Finite Group Databasebeta

← #10834004

Subgroups — #10834004

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2625C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic
3#82230C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic
5#71693C211yesmaximal, characteristic
6#69681C211yesmaximal, characteristic, complemented
7#69681C211yesmaximal, characteristic, complemented
8#69678C211yesmaximal, characteristic, complemented
9#69677C211yesmaximal, characteristic, complemented
10#69635C211yesmaximal, characteristic, complemented
11#69634C211yesmaximal, characteristic, complemented
12#69616C211yesmaximal, characteristic, complemented
13#69614C211yesmaximal, characteristic, complemented
14#69525C211yesmaximal, characteristic
15#69506C211yesmaximal, characteristic
16#24514—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C3×#400—21no—
24C3×#398—21no—
25C3×#397—21no—
26C3×#397—21no—
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5050—24no—
37#5000—24no—
38#4973—21no—
39#4973—21no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4943—21no—
49#4943—21no—
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
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×#451—31no—
63C2×#433—31no—
64#2625D311yescharacteristic
65#2310—31no—
66#2068—31no—
67#2066—31no—
68#2056—31no—
69#2056—31no—
70#1971—31no—
71#1969—31no—
72#1935—31no—
73#1933—31no—
74#1873—31no—
75#1854—31no—
76#1787—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6—21no—
81C22×Dic6—22no—
82C22×Dic6—22no—
83C22×Dic6—22no—
84C22×Dic6—22no—
85C22×Dic6—41no—
86C22×Dic6—41no—
87C22×Dic6C2311yescharacteristic
88C22×Dic6C2311yescharacteristic
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×#117—22no—
92C3×#117—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108C2×#216—22no—
109C2×#216—22no—
110C2×#216—22no—
111C2×#216—22no—
112C2×#216—41no—
113C2×#216—41no—
114#926—41no—
115#926—41no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—41no—
131#835—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#830D414yes—
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)—22no—
152#803—22no—
153C4×C3⋊C8—22no—
154C23×Q8—31no—
155C22×Q16—31no—
156C22×Q16—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×#117—31no—
159C2×#117C2×D311yescharacteristic
160#451—64no—
161#433—64no—
162C2×#107—31no—
163C2×#106—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8—31no—
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—21no—
179#398—21no—
180#397—21no—
181#397—21no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—31no—
186#391—61no—
187#391—61no—
188#391C2×D311yescharacteristic
189#391C2×D311yescharacteristic
190#389C2×D311yescharacteristic
191#389C2×D311yescharacteristic
192#381—61no—
193#381—61no—
194#380C2×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—21no—
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×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4112yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—24no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
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—24no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—44no—
241C2×Dic6—44no—
242C2×Dic6—44no—
243C2×Dic6—44no—
244C2×Dic6—42no—
245C2×Dic6—42no—
246C2×Dic6#4112yes—
247C2×Dic6#4112yes—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—81no—
258C3×C4⋊C4—81no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—42no—
262C3×C22⋊C4—81no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267#217—42no—
268#216—44no—
269#216—44no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4×Dic3—42no—
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⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8C2×D412yes—
287C22×Q8—31no—
288C22×Q8—31no—
289C22×Q8—62no—
290C22×Q8—62no—
291C22×Q8—61no—
292C22×Q8—62no—
293C22×Q8—62no—
294C22×Q8—121no—
295C22×Q8—121no—
296C22×Q8C22×D311yescharacteristic
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299C2×Q16—62no—
300C2×Q16—62no—
301C2×Q16—62no—
302C2×Q16—62no—
303C2×Q16—121no—
304C2×Q16—121no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8—31no—
308C22×C8—31no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311#117—22no—
312#117—22no—
313#117—121no—
314#117—121no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4#6711yescharacteristic
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4#6711yescharacteristic
329C2×C22⋊C4#6711yescharacteristic
330C2×C42—61no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—121no—
334#106—121no—
335C4⋊C8—62no—
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—34no—
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—21no—
355#95—21no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3#12112yes—
371C23×C3C22×D411yescharacteristic
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×Q8—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
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—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4#12112yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—48no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6—22no—
430Dic6—48no—
431Dic6—41no—
432Dic6—41no—
433Dic6—41no—
434Dic6—41no—
435C3⋊C8—22no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C24C2×#6711yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—32no—
447C2×Q8—32no—
448C2×Q8—64no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—124no—
460C2×Q8—124no—
461C2×Q8—124no—
462C2×Q8—124no—
463C2×Q8—122no—
464C2×Q8—122no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—64no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4D3×D412yes—
481C22×C4#23712yes—
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C23×D311yescharacteristic
488Q16—124no—
489Q16—124no—
490C2×C8—32no—
491C2×C8—32no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—122no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—122no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3C2×#12111yescharacteristic
535C22×C3#52511yescharacteristic
536C22×C3#53511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57811yescharacteristic
539C22×C3#57811yescharacteristic
540C22×C3#57811yescharacteristic
541C3×C4—22no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4#54611yescharacteristic
551C3×C4#54611yescharacteristic
552C3×C4#55111yescharacteristic
553C3×C4#55111yescharacteristic
554Dic3—42no—
555Dic3—42no—
556Dic3—81no—
557Dic3—81no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#93912yes—
565C23C22×#6711yescharacteristic
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—62no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—128no—
575Q8—121no—
576Q8—121no—
577Q8—121no—
578Q8—121no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—128no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4#93912yes—
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C2×C4#100911yescharacteristic
628C2×C4#100911yescharacteristic
629C8—62no—
630C8—62no—
631C8—121no—
632C8—121no—
633C2×C3—42no—
634C2×C3#241111yescharacteristic
635C2×C3#305911yescharacteristic
636C2×C3#306311yescharacteristic
637C2×C3#312111yescharacteristic
638C2×C3#312711yescharacteristic
639C2×C3#322311yescharacteristic
640C2×C3#322511yescharacteristic
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22C2×#93911yescharacteristic
649C22#582811yescharacteristic
650C22#583011yescharacteristic
651C22#583611yescharacteristic
652C22#583811yescharacteristic
653C22#583811yescharacteristic
654C22#583811yescharacteristic
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—122no—
665C4—122no—
666C4—241no—
667C4—241no—
668C4#565411yescharacteristic
669C4#565411yescharacteristic
670C4#565811yescharacteristic
671C4#565811yescharacteristic
672C3#2451411yescharacteristic, complemented, Sylow
673C2—42no—
674C2#8289411yescharacteristic
675C2#8339911yescharacteristic
676C2#8342311yescharacteristic
677C2#8345111yescharacteristic
678C2#8345211yescharacteristic
679C2#8346411yescharacteristic
680C2#8346711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2625D3
hypocenterC3#24514
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
6. #69681C2semidirect
7. #69681C2semidirect
8. #69678C2semidirect
9. #69677C2semidirect
10. #69635C2semidirect
11. #69634C2semidirect
12. #69616C2semidirect
13. #69614C2semidirect
672. C3#24514semidirect

Maximal subgroups

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

subgroupindex
C3×#26252
C2×#50502
#822302
C2×#50002
#716932
#696812
#696782
#696772
#696352
#696342
#696162
#696142
#695252
#695062
#245143