Finite Group Databasebeta

← #10876210

Subgroups — #10876210

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84044C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic, complemented
3C3×#1786C211yesmaximal, characteristic
4#72128C211yesmaximal, characteristic
5#70182C211yesmaximal, characteristic, complemented
6#70181C211yesmaximal, characteristic, complemented
7#70174C211yesmaximal, characteristic, complemented
8#70173C211yesmaximal, characteristic, complemented
9#70165C211yesmaximal, characteristic, complemented
10#70165C211yesmaximal, characteristic, complemented
11#70156C211yesmaximal, characteristic, complemented
12#70155C211yesmaximal, characteristic, complemented
13#69929C211yesmaximal, characteristic, complemented
14#69911C211yesmaximal, characteristic, complemented
15#69506C211yesmaximal, characteristic, complemented
16#27591—31nomaximal, Sylow
17C23×C3×D4C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5247—21no—
27#5247—21no—
28#5221—21no—
29#5221—21no—
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5212C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#818C2211yescharacteristic, complemented
41C2×#817C2211yescharacteristic, complemented
42#4977C2211yescharacteristic, complemented
43#4975C2211yescharacteristic, complemented
44Dic3×C4⋊C4C2211yescharacteristic, complemented
45Dic3×C4⋊C4C2211yescharacteristic, complemented
46C2×#808C2211yescharacteristic, complemented
47C2×#807C2211yescharacteristic, complemented
48#4945—21no—
49#4944—21no—
50#4654—21no—
51#4653—21no—
52#4551C2211yescharacteristic, complemented
53#4551C2211yescharacteristic, complemented
54#4550C2211yescharacteristic, complemented
55#4550C2211yescharacteristic, complemented
56#4550C2211yescharacteristic, complemented
57#4550C2211yescharacteristic, complemented
58#4529C2211yescharacteristic, complemented
59#4529C2211yescharacteristic, complemented
60#2314—31no—
61#2309—31no—
62#2092—31no—
63#2092—31no—
64#2071—31no—
65#2071—31no—
66#1926—31no—
67#1926—31no—
68#1925—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1818—31no—
73#1786—31no—
74#1786D311yescharacteristic
75C25×C3—21no—
76C22×C3×D4—22no—
77C22×C3×D4—22no—
78C22×C3×D4—21no—
79C22×C3×D4—22no—
80C22×C3×D4—22no—
81C22×C3×D4—41no—
82C22×C3×D4—41no—
83C22×C3×D4C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217D411yescharacteristic
102C2×#217D411yescharacteristic
103#926—22no—
104#926—22no—
105#926—41no—
106#926—41no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—41no—
125C3×#102—41no—
126C3×C22⋊C8—22no—
127C3×#95—21no—
128#832—22no—
129#832—22no—
130#832—41no—
131#832—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830—22no—
138#818—41no—
139#817—41no—
140#808—41no—
141#807—41no—
142C23×D4C2×D311yescharacteristic
143C2×#117—31no—
144C2×#117—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102C2×D311yescharacteristic
154C2×#102C2×D311yescharacteristic
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#390—31no—
160#390—31no—
161#390—31no—
162#390—31no—
163#390—61no—
164#390—61no—
165#388—61no—
166#388—61no—
167#387—61no—
168#387—61no—
169#384—31no—
170#382—31no—
171#380—31no—
172#380C2×D311yescharacteristic
173#379—61no—
174#379—61no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3—24no—
186C24×C3—21no—
187C24×C3—21no—
188C24×C3—42no—
189C24×C3—42no—
190C24×C3—81no—
191C24×C3—81no—
192C24×C3—81no—
193C24×C3—81no—
194C24×C3C2×D411yescharacteristic
195C2×C3×D4—24no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—44no—
207C2×C3×D4—44no—
208C2×C3×D4—44no—
209C2×C3×D4—44no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4#4112yes—
213C2×C3×D4#4112yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4112yes—
222C22×C3×C4#4112yes—
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—81no—
243#217—22no—
244#217—22no—
245#217—42no—
246#217—81no—
247#217—81no—
248#217—81no—
249#217—81no—
250#217—81no—
251#217—81no—
252#217—81no—
253#217—81no—
254#217—81no—
255C4⋊Dic3—22no—
256C4⋊Dic3—22no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C4×Dic3—81no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C25—21no—
281C22×D4—22no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—22no—
286C22×D4—41no—
287C22×D4—41no—
288C22×D4C22×D311yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4C22×D311yescharacteristic
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#117—62no—
296#117—62no—
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—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42—61no—
327#107—121no—
328#107—121no—
329#106—121no—
330#106—121no—
331#102—22no—
332#102—22no—
333#102—41no—
334#102—41no—
335#102—62no—
336#102—62no—
337#102—121no—
338#102—121no—
339C22⋊C8—22no—
340C22⋊C8—62no—
341#95—21no—
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—22no—
355C23×C3—48no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—84no—
369C23×C3—84no—
370C23×C3—84no—
371C23×C3—84no—
372C23×C3—84no—
373C23×C3—84no—
374C23×C3#12312yes—
375C23×C3C22×D411yescharacteristic
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—48no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
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#12312yes—
398C2×C3×C4C2×#4111yescharacteristic
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—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
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—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—24no—
433C24—21no—
434C24—21no—
435C24—42no—
436C24—42no—
437C24—81no—
438C24—81no—
439C24—81no—
440C24—81no—
441C24D3×D411yescharacteristic
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4#23712yes—
460C2×D4#23712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4#23512yes—
476C22×C4D3×D411yescharacteristic
477C22×C4#23712yes—
478C22×C4C23×D311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—241no—
506C4⋊C4—241no—
507C22⋊C4—62no—
508C22⋊C4—62no—
509C22⋊C4—81no—
510C22⋊C4—122no—
511C22⋊C4—241no—
512C22⋊C4—241no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—241no—
526C42—241no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—816no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3C2×#12311yescharacteristic
550C22×C3#52511yescharacteristic
551C22×C3#53511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57611yescharacteristic
554C22×C3#57711yescharacteristic
555C22×C3#57711yescharacteristic
556C3×C4—22no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4#55111yescharacteristic
560C3×C4#55111yescharacteristic
561C3×C4#55111yescharacteristic
562C3×C4#55111yescharacteristic
563Dic3—41no—
564Dic3—41no—
565Dic3—41no—
566Dic3—41no—
567Dic3—81no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—48no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23#88312yes—
599C23C2×D3×D411yescharacteristic
600D4—22no—
601D4—22no—
602D4—22no—
603D4—22no—
604D4—48no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4#88312yes—
646C2×C4C2×#23511yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4C2×#23711yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4#100411yescharacteristic
652C8—41no—
653C8—41no—
654C8—121no—
655C8—121no—
656C2×C3—42no—
657C2×C3—82no—
658C2×C3—82no—
659C2×C3#248211yescharacteristic
660C2×C3#309611yescharacteristic
661C2×C3#309611yescharacteristic
662C2×C3#312211yescharacteristic
663C2×C3#312311yescharacteristic
664C2×C3#322411yescharacteristic
665C2×C3#322411yescharacteristic
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—816no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22C2×#88311yescharacteristic
689C22#558611yescharacteristic
690C22#559211yescharacteristic
691C22#575711yescharacteristic
692C22#576211yescharacteristic
693C22#577411yescharacteristic
694C22#577511yescharacteristic
695C4—22no—
696C4—81no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—121no—
701C4—121no—
702C4—241no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#549511yescharacteristic
707C4#549511yescharacteristic
708C4#558711yescharacteristic
709C4#558711yescharacteristic
710C3#2759111yescharacteristic, complemented, Sylow
711C2—42no—
712C2—82no—
713C2—82no—
714C2#7383211yescharacteristic
715C2#7875611yescharacteristic
716C2#7891911yescharacteristic
717C2#7999311yescharacteristic
718C2#7999911yescharacteristic
719C2#8013711yescharacteristic
720C2#8014311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#1786D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #73868C2semidirect
5. #70182C2semidirect
6. #70181C2semidirect
7. #70174C2semidirect
8. #70173C2semidirect
9. #70165C2semidirect
10. #70165C2semidirect
11. #70156C2semidirect
12. #70155C2semidirect
13. #69929C2semidirect
14. #69911C2semidirect
15. #69506C2semidirect
40. C2×#818C22semidirect
41. C2×#817C22semidirect
42. #4977C22semidirect
43. #4975C22semidirect
44. Dic3×C4⋊C4C22semidirect
45. Dic3×C4⋊C4C22semidirect
46. C2×#808C22semidirect
47. C2×#807C22semidirect
52. #4551C22semidirect
53. #4551C22semidirect
54. #4550C22semidirect
55. #4550C22semidirect
56. #4550C22semidirect
57. #4550C22semidirect
58. #4529C22semidirect
59. #4529C22semidirect
710. C3#27591semidirect

Maximal subgroups

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

subgroupindex
#840442
#738682
C3×#17862
#721282
#701822
#701812
#701742
#701732
#701652
#701562
#701552
#699292
#699112
#695062
#275913