Finite Group Databasebeta

← #10876136

Subgroups — #10876136

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84086C211yesmaximal, characteristic
2#73864C211yesmaximal, characteristic, complemented
3C3×#1788C211yesmaximal, characteristic
4#72135C211yesmaximal, characteristic
5#70249C211yesmaximal, characteristic, complemented
6#70241C211yesmaximal, characteristic, complemented
7#70233C211yesmaximal, characteristic, complemented
8#70223C211yesmaximal, characteristic, complemented
9#70184C211yesmaximal, characteristic, complemented
10#70173C211yesmaximal, characteristic, complemented
11#70165C211yesmaximal, characteristic, complemented
12#70155C211yesmaximal, characteristic, complemented
13#69929C211yesmaximal, characteristic, complemented
14#69910C211yesmaximal, characteristic, complemented
15#69506C211yesmaximal, characteristic, complemented
16#27584—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C2×#877—21no—
21C2×C3×#103C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5220—21no—
34#5220—21no—
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5212C2211yescharacteristic
38C4×#217—21no—
39C4×#217—21no—
40C2×#830C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43#4975C2211yescharacteristic
44#4975C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4942—21no—
50#4650—21no—
51#4649—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2297—31no—
61#2285—31no—
62#2095—31no—
63#2094—31no—
64#2072—31no—
65#2071—31no—
66#1927—31no—
67#1927—31no—
68#1925—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1817—31no—
73#1788—31no—
74#1788D311yescharacteristic
75C2×C3×#41—22no—
76C2×C3×#41—22no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41—41no—
80C2×C3×#41—41no—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4—21no—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C23×Dic3D411yescharacteristic
87C23×Dic3D411yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99#926—41no—
100#926—41no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#877—44no—
121C3×#103—22no—
122C3×#103—41no—
123C3×#102—22no—
124C3×#102—41no—
125C3×C22⋊C8—22no—
126C3×#95—21no—
127#835—22no—
128#835—41no—
129#832—22no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#830—22no—
139#818—41no—
140#817—41no—
141#808—41no—
142#807—41no—
143C22×#41C2×D311yescharacteristic
144C2×#117—31no—
145C2×#117—61no—
146C22×C42—31no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×#102C2×D311yescharacteristic
159C2×C22⋊C8—31no—
160C2×C22⋊C8C2×D311yescharacteristic
161#391—31no—
162#391—31no—
163#390—31no—
164#390—31no—
165#386—61no—
166#386—61no—
167#384—31no—
168#384—31no—
169#382—61no—
170#380—31no—
171#380—61no—
172#380C2×D311yescharacteristic
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—61no—
176C4×C22⋊C4—61no—
177C4×C22⋊C4—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C24×C3C2×D411yescharacteristic
188C3×#41—44no—
189C3×#41—44no—
190C3×#41—44no—
191C3×#41—44no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8#4112yes—
198C2×C3×D4—24no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—42no—
205C2×C3×D4—42no—
206C2×C3×D4#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—24no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4#4112yes—
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—22no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—41no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C22⋊C4—42no—
246#217—42no—
247#217—42no—
248#217—42no—
249#217—42no—
250#217—42no—
251#217—42no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—22no—
265C4×Dic3—22no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×#41—22no—
280C2×#41—22no—
281C2×#41—22no—
282C2×#41—22no—
283C2×#41—41no—
284C2×#41—41no—
285C22×Q8C22×D311yescharacteristic
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C23×C4—21no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4C22×D311yescharacteristic
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#117—124no—
297#117—121no—
298#117—121no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—62no—
321C2×C42—62no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42—61no—
325#107—121no—
326#107—121no—
327#106—121no—
328#106—121no—
329#103—22no—
330#103—41no—
331#103—62no—
332#103—121no—
333#102—22no—
334#102—41no—
335#102—62no—
336#102—121no—
337C22⋊C8—22no—
338C22⋊C8—62no—
339#95—21no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3#12312yes—
361C23×C3C22×D411yescharacteristic
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—48no—
369C3×D4—41no—
370C3×D4—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—48no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4#12312yes—
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×Dic3—28no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—44no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3#13512yes—
418C2×Dic3AGL(1,ℤ/8)12yes—
419C2×Dic3#13712yes—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24—21no—
425C24D3×D411yescharacteristic
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8#23712yes—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4#23712yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—121no—
469C22×C4#23512yes—
470C22×C4D3×D411yescharacteristic
471C22×C4#23712yes—
472C22×C4C23×D311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C22⋊C4—42no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C42—62no—
509C42—62no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—122no—
515C42—122no—
516C42—122no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3C2×#12311yescharacteristic
532C22×C3#52511yescharacteristic
533C22×C3#53511yescharacteristic
534C22×C3C2×#13511yescharacteristic
535C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
536C22×C3C2×#13711yescharacteristic
537C22×C3#57511yescharacteristic
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4#55111yescharacteristic
543C3×C4#55111yescharacteristic
544C3×C4#55111yescharacteristic
545C3×C4#55111yescharacteristic
546Dic3—22no—
547Dic3—22no—
548Dic3—22no—
549Dic3—41no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23#88312yes—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—41no—
577D4—41no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—32no—
586C2×C4—32no—
587C2×C4—32no—
588C2×C4—48no—
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—68no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—124no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
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#88312yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4C2×#23711yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4#100411yescharacteristic
626C2×C4#100411yescharacteristic
627C8—41no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3—42no—
632C2×C3—82no—
633C2×C3#248211yescharacteristic
634C2×C3#309111yescharacteristic
635C2×C3#309211yescharacteristic
636C2×C3#310611yescharacteristic
637C2×C3#311611yescharacteristic
638C2×C3#321011yescharacteristic
639C2×C3#321411yescharacteristic
640C22—22no—
641C22—22no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22C2×#88311yescharacteristic
655C22#558611yescharacteristic
656C22#559211yescharacteristic
657C22D3×#13511yescharacteristic
658C22#576811yescharacteristic
659C22D3×AGL(1,ℤ/8)11yescharacteristic
660C22D3×#13711yescharacteristic
661C4—22no—
662C4—42no—
663C4—62no—
664C4—62no—
665C4—62no—
666C4—81no—
667C4—81no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#549511yescharacteristic
674C4#549511yescharacteristic
675C4#558711yescharacteristic
676C4#558711yescharacteristic
677C3#2758411yescharacteristic, complemented, Sylow
678C2—42no—
679C2—82no—
680C2#7383211yescharacteristic
681C2#7884211yescharacteristic
682C2#7889211yescharacteristic
683C2#8006911yescharacteristic
684C2#8007811yescharacteristic
685C2#8021511yescharacteristic
686C2#8022011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#1788D3
hypocenterC3#27584
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #73864C2semidirect
5. #70249C2semidirect
6. #70241C2semidirect
7. #70233C2semidirect
8. #70223C2semidirect
9. #70184C2semidirect
10. #70173C2semidirect
11. #70165C2semidirect
12. #70155C2semidirect
13. #69929C2semidirect
14. #69910C2semidirect
15. #69506C2semidirect
677. C3#27584semidirect

Maximal subgroups

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

subgroupindex
#840862
#738642
C3×#17882
#721352
#702492
#702412
#702332
#702232
#701842
#701732
#701652
#701552
#699292
#699102
#695062
#275843