Finite Group Databasebeta

← #10844652

Subgroups — #10844652

Order 768.

760 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×#2370C211yesmaximal, characteristic
2C2×#5063C211yesmaximal, characteristic
3C2×#5069C211yesmaximal, characteristic
4#73711C211yesmaximal, characteristic
5#69635C211yesmaximal, characteristic
6#69634C211yesmaximal, characteristic
7#69633C211yesmaximal, characteristic
8#69633C211yesmaximal, characteristic
9#69486C211yesmaximal, characteristic
10#69486C211yesmaximal, characteristic
11#69486C211yesmaximal, characteristic
12#69485C211yesmaximal, characteristic
13#69475C211yesmaximal, characteristic
14#69465C211yesmaximal, characteristic
15#69464C211yesmaximal, characteristic
16#25290—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#869C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#389—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#387—21no—
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#5069—24no—
33#5063—24no—
34#4984—21no—
35#4984—21no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4661—21no—
57#4661—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#479—31no—
64#2546—31no—
65#2370D311yescharacteristic
66#2068—31no—
67#2066—31no—
68#2043—31no—
69#2043—31no—
70#1981—31no—
71#1981—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1763—31no—
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C3×#125—22no—
81C3×#125D414yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#869—22no—
112#869—22no—
113#869—22no—
114#869—22no—
115#869—41no—
116#869—41no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#810—22no—
131#810—22no—
132#810—22no—
133#810—22no—
134#810—22no—
135#810—22no—
136#810—22no—
137#810—22no—
138#810—22no—
139#810—22no—
140#810—41no—
141#810—41no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8—31no—
151C2×#125C2×D311yescharacteristic
152C2×C4×Q8—31no—
153#491—64no—
154#479—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#97—31no—
173#398—61no—
174#398—61no—
175#391—31no—
176#391—31no—
177#389—21no—
178#389—61no—
179#389—61no—
180#389C2×D311yescharacteristic
181#389C2×D311yescharacteristic
182#387—21no—
183#387—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C4⋊C4C2×D311yescharacteristic
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C4⋊C4#4112yes—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—42no—
245C3×C42—81no—
246C3×C42—81no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3—42no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257Dic3⋊C4—42no—
258C4×Dic3—42no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—31no—
272C22×Q8—31no—
273C22×Q8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—31no—
278C4⋊Q8—62no—
279C4⋊Q8—62no—
280C4⋊Q8—62no—
281C4⋊Q8—62no—
282C4⋊Q8—121no—
283C4⋊Q8—121no—
284#125—22no—
285#125D1214yes—
286C4×Q8—62no—
287C4×Q8—62no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4#6711yescharacteristic
301C2×C4⋊C4#6711yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—62no—
315C4⋊C8—62no—
316C4⋊C8—121no—
317C4⋊C8—121no—
318#103—62no—
319#103—62no—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#103—121no—
330#97—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342C23×C3C22×D411yescharacteristic
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×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4#12112yes—
370C2×C3×C4#12112yes—
371C2×C3×C4#13712yes—
372C2×C3×C4#13712yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C42−1+411yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391Dic6—22no—
392Dic6—22no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402Dic6—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C2×Q8—32no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—81no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4C2×D1212yes—
473C4⋊C4C2×D1212yes—
474C4⋊C4#23912yes—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—81no—
481C42—81no—
482C42—122no—
483C42C2×D1212yes—
484C42#23512yes—
485C22×C3C2×#12111yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3C2×#13711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—81no—
502C3×C4#52911yescharacteristic
503C3×C4#54211yescharacteristic
504C3×C4#54811yescharacteristic
505C3×C4#55011yescharacteristic
506Dic3—42no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23C22×#6711yescharacteristic
511Q8—62no—
512Q8—62no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8—121no—
517Q8—121no—
518Q8—121no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—62no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4#93012yes—
562C2×C4#93012yes—
563C2×C4#95112yes—
564C2×C4#95112yes—
565C2×C4C22×D1211yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×#23911yescharacteristic
568C2×C4#100711yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#100911yescharacteristic
571C8—121no—
572C8—121no—
573C8—121no—
574C8—121no—
575C2×C3#242611yescharacteristic
576C2×C3#303911yescharacteristic
577C2×C3#304611yescharacteristic
578C2×C3#331211yescharacteristic
579C2×C3#332011yescharacteristic
580C2×C3#337211yescharacteristic
581C2×C3#337411yescharacteristic
582C22C2×#93011yescharacteristic
583C22#581711yescharacteristic
584C22C2×#95111yescharacteristic
585C22C2×#95111yescharacteristic
586C22#582911yescharacteristic
587C22#583611yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—81no—
599C4—122no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4#557611yescharacteristic
604C4#557711yescharacteristic
605C4#558011yescharacteristic
606C4#558211yescharacteristic
607C3#2529011yescharacteristic, complemented, Sylow
608C2#8249511yescharacteristic
609C2#8253311yescharacteristic
610C2#8257211yescharacteristic
611C2#8360311yescharacteristic
612C2#8362011yescharacteristic
613C2#8364411yescharacteristic
614C2#8369811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2370C2
socle, prime radicalC23×C3C22×D4
hypercenter#2370D3
hypocenterC3#25290
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
607. C3#25290semidirect

Maximal subgroups

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

subgroupindex
C3×#23702
C2×#50632
C2×#50692
#737112
#696352
#696342
#696332
#694862
#694852
#694752
#694652
#694642
#252903