Finite Group Databasebeta

← #10877336

Subgroups — #10877336

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2317C211yesmaximal, characteristic
2#82235C211yesmaximal, characteristic
3#71585C211yesmaximal, characteristic
4#71583C211yesmaximal, characteristic
5#71576C211yesmaximal, characteristic
6#71559C211yesmaximal, characteristic
7#71555C211yesmaximal, characteristic
8#69590C211yesmaximal, characteristic, complemented
9#69588C211yesmaximal, characteristic, complemented
10#69571C211yesmaximal, characteristic, complemented
11#69567C211yesmaximal, characteristic, complemented
12#69561C211yesmaximal, characteristic, complemented
13#69559C211yesmaximal, characteristic, complemented
14#69542C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#27690—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#389—21no—
23C3×#389—21no—
24C3×#385—21no—
25C3×#385—21no—
26C3×#384C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C3×#382C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#4978—21no—
34#4976—21no—
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4945—21no—
44#4944—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46#4929C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2323—31no—
61#2317D311yescharacteristic
62#2101—31no—
63#2099—31no—
64#2097—31no—
65#2095—31no—
66#1949—31no—
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1854—31no—
71#1852—31no—
72#1805—31no—
73#1803—31no—
74#1798—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C3×#117—22no—
79C3×#117—22no—
80C3×#117—41no—
81C3×#117—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—41no—
88C2×C3×C4⋊C4—41no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924Q814yes—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—22no—
130#810—41no—
131#810—41no—
132#810—41no—
133#810—41no—
134#808—41no—
135#807—41no—
136#807—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×C22⋊Q8—31no—
141C2×C4×Q8—31no—
142C2×#117C2×D311yescharacteristic
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4C2×D311yescharacteristic
145C2×#107—31no—
146C2×#107—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#391—31no—
158#389—21no—
159#389—21no—
160#389—31no—
161#389—61no—
162#388—61no—
163#388—61no—
164#387—61no—
165#385—21no—
166#385—21no—
167#384C2×D311yescharacteristic
168#384C2×D311yescharacteristic
169#382C2×D311yescharacteristic
170#382C2×D311yescharacteristic
171#380—31no—
172C4×C4⋊C4—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#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—81no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×Q812yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4—81no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4—81no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—42no—
234C3×C42#4112yes—
235C3×C42#4112yes—
236#217—24no—
237#217—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3C2×Q812yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247C4×Dic3—42no—
248C4×Dic3—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×Q8—31no—
262C23×C4—21no—
263C23×C4C22×D311yescharacteristic
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22⋊Q8—34no—
269C22⋊Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272#117—22no—
273#117—22no—
274#117—41no—
275#117—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4#6711yescharacteristic
300C2×C42—21no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#107—121no—
307#107—121no—
308#106—121no—
309C4⋊C8—62no—
310C4⋊C8—62no—
311#103—121no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3C22⋊Q812yes—
333C23×C3C22×D411yescharacteristic
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
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×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—84no—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4#12312yes—
365C2×C3×C4#12312yes—
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384Dic6—22no—
385Dic6—22no—
386Dic6—41no—
387Dic6—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24C2×#6711yescharacteristic
393C2×Q8—32no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4D3×Q812yes—
415C22×C4#23912yes—
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—32no—
445C4⋊C4—42no—
446C4⋊C4—64no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—42no—
461C22⋊C4—64no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—122no—
467C42—241no—
468C42#23512yes—
469C42#23512yes—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C22⋊Q811yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#57611yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C22×C3#57811yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#55311yescharacteristic
494C3×C4#55311yescharacteristic
495C3×C4#55411yescharacteristic
496C3×C4#55411yescharacteristic
497Dic3—42no—
498Dic3—42no—
499Dic3—81no—
500Dic3—81no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23#94612yes—
508C23C22×#6711yescharacteristic
509Q8—62no—
510Q8—62no—
511Q8—121no—
512Q8—121no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—84no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4#92912yes—
555C2×C4#92912yes—
556C2×C4#94612yes—
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×#23511yescharacteristic
559C2×C4#100411yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4C2×D3×Q811yescharacteristic
562C2×C4C2×#23911yescharacteristic
563C8—121no—
564C8—121no—
565C8—121no—
566C8—121no—
567C2×C3—42no—
568C2×C3#248711yescharacteristic
569C2×C3#308211yescharacteristic
570C2×C3#308311yescharacteristic
571C2×C3#309711yescharacteristic
572C2×C3#309811yescharacteristic
573C2×C3#322311yescharacteristic
574C2×C3#322411yescharacteristic
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×#92911yescharacteristic
583C22#580511yescharacteristic
584C22C2×#94611yescharacteristic
585C22#583511yescharacteristic
586C22#583611yescharacteristic
587C22#583811yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—122no—
599C4—122no—
600C4—241no—
601C4—241no—
602C4#551111yescharacteristic
603C4#551111yescharacteristic
604C4#551411yescharacteristic
605C4#551411yescharacteristic
606C3#2769011yescharacteristic, complemented, Sylow
607C2—42no—
608C2#8219711yescharacteristic
609C2#8256711yescharacteristic
610C2#8257311yescharacteristic
611C2#8338911yescharacteristic
612C2#8341411yescharacteristic
613C2#8347511yescharacteristic
614C2#8348011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2317D3
hypocenterC3#27690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
8. #69590C2semidirect
9. #69588C2semidirect
10. #69571C2semidirect
11. #69567C2semidirect
12. #69561C2semidirect
13. #69559C2semidirect
14. #69542C2semidirect
15. #69540C2semidirect
606. C3#27690semidirect

Maximal subgroups

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

subgroupindex
C3×#23172
#822352
#715852
#715832
#715762
#715592
#715552
#695902
#695882
#695712
#695672
#695612
#695592
#695422
#695402
#276903