Finite Group Databasebeta

← #10855117

Subgroups — #10855117

Order 768.

854 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×#5121C211yesmaximal, characteristic, complemented
2#82353C211yesmaximal, characteristic
3#82337C211yesmaximal, characteristic
4C2×#4723C211yesmaximal, characteristic, complemented
5C3×#1870C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic, complemented
7#69940C211yesmaximal, characteristic, complemented
8#69936C211yesmaximal, characteristic, complemented
9#69931C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic, complemented
11#69928C211yesmaximal, characteristic, complemented
12#68692C211yesmaximal, characteristic, complemented
13#68640C211yesmaximal, characteristic, complemented
14#68628C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#26155—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#5221—21no—
28#5220—21no—
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2211yescharacteristic
32#5121—24no—
33C2×#818C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#4982—21no—
44Dic3×C4⋊C4—21no—
45#4951—21no—
46#4951—21no—
47#4948—21no—
48#4942—21no—
49#4935C2211yescharacteristic
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#2417—31no—
65#2354—31no—
66#2096—31no—
67#2084—31no—
68#2077—31no—
69#2035—31no—
70#1928—31no—
71#1921—31no—
72#1894—31no—
73#1873—31no—
74#1870D311yescharacteristic
75#1854—31no—
76#1853—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—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118#924—22no—
119#924—22no—
120#924—22no—
121#924—41no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C3×C22⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
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#818D414yescomplemented
140#817—22no—
141#817—22no—
142#817—22no—
143#817—41no—
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×#106—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#398—61no—
171#396—61no—
172#391—31no—
173#390—61no—
174#389—31no—
175#388—61no—
176#388—61no—
177#386—61no—
178#384—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—61no—
183C4×C4⋊C4—31no—
184C4×C4⋊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—22no—
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—81no—
255#217—81no—
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⋊Dic3—42no—
271C4⋊Dic3—81no—
272C4⋊Dic3C2×D412yes—
273C4⋊Dic3C2×D412yes—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—81no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280C22×Q8—31no—
281C22×D4D1211yescharacteristic
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22⋊Q8—62no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—121no—
296C4×D4—22no—
297C4×D4—22no—
298#117—62no—
299#117—62no—
300#117—121no—
301#117—121no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—121no—
317C2×C4⋊C4D1211yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4D1211yescharacteristic
326C2×C22⋊C4D1211yescharacteristic
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—34no—
332#107—62no—
333#106—62no—
334#106—62no—
335#106—62no—
336#106—121no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—121no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
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—21no—
363C23×C3—21no—
364C23×C3#12312yes—
365C23×C3C22×D411yescharacteristic
366C3×D4—44no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12312yes—
386C2×C3×C4#13712yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—22no—
394C2×Dic3—22no—
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—41no—
416C2×Dic3—41no—
417C2×Dic3—84no—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422C3×C8—22no—
423C3×C8—22no—
424C3×C8—41no—
425C3×C8—41no—
426C24C2×D1211yescharacteristic
427C24C2×D1211yescharacteristic
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—41no—
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—21no—
446C22×C4—21no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4#23712yes—
463C22×C4#23712yes—
464C22×C4C23×D311yescharacteristic
465M16—28no—
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×C8—21no—
475C2×C8—21no—
476C2×C8D3×D412yes—
477C2×C8#23712yes—
478C4⋊C4—22no—
479C4⋊C4—32no—
480C4⋊C4—32no—
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—61no—
492C4⋊C4—61no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—21no—
509C42—21no—
510C42—122no—
511C42—122no—
512C42—122no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3C2×#12311yescharacteristic
528C22×C3D4211yescharacteristic
529C22×C3#54711yescharacteristic
530C22×C3C2×#13711yescharacteristic
531C22×C3#57511yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57711yescharacteristic
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4#52511yescharacteristic
539C3×C4#53911yescharacteristic
540C3×C4#54811yescharacteristic
541C3×C4#55311yescharacteristic
542Dic3—42no—
543Dic3—42no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23#88612yes—
564C23C22×D1211yescharacteristic
565Q8—121no—
566Q8—121no—
567Q8—121no—
568Q8—121no—
569D4—44no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—62no—
589C2×C4—62no—
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—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—244no—
613C2×C4#88612yes—
614C2×C4#90912yes—
615C2×C4C2×D3×D411yescharacteristic
616C2×C4C2×#23711yescharacteristic
617C2×C4C2×#23711yescharacteristic
618C2×C4C2×#23711yescharacteristic
619C2×C4#100911yescharacteristic
620C2×C4#101011yescharacteristic
621C8—22no—
622C8—22no—
623C8—41no—
624C8—41no—
625C2×C3—42no—
626C2×C3—81no—
627C2×C3#244311yescharacteristic
628C2×C3#309511yescharacteristic
629C2×C3#309911yescharacteristic
630C2×C3#326711yescharacteristic
631C2×C3#327811yescharacteristic
632C2×C3#334811yescharacteristic
633C2×C3#335611yescharacteristic
634C22—22no—
635C22—22no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22C2×#88611yescharacteristic
649C22D4×D1211yescharacteristic
650C22#555511yescharacteristic
651C22C2×#90911yescharacteristic
652C22#574811yescharacteristic
653C22#575111yescharacteristic
654C22#575111yescharacteristic
655C4—22no—
656C4—22no—
657C4—41no—
658C4—41no—
659C4—122no—
660C4—122no—
661C4—241no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4#558611yescharacteristic
668C4#559811yescharacteristic
669C4#561811yescharacteristic
670C4#566511yescharacteristic
671C3#2615511yescharacteristic, complemented, Sylow
672C2—42no—
673C2—81no—
674C2#7470311yescharacteristic
675C2#7862511yescharacteristic
676C2#7865411yescharacteristic
677C2#7947411yescharacteristic
678C2#7949611yescharacteristic
679C2#7953411yescharacteristic
680C2#7956011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1870C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1870D3
hypocenterC3#26155
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. #69940C2semidirect
8. #69936C2semidirect
9. #69931C2semidirect
10. #69929C2semidirect
11. #69928C2semidirect
12. #68692C2semidirect
13. #68640C2semidirect
14. #68628C2semidirect
15. #68624C2semidirect
139. #818D4semidirect
671. C3#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
#823532
#823372
C2×#47232
C3×#18702
#719892
#699402
#699362
#699312
#699292
#699282
#686922
#686402
#686282
#686242
#261553