Finite Group Databasebeta

← #10834005

Subgroups — #10834005

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
1C2×#5186C211yesmaximal, characteristic
2C2×#4840C211yesmaximal, characteristic
3#76642C211yesmaximal, characteristic
4#76610C211yesmaximal, characteristic
5C3×#2056C211yesmaximal, characteristic, complemented
6#72067C211yesmaximal, characteristic, complemented
7#70488C211yesmaximal, characteristic, complemented
8#70269C211yesmaximal, characteristic
9#70266C211yesmaximal, characteristic
10#70258C211yesmaximal, characteristic, complemented
11#70253C211yesmaximal, characteristic, complemented
12#69683C211yesmaximal, characteristic, complemented
13#69681C211yesmaximal, characteristic, complemented
14#68876C211yesmaximal, characteristic
15#68855C211yesmaximal, characteristic, complemented
16#24514—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C22×C3×Q16C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5186—24no—
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4987C2211yescharacteristic
42#4984C2211yescharacteristic
43#4973C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4929C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49#4840—24no—
50#4675—21no—
51#4674—21no—
52#4664—21no—
53#4664—21no—
54#4658—21no—
55#4657—21no—
56#4646—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#451—31no—
63C2×#433—31no—
64#2625—31no—
65#2310—31no—
66#2068—31no—
67#2066—31no—
68#2056—31no—
69#2056D311yescharacteristic, complemented
70#1971—31no—
71#1969—31no—
72#1935—31no—
73#1933—31no—
74#1873—31no—
75#1854—31no—
76#1787—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C2×D3×Q8—22no—
80C2×D3×Q8—22no—
81C2×D3×Q8—22no—
82C2×D3×Q8—22no—
83C2×D3×Q8—41no—
84C2×D3×Q8—41no—
85C22×C4×D3—21no—
86C22×C4×D3C2311yescharacteristic
87C22×Dic6—21no—
88C22×Dic6C2311yescharacteristic
89C2×C3×Q16—22no—
90C2×C3×Q16—22no—
91C2×C3×Q16—41no—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#946—22no—
98#946—22no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—41no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#892—41no—
120#892—41no—
121#872—22no—
122#872—22no—
123C3×C4⋊C8—22no—
124C3×#103—22no—
125C3×#103—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—41no—
132#835—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8D414yes—
146#817—41no—
147#816—22no—
148#816—41no—
149#815—22no—
150C8×Dic3—22no—
151#810—22no—
152#807—41no—
153C4⋊(C3⋊C8)—22no—
154C23×Q8—31no—
155C22×Q16—31no—
156C22×Q16C2×D311yescharacteristic
157C2×C22⋊Q8—31no—
158C2×#117—31no—
159C2×#117—31no—
160#451—64no—
161#433—64no—
162C2×#107—31no—
163C2×#106—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103C2×D311yescharacteristic
173C2×#103C2×D311yescharacteristic
174C2×C22⋊C8—31no—
175C2×C22⋊C8—31no—
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—61no—
179#398—61no—
180#397—61no—
181#397—61no—
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#389—31no—
191#389—31no—
192#381—61no—
193#381—61no—
194#380—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C23×D3C2×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—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#4112yes—
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×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226D3×Q8—44no—
227D3×Q8—44no—
228D3×Q8—44no—
229D3×Q8—44no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—24no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3C2×D412yes—
238C2×C4×D3#4112yes—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—24no—
244C2×Dic6—21no—
245C2×Dic6—42no—
246C2×Dic6—42no—
247C2×Dic6#4112yes—
248C3×Q16—44no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8C2×D412yes—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C42—21no—
260C3×C42—21no—
261#216—44no—
262(C2×D3)⋊C4—22no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—81no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—42no—
277Dic3⋊C4—81no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
287C22×Q8—31no—
288C22×Q8—61no—
289C22×Q8—62no—
290C22×Q8—62no—
291C22×Q8—62no—
292C22×Q8—62no—
293C22×Q8—121no—
294C22×Q8—121no—
295C22×Q8C22×D311yescharacteristic
296C22×Q8C22×D311yescharacteristic
297C23×C4—31no—
298C23×C4—61no—
299C2×Q16—22no—
300C2×Q16—22no—
301C2×Q16—41no—
302C2×Q16—62no—
303C2×Q16—62no—
304C2×Q16—121no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C22⋊Q8—62no—
310C22⋊Q8—62no—
311#117—62no—
312#117—62no—
313#117—121no—
314#117—121no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—31no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—121no—
334#106—121no—
335C4⋊C8—22no—
336C4⋊C8—62no—
337#103—22no—
338#103—41no—
339#103—62no—
340#103—62no—
341#103—62no—
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—61no—
354#95—61no—
355#95—61no—
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×C3C22×D411yescharacteristic
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3#12112yes—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—41no—
375C3×Q8—41no—
376C3×Q8—41no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
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×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
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#12112yes—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—48no—
430Dic6—22no—
431Dic6—22no—
432Dic6—48no—
433Dic6—41no—
434Dic6—41no—
435C3×C8—22no—
436C3×C8—41no—
437C3⋊C8—22no—
438C3⋊C8—41no—
439C24—31no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—32no—
452C2×Q8—64no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—122no—
459C2×Q8—122no—
460C2×Q8—124no—
461C2×Q8—124no—
462C2×Q8—124no—
463C2×Q8—124no—
464C2×Q8#23712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—64no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4D3×D411yescharacteristic
486C22×C4D3×D411yescharacteristic
487C22×C4C23×D311yescharacteristic
488Q16—44no—
489Q16—124no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—32no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8D3×D412yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—122no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C22⋊C4—62no—
518C22⋊C4—62no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C42—21no—
523C42—21no—
524C42—61no—
525C42—61no—
526C42—122no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#52511yescharacteristic
529C22×C3#53511yescharacteristic
530C22×C3#57711yescharacteristic
531C22×C3#57811yescharacteristic
532C22×C3#57811yescharacteristic
533C22×C3#57811yescharacteristic
534C2×D3—22no—
535C2×D3—22no—
536C2×D3—22no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—41no—
540C2×D3—41no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4#54611yescharacteristic
548C3×C4#54611yescharacteristic
549C3×C4#55111yescharacteristic
550C3×C4#55111yescharacteristic
551Dic3—22no—
552Dic3—42no—
553Dic3—41no—
554Dic3—41no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558C23—32no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23C2×D3×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—62no—
575Q8—62no—
576Q8—128no—
577Q8—121no—
578Q8—121no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—32no—
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—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
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—121no—
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4#100411yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4#100911yescharacteristic
629C8—22no—
630C8—41no—
631C8—62no—
632C8—121no—
633C2×C3#241111yescharacteristic
634C2×C3#305911yescharacteristic
635C2×C3#306311yescharacteristic
636C2×C3#312111yescharacteristic
637C2×C3#312711yescharacteristic
638C2×C3#322311yescharacteristic
639C2×C3#322511yescharacteristic
640D3—42no—
641C22—62no—
642C22—62no—
643C22—62no—
644C22—121no—
645C22—121no—
646C22—121no—
647C22—121no—
648C22D3×#12111yescharacteristic
649C22#560611yescharacteristic
650C22#560711yescharacteristic
651C22#576611yescharacteristic
652C22#577011yescharacteristic
653C22#577911yescharacteristic
654C22#577911yescharacteristic
655C4—41no—
656C4—42no—
657C4—41no—
658C4—62no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—121no—
663C4—122no—
664C4—121no—
665C4—241no—
666C4—241no—
667C4—241no—
668C4#559111yescharacteristic
669C4#565811yescharacteristic
670C4#565911yescharacteristic
671C4#567611yescharacteristic
672C3#2451411yescharacteristic, complemented, Sylow
673C2—122no—
674C2#7710911yescharacteristic
675C2#8046111yescharacteristic
676C2#8048411yescharacteristic
677C2#8057211yescharacteristic
678C2#8061111yescharacteristic
679C2#8076211yescharacteristic
680C2#8090511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2056D3
hypocenterC3#24514
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
5. C3×#2056C2semidirect
6. #72067C2semidirect
7. #70488C2semidirect
10. #70258C2semidirect
11. #70253C2semidirect
12. #69683C2semidirect
13. #69681C2semidirect
15. #68855C2semidirect
69. #2056D3semidirect
672. C3#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#51862
C2×#48402
#766422
#766102
C3×#20562
#720672
#704882
#702692
#702662
#702582
#702532
#696832
#696812
#688762
#688552
#245143