Finite Group Databasebeta

← #10813607

Subgroups — #10813607

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82334C211yesmaximal, characteristic
2C2×C4×#214C211yesmaximal, characteristic
3C2×#4758C211yesmaximal, characteristic
4C2×#4758C211yesmaximal, characteristic
5#75982C211yesmaximal, characteristic, complemented
6C3×#1862C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic, complemented
8#70607C211yesmaximal, characteristic, complemented
9#70168C211yesmaximal, characteristic
10#70154C211yesmaximal, characteristic
11#69618C211yesmaximal, characteristic, complemented
12#68905C211yesmaximal, characteristic, complemented
13#68858C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68651C211yesmaximal, characteristic, complemented
16#22683—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5210—21no—
30C2×C3⋊#95—21no—
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37C4×#214—24no—
38#4984C2211yescharacteristic
39#4977C2211yescharacteristic
40#4977C2211yescharacteristic
41#4976—21no—
42Dic3×C4⋊C4—21no—
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4946—21no—
49#4942—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4758—24no—
54#4758C418yescomplemented
55#4650—21no—
56#4647—21no—
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×C4×SD16—31no—
64C2×C4×SD16—31no—
65C2×C4×SD16—31no—
66#2351—31no—
67#2349—31no—
68#1941—31no—
69#1941—31no—
70#1863—31no—
71#1863—31no—
72#1862—31no—
73#1862D311yescharacteristic
74#1846—31no—
75#1805—31no—
76#1804—31no—
77C8×C4⋊C4—31no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97D4×Dic3—41no—
98D4×Dic3C2×C414yes—
99C2×#214C2×C414yes—
100C2×#214D414yes—
101C2×#214Q814yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#887—22no—
121#887—22no—
122#887—41no—
123#887C2×C414yescomplemented
124C4×Dic6—22no—
125C4×Dic6—41no—
126C3×#107—41no—
127C3×#102—22no—
128C3×#102C2×C414yes—
129C3×#95—21no—
130#832—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#818—41no—
141#816—22no—
142#816C2×C414yescomplemented
143Dic3⋊C8—22no—
144C8×Dic3—22no—
145C8×Dic3C2×C414yescomplemented
146#810—22no—
147#808—22no—
148#808—22no—
149#808C2×C414yescomplemented
150C4×C3⋊C8—22no—
151C22×SD16—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157C2×C4×D4C2×D311yescharacteristic
158C4×SD16—38no—
159C4×SD16—64no—
160C4×SD16—64no—
161C2×#107—31no—
162C2×#107C2×D311yescharacteristic
163C2×#106—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#102—31no—
171C2×#102C2×D311yescharacteristic
172C2×#102C2×D311yescharacteristic
173C2×C4×C8—31no—
174C2×C4×C8—31no—
175C2×C4×C8—31no—
176#389—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#389—61no—
181#389C2×D311yescharacteristic
182#385—61no—
183#385—61no—
184#382—31no—
185#382—31no—
186#382—61no—
187#380—61no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C2×#95—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336C2×D311yescharacteristic
197C24×C3—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C22×C412yes—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×Q812yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—81no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—24no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—81no—
221C22×Dic3—81no—
222C22×Dic3—81no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C22×C412yes—
231C2×Dic6C2×D412yes—
232C2×Dic6C2×Q812yes—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C22×C412yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4C22×C412yes—
246C3×C22⋊C4—42no—
247C3×C42—21no—
248C3×C42—21no—
249#217—24no—
250#217—42no—
251#214C4⋊C4116yescomplemented
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3C22×C412yes—
259Dic3⋊C4—22no—
260Dic3⋊C4—24no—
261Dic3⋊C4—22no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—24no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3C22×C412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C22×C412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8C2×Q812yes—
279C22×Q8—31no—
280C22×D4C22×D311yescharacteristic
281C23×C4—21no—
282C23×C4—61no—
283C23×C4—61no—
284C2×SD16—34no—
285C2×SD16—34no—
286C2×SD16—34no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—121no—
296C4×Q8—121no—
297C4×D4—22no—
298C4×D4—34no—
299C4×D4—41no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—121no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—41no—
328#107—121no—
329#106—34no—
330#106—62no—
331#106—62no—
332C4⋊C8—62no—
333#103—34no—
334#103—62no—
335#103—62no—
336#102—22no—
337#102—62no—
338#102C4×D314yes—
339C4×C8—34no—
340C4×C8—62no—
341C4×C8—62no—
342#95—21no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4C2×C4⋊C414yes—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—84no—
379C2×C3×C4#13512yes—
380C2×C3×C4C23×C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
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—84no—
408C2×Dic3—84no—
409C2×Dic3C4×D412yes—
410C2×Dic3C4×D412yes—
411C2×Dic3#13512yes—
412Dic6—22no—
413Dic6—22no—
414Dic6—22no—
415Dic6C2×C4⋊C414yes—
416C3×C8—22no—
417C3×C8—41no—
418C3⋊C8—41no—
419C3⋊C8C2×C4⋊C414yes—
420C24—21no—
421C2×Q8—32no—
422C2×Q8—32no—
423C2×Q8—32no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D312yes—
431C2×D4D3×D412yes—
432C2×D4D3×Q812yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—81no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4—241no—
452C22×C4—241no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—316no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8C2×C4×D312yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—32no—
475C4⋊C4—42no—
476C4⋊C4—62no—
477C4⋊C4—64no—
478C4⋊C4—62no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—241no—
488C4⋊C4—241no—
489C4⋊C4C2×C4×D312yes—
490C22⋊C4—42no—
491C22⋊C4—64no—
492C22⋊C4—122no—
493C42—21no—
494C42—21no—
495C42—32no—
496C42—32no—
497C42—64no—
498C42—61no—
499C42—61no—
500C42—122no—
501C42—122no—
502C22×C3—24no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3C2×C4×D411yescharacteristic
510C22×C3D4211yescharacteristic
511C22×C3D4×Q811yescharacteristic
512C22×C3C2×#13511yescharacteristic
513C22×C3C2×#13511yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57711yescharacteristic
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4C22×C4⋊C411yescharacteristic
522C3×C4#52011yescharacteristic
523C3×C4#54711yescharacteristic
524C3×C4#55411yescharacteristic
525Dic3—24no—
526Dic3—22no—
527Dic3—22no—
528Dic3—22no—
529Dic3—42no—
530Dic3—41no—
531Dic3—81no—
532Dic3—81no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×D3×D411yescharacteristic
541Q8—34no—
542Q8—62no—
543Q8—62no—
544Q8—62no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4D3×C4⋊C414yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—84no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—244no—
589C2×C4—244no—
590C2×C4#91212yes—
591C2×C4C22×C4×D311yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4#100411yescharacteristic
594C2×C4C2×D3×Q811yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#101011yescharacteristic
597C8—22no—
598C8—34no—
599C8—41no—
600C8—121no—
601C2×C3—44no—
602C2×C3D4×C4⋊C411yescharacteristic
603C2×C3C4×#13511yescharacteristic
604C2×C3#294911yescharacteristic
605C2×C3#328211yescharacteristic
606C2×C3#328211yescharacteristic
607C2×C3#338111yescharacteristic
608C2×C3#338111yescharacteristic
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22C4×D3×D411yescharacteristic
617C22#559011yescharacteristic
618C22#563211yescharacteristic
619C22C2×#91211yescharacteristic
620C22D3×#13511yescharacteristic
621C22#577511yescharacteristic
622C22#577811yescharacteristic
623C4—22no—
624C4—41no—
625C4—42no—
626C4—62no—
627C4—64no—
628C4—62no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—121no—
633C4—122no—
634C4—241no—
635C4—241no—
636C4C2×D3×C4⋊C411yescharacteristic
637C4#554611yescharacteristic
638C4#563611yescharacteristic
639C4#565511yescharacteristic
640C3#2268311yescharacteristic, complemented, Sylow
641C2—44no—
642C2#7439311yescharacteristic
643C2C4×#91211yescharacteristic
644C2#7977711yescharacteristic
645C2#8002711yescharacteristic
646C2#8024411yescharacteristic
647C2#8095811yescharacteristic
648C2#8115711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1862C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1862D3
hypocenterC3#22683
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
5. #75982C2semidirect
7. #70642C2semidirect
8. #70607C2semidirect
11. #69618C2semidirect
12. #68905C2semidirect
13. #68858C2semidirect
14. #68683C2semidirect
15. #68651C2semidirect
54. #4758C4semidirect
123. #887C2×C4semidirect
142. #816C2×C4semidirect
145. C8×Dic3C2×C4semidirect
149. #808C2×C4semidirect
251. #214C4⋊C4semidirect
640. C3#22683semidirect

Maximal subgroups

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

subgroupindex
#823342
C2×C4×#2142
C2×#47582
#759822
C3×#18622
#706422
#706072
#701682
#701542
#696182
#689052
#688582
#686832
#686512
#226833