Finite Group Databasebeta

← #10845219

Subgroups — #10845219

Order 768.

776 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×#2565C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic
3C2×#5031C211yesmaximal, characteristic
4#73651C211yesmaximal, characteristic
5#71738C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic
7#69690C211yesmaximal, characteristic
8#69690C211yesmaximal, characteristic
9#69681C211yesmaximal, characteristic
10#69681C211yesmaximal, characteristic
11#69589C211yesmaximal, characteristic
12#69588C211yesmaximal, characteristic
13#69560C211yesmaximal, characteristic
14#69559C211yesmaximal, characteristic
15#69485C211yesmaximal, characteristic
16#25333—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#398—21no—
25C3×#398—21no—
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#387—21no—
29C3×#387—21no—
30C3×#384C2211yescharacteristic
31C3×#382C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5069—24no—
36#5031—24no—
37Dic3×C4⋊C4—21no—
38#4972—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4654—21no—
53#4654—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#462—31no—
64#2565D311yescharacteristic
65#2365—31no—
66#2074—31no—
67#2072—31no—
68#2056—31no—
69#2056—31no—
70#1989—31no—
71#1940—31no—
72#1878—31no—
73#1878—31no—
74#1805—31no—
75#1803—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C3×#125—22no—
81C3×#125—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#216—22no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—22no—
102C2×#216—41no—
103C2×#216—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#869—22no—
118#869D414yes—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C4×Dic6—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131#835—22no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#810—22no—
139#810—22no—
140#810—22no—
141#810—22no—
142#810—22no—
143#810—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C22×Q16—31no—
149C22×Q16—31no—
150C2×C4⋊Q8—31no—
151C2×#125C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#491—64no—
156#462—64no—
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×#97—31no—
169#398—21no—
170#398—21no—
171#391—31no—
172#391—31no—
173#391C2×D311yescharacteristic
174#391C2×D311yescharacteristic
175#389—61no—
176#387—21no—
177#387—21no—
178#387—61no—
179#387—61no—
180#384C2×D311yescharacteristic
181#382C2×D311yescharacteristic
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8#4112yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—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×Dic6C2×D412yes—
225C2×Dic6C2×D412yes—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#216—44no—
247#216—44no—
248C4⋊Dic3—24no—
249C4⋊Dic3—24no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—42no—
257C4×Dic3—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×Q8C22×D311yescharacteristic
274C2×Q16—62no—
275C2×Q16—62no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—121no—
279C2×Q16—121no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286#125—22no—
287#125—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—121no—
293C4×Q8—121no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4#6711yescharacteristic
306C2×C4⋊C4#6711yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C42—21no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317C4⋊C8—62no—
318C4⋊C8—121no—
319C4⋊C8—121no—
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#97—62no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342C23×C3C22×D411yescharacteristic
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—41no—
346C3×Q8—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
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—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4#13712yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C42−1+411yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395Dic6—22no—
396Dic6—22no—
397Dic6—22no—
398Dic6—22no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402Dic6—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—32no—
413C2×Q8—32no—
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#23912yes—
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
440Q16—124no—
441Q16—124no—
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—
452C2×C8—61no—
453C2×C8—61no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—32no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—122no—
481C42—122no—
482C42—122no—
483C42D3×D412yes—
484C42#23712yes—
485C22×C3C2×#12111yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57811yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4#52911yescharacteristic
503C3×C4#54211yescharacteristic
504C3×C4#54611yescharacteristic
505C3×C4#54711yescharacteristic
506Dic3—42no—
507Dic3—42no—
508Dic3—42no—
509Dic3—81no—
510C23C22×#6711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515Q8—62no—
516Q8—62no—
517Q8—62no—
518Q8—62no—
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—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
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—41no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
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—121no—
562C2×C4#93812yes—
563C2×C4#93812yes—
564C2×C4#95112yes—
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×#23711yescharacteristic
567C2×C4C2×#23911yescharacteristic
568C2×C4#100711yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#100911yescharacteristic
571C8—121no—
572C8—121no—
573C8—121no—
574C8—121no—
575C2×C3#242711yescharacteristic
576C2×C3#303711yescharacteristic
577C2×C3#304411yescharacteristic
578C2×C3#331211yescharacteristic
579C2×C3#331511yescharacteristic
580C2×C3#337411yescharacteristic
581C2×C3#337711yescharacteristic
582C22C2×#93811yescharacteristic
583C22#581711yescharacteristic
584C22#582211yescharacteristic
585C22C2×#95111yescharacteristic
586C22#582911yescharacteristic
587C22#583811yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—122no—
600C4—122no—
601C4—122no—
602C4—241no—
603C4#562411yescharacteristic
604C4#563011yescharacteristic
605C4#563111yescharacteristic
606C4#564311yescharacteristic
607C3#2533311yescharacteristic, complemented, Sylow
608C2#8275611yescharacteristic
609C2#8331411yescharacteristic
610C2#8341211yescharacteristic
611C2#8360011yescharacteristic
612C2#8360311yescharacteristic
613C2#8369811yescharacteristic
614C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2565C2
socle, prime radicalC23×C3C22×D4
hypercenter#2565D3
hypocenterC3#25333
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2565

Complements

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

NH ≅ G/Ntype
607. C3#25333semidirect

Maximal subgroups

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

subgroupindex
C3×#25652
C2×#50692
C2×#50312
#736512
#717382
#716312
#696902
#696812
#695892
#695882
#695602
#695592
#694852
#253333