Finite Group Databasebeta

← #10835916

Subgroups — #10835916

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2C2×#5031C211yesmaximal, characteristic, complemented
3#82262C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic
5#71700C211yesmaximal, characteristic
6#71700C211yesmaximal, characteristic
7#71627C211yesmaximal, characteristic, complemented
8#69678C211yesmaximal, characteristic, complemented
9#69677C211yesmaximal, characteristic, complemented
10#69616C211yesmaximal, characteristic, complemented
11#69614C211yesmaximal, characteristic, complemented
12#69527C211yesmaximal, characteristic
13#69526C211yesmaximal, characteristic
14#69486C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#24664—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×#387—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5031—24no—
38#5000—24no—
39#4981—21no—
40#4981—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4953—21no—
50#4953—21no—
51#4936C2211yescharacteristic
52#4929C2211yescharacteristic
53#4929C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×#433—31no—
64#2621—31no—
65#2323D311yescharacteristic
66#2036—31no—
67#2035—31no—
68#2001—31no—
69#2001—31no—
70#1981—31no—
71#1971—31no—
72#1969—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1805—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—21no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#216—22no—
109C2×#216—41no—
110C2×#216—41no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115#924—41no—
116#924—41no—
117#924—41no—
118#924—41no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#869—22no—
126#869—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—41no—
133#835—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#830—22no—
142#810—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#810—41no—
147#810—41no—
148#808—41no—
149#807—41no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)D414yes—
152#803—22no—
153C4×C3⋊C8—22no—
154C22×Q16—31no—
155C22×Q16—31no—
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8C2×D311yescharacteristic
160C2×C4×Q8C2×D311yescharacteristic
161C2×#117C2×D311yescharacteristic
162#462—64no—
163#433—64no—
164C2×#107—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8—31no—
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391—31no—
186#391C2×D311yescharacteristic
187#389—21no—
188#389C2×D311yescharacteristic
189#388—21no—
190#388—21no—
191#387—21no—
192#382—31no—
193#380C2×D311yescharacteristic
194C4×C4⋊C4C2×D311yescharacteristic
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42—81no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259#217—42no—
260#217—42no—
261#216—44no—
262#216—44no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—42no—
268C4⋊Dic3—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8C2×D412yes—
282C2×C3⋊C8C2×D412yes—
283C22×Q8—31no—
284C22×Q8—31no—
285C22×Q8C22×D311yescharacteristic
286C23×C4C22×D311yescharacteristic
287C2×Q16—62no—
288C2×Q16—62no—
289C2×Q16—62no—
290C2×Q16—62no—
291C2×Q16—121no—
292C2×Q16—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—121no—
302C22⋊Q8—121no—
303C22⋊Q8—121no—
304C22⋊Q8—121no—
305C4×Q8—22no—
306C4×Q8—22no—
307#117—22no—
308#117—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4#6711yescharacteristic
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4#6711yescharacteristic
328C2×C22⋊C4#6711yescharacteristic
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—121no—
334#106—121no—
335C4⋊C8—34no—
336C4⋊C8—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103—121no—
342#103—121no—
343#103—121no—
344#103—121no—
345C22⋊C8—62no—
346C22⋊C8—62no—
347#97—62no—
348C4×C8—62no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4#12112yes—
402C2×C3×C4#12112yes—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4H(5,𝔽2)11yescharacteristic
408C2×C3×C4H(5,𝔽2)11yescharacteristic
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3⋊C8—22no—
430C3⋊C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24C2×#6711yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8#23512yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C23×D311yescharacteristic
472Q16—124no—
473Q16—124no—
474C2×C8—32no—
475C2×C8—32no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—81no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4#23512yes—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—81no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—81no—
515C42D3×D412yes—
516C42#23712yes—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53511yescharacteristic
527C22×C3#57711yescharacteristic
528C22×C3#57811yescharacteristic
529C22×C3#57811yescharacteristic
530C22×C3#57811yescharacteristic
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—42no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4#54611yescharacteristic
541C3×C4#55011yescharacteristic
542C3×C4#55311yescharacteristic
543C3×C4#55511yescharacteristic
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23C22×#6711yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—121no—
561Q8—121no—
562Q8—121no—
563Q8—121no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4#93912yes—
610C2×C4#93912yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×#23511yescharacteristic
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4#100911yescharacteristic
617C8—62no—
618C8—62no—
619C8—121no—
620C8—121no—
621C2×C3—81no—
622C2×C3#241411yescharacteristic
623C2×C3#305911yescharacteristic
624C2×C3#306311yescharacteristic
625C2×C3#312011yescharacteristic
626C2×C3#312711yescharacteristic
627C2×C3#322311yescharacteristic
628C2×C3#322511yescharacteristic
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22#579111yescharacteristic
637C22C2×#93911yescharacteristic
638C22#583011yescharacteristic
639C22#583611yescharacteristic
640C22#583811yescharacteristic
641C22#583811yescharacteristic
642C22#583811yescharacteristic
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4#555011yescharacteristic
657C4#555311yescharacteristic
658C4#556311yescharacteristic
659C4#556511yescharacteristic
660C3#2466411yescharacteristic, complemented, Sylow
661C2—81no—
662C2#8235011yescharacteristic
663C2#8255511yescharacteristic
664C2#8257511yescharacteristic
665C2#8339911yescharacteristic
666C2#8342311yescharacteristic
667C2#8345211yescharacteristic
668C2#8346411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2323D3
hypocenterC3#24664
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#5031C2semidirect
7. #71627C2semidirect
8. #69678C2semidirect
9. #69677C2semidirect
10. #69616C2semidirect
11. #69614C2semidirect
14. #69486C2semidirect
15. #69468C2semidirect
660. C3#24664semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50312
#822622
C2×#50002
#717002
#716272
#696782
#696772
#696162
#696142
#695272
#695262
#694862
#694682
#246643