Finite Group Databasebeta

← #10823592

Subgroups — #10823592

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5625C211yesmaximal, characteristic
2C2×#5184C211yesmaximal, characteristic, complemented
3#82744C211yesmaximal, characteristic
4C2×#4790C211yesmaximal, characteristic, complemented
5C3×#2001C211yesmaximal, characteristic
6#72073C211yesmaximal, characteristic, complemented
7#71700C211yesmaximal, characteristic
8#71695C211yesmaximal, characteristic
9#70246C211yesmaximal, characteristic, complemented
10#70232C211yesmaximal, characteristic, complemented
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69936C211yesmaximal, characteristic
13#69907C211yesmaximal, characteristic
14#68828C211yesmaximal, characteristic, complemented
15#68783C211yesmaximal, characteristic, complemented
16#23586—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#944C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26#5625—24no—
27#5625—24no—
28C2×#890—21no—
29C2×#890C2211yescharacteristic
30C2×#889—21no—
31C2×#887C2211yescharacteristic
32C2×#879—21no—
33C2×#879—21no—
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37#5229C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41#5212C2211yescharacteristic
42C2×#830C2211yescharacteristic
43#5184—24no—
44C2×#817C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#815C2211yescharacteristic
48C2×C8×Dic3C2211yescharacteristic
49#4984—21no—
50#4984C2211yescharacteristic
51#4980—21no—
52Dic3×C4⋊C4C2211yescharacteristic
53C2×#810C2211yescharacteristic
54C2×#807C2211yescharacteristic
55#4951—21no—
56#4951—21no—
57#4790—24no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#541—31no—
65C2×#441—31no—
66C2×C4×Q16—31no—
67#2323—31no—
68#2059—31no—
69#2043—31no—
70#2036—31no—
71#2035—31no—
72#2026—31no—
73#2016—31no—
74#2001—31no—
75#2001D311yescharacteristic
76#1763—31no—
77C4×#103—31no—
78#1750—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C2×C3×Q16—22no—
84C2×C3×Q16—22no—
85C2×C3×Q16—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101C2×#217D411yescharacteristic
102Q8×Dic3—22no—
103Q8×Dic3—22no—
104Q8×Dic3—22no—
105Q8×Dic3D414yes—
106#944—22no—
107#944—22no—
108C2×#216—22no—
109C2×#216—22no—
110C2×#216—22no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3D411yescharacteristic
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122#924—22no—
123#924—22no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C2×C4×Dic3—21no—
130C2×C4×Dic3—21no—
131C2×C4×Dic3—21no—
132C2×C4×Dic3—21no—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135C22×C3⋊C8C2311yescharacteristic
136C22×C3⋊C8C2311yescharacteristic
137#890—28no—
138#890—22no—
139#890—22no—
140#889—28no—
141#887—22no—
142#887D414yes—
143#879—28no—
144#879—28no—
145C3×#103—22no—
146C3×#103—22no—
147C3×C22⋊C8—22no—
148#835—22no—
149#835—22no—
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156#830—22no—
157#817—22no—
158#816—22no—
159#816—22no—
160#815—22no—
161C8×Dic3—22no—
162#810—22no—
163#807—22no—
164C22×Q16—31no—
165C22×Q16C2×D311yescharacteristic
166#541—64no—
167#541—64no—
168C2×C4⋊Q8—31no—
169C2×#126—61no—
170C2×#126—61no—
171C2×#125—31no—
172C2×#125—61no—
173C2×#125—61no—
174C2×C22⋊Q8—31no—
175C2×C22⋊Q8C2×D311yescharacteristic
176C2×C22⋊Q8C2×D311yescharacteristic
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×Q8—31no—
180C2×#117—31no—
181#441—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×#106—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103C2×D311yescharacteristic
192C2×#103C2×D311yescharacteristic
193C2×C22⋊C8—31no—
194C2×C22⋊C8C2×D311yescharacteristic
195C2×#97—31no—
196C2×C4×C8—31no—
197#391—31no—
198#389—31no—
199#389—61no—
200#388—61no—
201#388—61no—
202#387—61no—
203#380—31no—
204C4×C4⋊C4—31no—
205#340—31no—
206#340—31no—
207#340—31no—
208#340C2×D311yescharacteristic
209#336—31no—
210#336—31no—
211C24×C3C2×D411yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8C2×D412yes—
223C2×C3×Q8#4112yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6C2×D412yes—
250C3×Q16—28no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8#4112yes—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4C2×D412yes—
262C3×C4⋊C4#4112yes—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—24no—
265#217—24no—
266#217—22no—
267#217—22no—
268#217—81no—
269#216—28no—
270C4⋊Dic3—24no—
271C4⋊Dic3—21no—
272C4⋊Dic3—24no—
273C4⋊Dic3—22no—
274C4⋊Dic3—22no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—42no—
280C4⋊Dic3—81no—
281C4⋊Dic3C2×D412yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—42no—
292C4×Dic3—81no—
293C4×Dic3C2×D412yes—
294C4×Dic3C2×D412yes—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8#4112yes—
301C22×Q8—31no—
302C22×Q8C22×D311yescharacteristic
303C22×Q8C22×D311yescharacteristic
304C23×C4C22×D311yescharacteristic
305C2×Q16—22no—
306C2×Q16—22no—
307C2×Q16—22no—
308C2×Q16—62no—
309C2×Q16—62no—
310C2×Q16—62no—
311C22×C8—31no—
312C22×C8—31no—
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—62no—
316C4⋊Q8—62no—
317#126—68no—
318#126—68no—
319#125—62no—
320#125—68no—
321#125—68no—
322#125—62no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329C4×Q8—34no—
330C4×Q8—34no—
331C4×Q8—62no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×Q8—62no—
335#117—62no—
336#117—62no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—31no—
358C2×C22⋊C4—31no—
359C2×C22⋊C4—61no—
360C2×C22⋊C4—61no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—61no—
364C2×C42—61no—
365C2×C42—61no—
366C2×C42—61no—
367#107—62no—
368#106—62no—
369#103—22no—
370#103—22no—
371#103—62no—
372#103—62no—
373#103—62no—
374#103—62no—
375#103—62no—
376#103—62no—
377C22⋊C8—22no—
378C22⋊C8—62no—
379#97—62no—
380C4×C8—62no—
381#95—61no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386C23×C3—21no—
387C23×C3—21no—
388C23×C3—21no—
389C23×C3—21no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—21no—
393C23×C3C22×D411yescharacteristic
394C3×Q8—22no—
395C3×Q8—22no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—41no—
399C3×Q8—41no—
400C3×Q8—41no—
401C3×Q8—41no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—22no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3#12012yes—
450C2×Dic3#12012yes—
451Dic6—22no—
452Dic6—22no—
453Dic6—41no—
454Dic6—41no—
455C3×C8—22no—
456C3×C8—41no—
457C3⋊C8—22no—
458C3⋊C8—41no—
459C24D3×D411yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—32no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8D3×D412yes—
477C2×Q8#23712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—31no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4D3×D411yescharacteristic
497C22×C4C23×D311yescharacteristic
498Q16—28no—
499Q16—68no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—32no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8#23712yes—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—32no—
517C4⋊C4—62no—
518C4⋊C4—62no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—122no—
532C4⋊C4—241no—
533C4⋊C4D3×D412yes—
534C4⋊C4#23712yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—62no—
538C22⋊C4—62no—
539C22⋊C4—64no—
540C22⋊C4—241no—
541C42—32no—
542C42—32no—
543C42—64no—
544C42—64no—
545C42—61no—
546C42—61no—
547C42—122no—
548C42—241no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3C2×#12011yescharacteristic
557C22×C3#52511yescharacteristic
558C22×C3#53211yescharacteristic
559C22×C3#57711yescharacteristic
560C22×C3#57811yescharacteristic
561C22×C3#57811yescharacteristic
562C22×C3#57811yescharacteristic
563C3×C4—42no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4#52511yescharacteristic
569C3×C4#52511yescharacteristic
570C3×C4#53211yescharacteristic
571C3×C4#53211yescharacteristic
572Dic3—22no—
573Dic3—22no—
574Dic3—42no—
575Dic3—41no—
576Dic3—42no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23C2×D3×D411yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8—22no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—62no—
597Q8—62no—
598Q8—121no—
599Q8—121no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—32no—
608C2×C4—32no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—62no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4C2×D3×D411yescharacteristic
644C2×C4C2×D3×D411yescharacteristic
645C2×C4C2×#23711yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—22no—
650C8—41no—
651C8—62no—
652C8—121no—
653C2×C3—81no—
654C2×C3#238811yescharacteristic
655C2×C3#301111yescharacteristic
656C2×C3#301311yescharacteristic
657C2×C3#312011yescharacteristic
658C2×C3#312711yescharacteristic
659C2×C3#317711yescharacteristic
660C2×C3#317811yescharacteristic
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#558611yescharacteristic
669C22D3×#12011yescharacteristic
670C22#559611yescharacteristic
671C22#576611yescharacteristic
672C22#577011yescharacteristic
673C22#577911yescharacteristic
674C22#577911yescharacteristic
675C4—42no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—62no—
681C4—62no—
682C4—121no—
683C4—122no—
684C4—121no—
685C4—122no—
686C4—241no—
687C4—241no—
688C4#558611yescharacteristic
689C4#559611yescharacteristic
690C4#560611yescharacteristic
691C4#571611yescharacteristic
692C3#2358611yescharacteristic, complemented, Sylow
693C2—81no—
694C2#7666511yescharacteristic
695C2#8017711yescharacteristic
696C2#8018511yescharacteristic
697C2#8021611yescharacteristic
698C2#8023211yescharacteristic
699C2#8074411yescharacteristic
700C2#8089011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2001D3
hypocenterC3#23586
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
2. C2×#5184C2semidirect
4. C2×#4790C2semidirect
6. #72073C2semidirect
9. #70246C2semidirect
10. #70232C2semidirect
11. Dic3×#103C2semidirect
14. #68828C2semidirect
15. #68783C2semidirect
692. C3#23586semidirect

Maximal subgroups

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

subgroupindex
C2×#56252
C2×#51842
#827442
C2×#47902
C3×#20012
#720732
#717002
#716952
#702462
#702322
Dic3×#1032
#699362
#699072
#688282
#687832
#235863