Finite Group Databasebeta

← #10824630

Subgroups — #10824630

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic, complemented
3C4×#924C211yesmaximal, characteristic
4C2×#4925C211yesmaximal, characteristic
5#71700C211yesmaximal, characteristic
6#71694C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8#69682C211yesmaximal, characteristic, complemented
9#69681C211yesmaximal, characteristic, complemented
10#69634C211yesmaximal, characteristic, complemented
11#69622C211yesmaximal, characteristic, complemented
12#69526C211yesmaximal, characteristic
13#69518C211yesmaximal, characteristic
14C4×#808C211yesmaximal, characteristic, complemented
15C4×#810C211yesmaximal, characteristic, complemented
16#23699—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#385—21no—
27C3×#380C2211yescharacteristic
28C3×C4×C4⋊C4—21no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4—21no—
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5212C2211yescharacteristic
35#5210—21no—
36C4×#217—21no—
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C4×#216—24no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4933—21no—
48C4×Dic3⋊C4—21no—
49#4929C2211yescharacteristic
50#4928C2211yescharacteristic
51#4925—24no—
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×C22⋊Q8D311yescharacteristic
66#2086—31no—
67#2075—31no—
68#2066—31no—
69#2066—31no—
70#2056—31no—
71#2035—31no—
72#2035—31no—
73#2001—31no—
74C4×#106—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4D411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103C2×#217—21no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—22no—
111#924—22no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#810—22no—
139#810—22no—
140#810—22no—
141#808—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146C4×C3⋊C8—22no—
147C4×C3⋊C8D414yes—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C22×C42C2×D311yescharacteristic
154C4×Q16—64no—
155#431—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#389—31no—
174#389C2×D311yescharacteristic
175#387—21no—
176#387—61no—
177#385—21no—
178#385—61no—
179#380—31no—
180#380C2×D311yescharacteristic
181C4×C4⋊C4—21no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185C4×C22⋊C4—21no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4#4112yes—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—42no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42—42no—
249C3×C42C2×D412yes—
250C3×C42#4112yes—
251#217—24no—
252#217—42no—
253#216—28no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263C4×Dic3—42no—
264C4×Dic3—42no—
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⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8C2×D412yes—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C23×C4#6711yescharacteristic
280C23×C4#6711yescharacteristic
281C23×C4C22×D311yescharacteristic
282C2×Q16—62no—
283C2×Q16—62no—
284C2×Q16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—62no—
296C4×Q8—62no—
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⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42#6711yescharacteristic
320C2×C42#6711yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325#106—62no—
326C4⋊C8—62no—
327C4⋊C8—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334C22⋊C8—62no—
335C22⋊C8—62no—
336C4×C8—34no—
337C4×C8—62no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3#13512yes—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—28no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
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—44no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#12112yes—
389C2×C3×C4#12112yes—
390C2×C3×C4#13512yes—
391C2×C3×C4#13712yes—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×Dic3—22no—
400C2×Dic3—22no—
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—
411Dic6—22no—
412Dic6—22no—
413Dic6—41no—
414Dic6—41no—
415C3⋊C8—22no—
416C3⋊C8—22no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C24C2×#6711yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8#23512yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459C2×C8—32no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—64no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4#23512yes—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—64no—
492C22⋊C4—122no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—42no—
499C42—122no—
500C42—122no—
501C42D3×D412yes—
502C42#23712yes—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12111yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3C2×#13511yescharacteristic
514C22×C3C2×#13511yescharacteristic
515C22×C3C2×#13711yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4#47014yes—
526C3×C4#52511yescharacteristic
527C3×C4#52911yescharacteristic
528C3×C4#52911yescharacteristic
529C3×C4#53211yescharacteristic
530Dic3—42no—
531Dic3—42no—
532Dic3—81no—
533Dic3—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#95012yes—
541C23C22×#6711yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—62no—
579C2×C4—62no—
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—121no—
589C2×C4—121no—
590C2×C4#93912yes—
591C2×C4#93912yes—
592C2×C4#94212yes—
593C2×C4#95012yes—
594C2×C4#95112yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—62no—
602C8—62no—
603C8—121no—
604C8—121no—
605C2×C3—42no—
606C2×C3#239211yescharacteristic
607C2×C3#305011yescharacteristic
608C2×C3C2×#47011yescharacteristic
609C2×C3#310711yescharacteristic
610C2×C3#310711yescharacteristic
611C2×C3#310811yescharacteristic
612C2×C3#310811yescharacteristic
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22#579111yescharacteristic
621C22C2×#93911yescharacteristic
622C22C2×#94211yescharacteristic
623C22C2×#95011yescharacteristic
624C22C2×#95011yescharacteristic
625C22C2×#95111yescharacteristic
626C22#582811yescharacteristic
627C4—22no—
628C4—22no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—122no—
636C4—122no—
637C4—241no—
638C4—241no—
639C4#505214yes—
640C4#553811yescharacteristic
641C4#554711yescharacteristic
642C4#554811yescharacteristic
643C4#555711yescharacteristic
644C3#2369911yescharacteristic, complemented, Sylow
645C2—42no—
646C2#8234511yescharacteristic
647C2#8244311yescharacteristic
648C2#8253411yescharacteristic
649C2C2×#505211yescharacteristic
650C2#8292211yescharacteristic
651C2#8315211yescharacteristic
652C2#8335211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC4×C22⋊Q8D3
hypocenterC3#23699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C4×#216C2semidirect
7. C4×#835C2semidirect
8. #69682C2semidirect
9. #69681C2semidirect
10. #69634C2semidirect
11. #69622C2semidirect
14. C4×#808C2semidirect
15. C4×#810C2semidirect
644. C3#23699semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×C4×#2162
C4×#9242
C2×#49252
#717002
#716942
C4×#8352
#696822
#696812
#696342
#696222
#695262
#695182
C4×#8082
C4×#8102
#236993