Finite Group Databasebeta

← #10855102

Subgroups — #10855102

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic, complemented
2#82351C211yesmaximal, characteristic
3#82335C211yesmaximal, characteristic
4C2×#4723C211yesmaximal, characteristic, complemented
5C3×#1870C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic, complemented
7#69939C211yesmaximal, characteristic, complemented
8#69936C211yesmaximal, characteristic, complemented
9#69930C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic, complemented
11#69929C211yesmaximal, characteristic, complemented
12#68693C211yesmaximal, characteristic, complemented
13#68641C211yesmaximal, characteristic, complemented
14#68628C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#26154—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5222—21no—
28#5220—21no—
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2211yescharacteristic
32#5121—24no—
33C2×#818C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#4980—21no—
44#4976—21no—
45#4952—21no—
46#4951—21no—
47#4945—21no—
48Dic3×C22⋊C4—21no—
49#4936C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53#4723—24no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#485—31no—
63C2×#428—31no—
64#2415—31no—
65#2352—31no—
66#2097—31no—
67#2085—31no—
68#2076—31no—
69#2035—31no—
70#1928—31no—
71#1920—31no—
72#1894—31no—
73#1872—31no—
74#1870D311yescharacteristic
75#1854—31no—
76#1854—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×Dic3—21no—
81C22×Dic6C2311yescharacteristic
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102#926—22no—
103#926—22no—
104#926—41no—
105#926—41no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—41no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116#924—22no—
117#924—22no—
118#924—22no—
119#924—41no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3—21no—
126C3×C22⋊C8—22no—
127C3×C22⋊C8—22no—
128C3×C22⋊C8—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#818—22no—
139#818—22no—
140#818—22no—
141#817—22no—
142#817—41no—
143#817D414yescomplemented
144#816—22no—
145#816—22no—
146#816—22no—
147#816—41no—
148C22×M16C2×D311yescharacteristic
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×D4C2×D311yescharacteristic
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4—31no—
155#485—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8C2×D311yescharacteristic
168C2×C22⋊C8C2×D311yescharacteristic
169C2×C22⋊C8C2×D311yescharacteristic
170#397—61no—
171#394—61no—
172#391—31no—
173#389—31no—
174#389—61no—
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C2×C3×D4—22no—
196C2×C3×D4—41no—
197C2×C3×D4—41no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4#4112yes—
213C22×C3×C4#4112yes—
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C3×M16—28no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C42—21no—
249C3×C42—21no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—81no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—42no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3—81no—
278C22×Q8—31no—
279C22×D4D1211yescharacteristic
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C23×C4C22×D311yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C22⋊Q8—121no—
294C4×D4—22no—
295C4×D4—22no—
296#117—62no—
297#117—62no—
298#117—121no—
299#117—121no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—121no—
313C2×C4⋊C4D1211yescharacteristic
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⋊C4D1211yescharacteristic
322C2×C22⋊C4D1211yescharacteristic
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—62no—
329#107—62no—
330#106—34no—
331#106—62no—
332#106—121no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—121no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—22no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3#12312yes—
363C23×C3C22×D411yescharacteristic
364C3×D4—44no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12312yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
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—84no—
416Dic6—41no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—22no—
421C3×C8—22no—
422C3×C8—41no—
423C3×C8—41no—
424C24C2×D1211yescharacteristic
425C24C2×D1211yescharacteristic
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4C2×D1211yescharacteristic
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4#23712yes—
461C22×C4#23712yes—
462C22×C4C23×D311yescharacteristic
463M16—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8D3×D412yes—
475C2×C8#23712yes—
476C4⋊C4—22no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—122no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—122no—
507C42—122no—
508C42—241no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3C2×#12311yescharacteristic
524C22×C3D4211yescharacteristic
525C22×C3#54711yescharacteristic
526C22×C3#57511yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57811yescharacteristic
530C3×C4—22no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4#52511yescharacteristic
535C3×C4#53911yescharacteristic
536C3×C4#54811yescharacteristic
537C3×C4#55311yescharacteristic
538Dic3—42no—
539Dic3—42no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23#88612yes—
560C23C22×D1211yescharacteristic
561Q8—121no—
562Q8—121no—
563Q8—121no—
564Q8—121no—
565D4—44no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—244no—
610C2×C4#88612yes—
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4#101011yescharacteristic
617C8—22no—
618C8—22no—
619C8—41no—
620C8—41no—
621C2×C3—42no—
622C2×C3—81no—
623C2×C3#244311yescharacteristic
624C2×C3#309311yescharacteristic
625C2×C3#309711yescharacteristic
626C2×C3#326911yescharacteristic
627C2×C3#327511yescharacteristic
628C2×C3#335011yescharacteristic
629C2×C3#335211yescharacteristic
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22C2×#88611yescharacteristic
645C22D4×D1211yescharacteristic
646C22#555511yescharacteristic
647C22#574811yescharacteristic
648C22#574811yescharacteristic
649C22#575111yescharacteristic
650C22#575311yescharacteristic
651C4—22no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—122no—
656C4—122no—
657C4—241no—
658C4—241no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#558611yescharacteristic
664C4#559811yescharacteristic
665C4#561811yescharacteristic
666C4#566511yescharacteristic
667C3#2615411yescharacteristic, complemented, Sylow
668C2—42no—
669C2—81no—
670C2#7470311yescharacteristic
671C2#7862311yescharacteristic
672C2#7865211yescharacteristic
673C2#7947611yescharacteristic
674C2#7949811yescharacteristic
675C2#7953011yescharacteristic
676C2#7955111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1870C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1870D3
hypocenterC3#26154
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
1. C2×#5121C2semidirect
4. C2×#4723C2semidirect
6. #71989C2semidirect
7. #69939C2semidirect
8. #69936C2semidirect
9. #69930C2semidirect
10. #69929C2semidirect
11. #69929C2semidirect
12. #68693C2semidirect
13. #68641C2semidirect
14. #68628C2semidirect
15. #68623C2semidirect
143. #817D4semidirect
667. C3#26154semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
#823512
#823352
C2×#47232
C3×#18702
#719892
#699392
#699362
#699302
#699292
#686932
#686412
#686282
#686232
#261543