Finite Group Databasebeta

← #10858303

Subgroups — #10858303

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#5022C211yesmaximal, characteristic, complemented
3C2×C4×#215C211yesmaximal, characteristic
4#73677C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71633C211yesmaximal, characteristic
7#71628C211yesmaximal, characteristic, complemented
8#71628C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic
10#69626C211yesmaximal, characteristic
11#69619C211yesmaximal, characteristic, complemented
12#69606C211yesmaximal, characteristic, complemented
13#69586C211yesmaximal, characteristic
14#69570C211yesmaximal, characteristic, complemented
15#69482C211yesmaximal, characteristic, complemented
16#26379—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#389—21no—
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#384—21no—
28C3×#384C2211yescharacteristic
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C4×#215—24no—
35#5022—24no—
36#4993—21no—
37#4989—21no—
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4939C2211yescharacteristic
47#4939C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4676—21no—
53#4667—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×C4×SD16—31no—
64#2454—31no—
65Q8×C4⋊C4D311yescharacteristic, complemented
66#2093—31no—
67#2084—31no—
68#2067—31no—
69#2059—31no—
70#1942—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1864—31no—
75#1863—31no—
76#1863—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—41no—
85C3×C4×Q8—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—22no—
105C2×#215—22no—
106C2×#215Q814yes—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×D12—22no—
124C4×D12—41no—
125#870—22no—
126#870—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131#835—22no—
132#835—22no—
133#835D414yes—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#809—22no—
137#809—22no—
138#809—41no—
139#808—22no—
140#808—22no—
141#807—22no—
142#807—22no—
143#807—41no—
144C4⋊(C3⋊C8)—22no—
145C4×C3⋊C8—22no—
146C22×SD16—31no—
147C2×#125—31no—
148C2×C4×Q8C2×D311yescharacteristic
149C2×C4×Q8C2×D311yescharacteristic
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×D4—31no—
152#479—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8—31no—
167#399—61no—
168#396—61no—
169#390—31no—
170#390—31no—
171#389—21no—
172#389—21no—
173#389—31no—
174#389—31no—
175#389C2×D311yescharacteristic
176#389C2×D311yescharacteristic
177#385—61no—
178#384—21no—
179#384C2×D311yescharacteristic
180#380—61no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2×D311yescharacteristic
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191C23×D3—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8C2×D412yes—
196C2×C3×Q8C2×Q812yes—
197C2×C3×Q8#4112yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12C2×Q812yes—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—81no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4#4112yes—
236C3×C42—24no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—42no—
244C3×C42—42no—
245#215—28no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D412yes—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8C2×Q812yes—
273C22×Q8C22×D311yescharacteristic
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283#125—62no—
284#125—121no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—41no—
290C4×Q8—41no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
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⋊C4#6711yescharacteristic
308C2×C4⋊C4#6711yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#107—62no—
325#107—121no—
326#106—62no—
327#106—62no—
328C4⋊C8—62no—
329#103—34no—
330#103—62no—
331#103—62no—
332#102—62no—
333#102—62no—
334#102—121no—
335C4×C8—62no—
336#95—21no—
337#95—21no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8C22⋊Q814yes—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4#13512yes—
383C2×C3×C4#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×Q811yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399D12—22no—
400D12—22no—
401D12—41no—
402D12—41no—
403C4×D3—84no—
404C3⋊C8—22no—
405C3⋊C8—22no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C24—61no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8#23512yes—
413C2×Q8D3×D412yes—
414C2×Q8D3×Q812yes—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C23×D311yescharacteristic
438SD16—68no—
439C2×C8—32no—
440C2×C8—32no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—32no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—241no—
474C4⋊C4—241no—
475C4⋊C4#23512yes—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—42no—
490C22×C3#52511yescharacteristic
491C22×C3D4211yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57611yescharacteristic
496C22×C3#57711yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C3×C4—22no—
505C3×C4—24no—
506C3×C4—22no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4C2×C22⋊Q811yescharacteristic
515C3×C4#53911yescharacteristic
516C3×C4#54711yescharacteristic
517C3×C4#55411yescharacteristic
518Dic3—42no—
519Dic3—81no—
520Dic3—81no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23C22×#6711yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8#89614yes—
533D4—62no—
534D4—62no—
535D4—121no—
536D4—121no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—62no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—244no—
575C2×C4#94212yes—
576C2×C4#95012yes—
577C2×C4C2×#23511yescharacteristic
578C2×C4C2×#23511yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4#100411yescharacteristic
581C2×C4C2×D3×Q811yescharacteristic
582C2×C4#100711yescharacteristic
583C8—62no—
584C8—62no—
585C8—121no—
586C8—121no—
587C2×C3#244911yescharacteristic
588C2×C3#310811yescharacteristic
589C2×C3#311911yescharacteristic
590C2×C3#326611yescharacteristic
591C2×C3#328711yescharacteristic
592C2×C3#338111yescharacteristic
593C2×C3#338511yescharacteristic
594D3—82no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#579111yescharacteristic
603C22D4×#6711yescharacteristic
604C22C2×#94211yescharacteristic
605C22Q8×#6711yescharacteristic
606C22C2×#95011yescharacteristic
607C22#583511yescharacteristic
608C22#583711yescharacteristic
609C4—22no—
610C4—22no—
611C4—24no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—122no—
620C4—241no—
621C4—241no—
622C4#550711yescharacteristic
623C4C2×#89611yescharacteristic
624C4#551111yescharacteristic
625C4#551511yescharacteristic
626C3#2637911yescharacteristic, complemented, Sylow
627C2—242no—
628C2#8217311yescharacteristic
629C2#8244311yescharacteristic
630C2#8256811yescharacteristic
631C2#8355911yescharacteristic
632C2#8358511yescharacteristic
633C2#8369311yescharacteristic
634C2#8370611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×Q8×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterQ8×C4⋊C4D3
hypocenterC3#26379
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C4⋊C4C2semidirect
2. C2×#5022C2semidirect
7. #71628C2semidirect
8. #71628C2semidirect
11. #69619C2semidirect
12. #69606C2semidirect
14. #69570C2semidirect
15. #69482C2semidirect
65. Q8×C4⋊C4D3semidirect
626. C3#26379semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C4⋊C42
C2×#50222
C2×C4×#2152
#736772
#716332
#716282
#696302
#696262
#696192
#696062
#695862
#695702
#694822
#263793