
231#217—42no—
232#217—81no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246C4⋊Dic3C2×Q812yes—
247C4⋊Dic3#4112yes—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C22×Q8—31no—
261C23×C4C22×D311yescharacteristic
262C22×C8—22no—
263C22×C8—41no—
264C22×C8—41no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#125—62no—
270#125—121no—
271C22⋊Q8—34no—
272C22⋊Q8—62no—
273C4×Q8—62no—
274C4×Q8—121no—
275#117—62no—
276#117—62no—
277#117—121no—
278#117—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4D1211yescharacteristic
300C2×C22⋊C4D1211yescharacteristic
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307#107—121no—
308#106—62no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311#103—62no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—22no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95D1211yescharacteristic
324#95D1211yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3C22×D411yescharacteristic
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4#13512yes—
352C2×C3×C4C22×Q811yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382Dic6—22no—
383Dic6—22no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C24C2×D1211yescharacteristic
391C2×Q8—32no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4#23712yes—
415C22×C4D3×Q812yes—
416C22×C4C23×D311yescharacteristic
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—44no—
430C4⋊C4—32no—
431C4⋊C4—32no—
432C4⋊C4—32no—
433C4⋊C4—64no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C4⋊C4—122no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C22⋊C4—22no—
454C22⋊C4—41no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—122no—
464C42—122no—
465C42—122no—
466C42—122no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×C22⋊Q811yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54611yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57811yescharacteristic
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4#52911yescharacteristic
487C3×C4#53811yescharacteristic
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490Dic3—42no—
491Dic3—42no—
492Dic3—42no—
493Dic3—42no—
494Dic3—81no—
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#89712yes—
505C23C22×D1211yescharacteristic
506Q8—62no—
507Q8—62no—
508Q8—121no—
509Q8—121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4#89712yes—
552C2×C4#90412yes—
553C2×C4C2×#23711yescharacteristic
554C2×C4#100411yescharacteristic
555C2×C4#100411yescharacteristic
556C2×C4C2×D3×Q811yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4D3×#4111yescharacteristic
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C2×C3—42no—
564C2×C3#245011yescharacteristic
565C2×C3#307611yescharacteristic
566C2×C3#308211yescharacteristic
567C2×C3#330111yescharacteristic
568C2×C3#330211yescharacteristic
569C2×C3#331511yescharacteristic
570C2×C3#331711yescharacteristic
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C2×#89711yescharacteristic
579C22#555411yescharacteristic
580C22#555411yescharacteristic
581C22C2×#90411yescharacteristic
582C22#574811yescharacteristic
583C22#575111yescharacteristic
584C22#575311yescharacteristic
585C4—22no—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—122no—
591C4—122no—
592C4—122no—
593C4—122no—
594C4—241no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4#559311yescharacteristic
599C4#563611yescharacteristic
600C4#563611yescharacteristic
601C4#564911yescharacteristic
602C3#2641711yescharacteristic, complemented, Sylow
603C2—42no—
604C2#7442411yescharacteristic
605C2#7902511yescharacteristic
606C2#7910111yescharacteristic
607C2#7944211yescharacteristic
608C2#7945011yescharacteristic
609C2#7950411yescharacteristic
610C2#7951411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1850C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1850D3
hypocenterC3#26417
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
6. #70489C2semidirect
7. #70485C2semidirect
8. #70462C2semidirect
9. #70461C2semidirect
12. #68690C2semidirect
13. #68684C2semidirect
14. #68643C2semidirect
15. #68622C2semidirect
602. C3#26417semidirect

Maximal subgroups

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

subgroupindex
#734942
#734682
C3×#18502
#719722
#719692
#704892
#704852
#704622
#704612
#699312
#699292
#686902
#686842
#686432
#686222
#264173