Finite Group Databasebeta

← #10944020

Subgroups — #10944020

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×#5090C211yesmaximal, characteristic, complemented
4C2×#5083C211yesmaximal, characteristic, complemented
5C2×#5047C212yesmaximal
6#71693C212yesmaximal, complemented
7#71530C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69687C212yesmaximal, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#32907—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#216C2212yes—
18C22×#215C2212yes—
19C2×C4×#67—21no—
20C2×C4×#67—21no—
21C2×#926C2211yescharacteristic
22C4×#235—24no—
23C4×#235—24no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#872—21no—
27C2×C42×D3—21no—
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#391C2212yes—
31C3×#389—21no—
32C3×#389—21no—
33C3×#384C2211yescharacteristic
34C3×#380—21no—
35C3×#380—21no—
36C2×#835C2212yes—
37C2×#835C2212yes—
38C2×#830C2212yes—
39#5090—24no—
40#5083—24no—
41#5047—28no—
42C2×#810C2212yes—
43C2×#809C2212yes—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×C4×#41—31no—
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2857D311yescharacteristic, complemented
54#1878—32no—
55#1877—32no—
56#1873—32no—
57#1803—31no—
58#1802—31no—
59#1797—31no—
60C22×C3×Q8C2312yes—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235D414yes—
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×C4×D3—21no—
74C22×C4×D3—21no—
75C22×Dic6C2311yescharacteristic
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#216—24no—
97C2×#215—24no—
98C2×#215—24no—
99C2×#215—24no—
100C4×#67—28no—
101C4×#67—28no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104#926—22no—
105#926—22no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115C4×D12—22no—
116C4×D12—22no—
117C4×D12—22no—
118C4×D12D414yes—
119#872—28no—
120C42×D3—28no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6D414yes—
125C3×#95—22no—
126C3×#95—22no—
127#835—24no—
128#835—24no—
129#830—24no—
130#810—24no—
131#809—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×C22⋊Q8C2×D312yes—
139C4×#41—64no—
140C4×#41—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—31no—
148C2×#117—61no—
149C22×C42—61no—
150C22×C42C2×D311yescharacteristic
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#391C2×D312yes—
162#389—21no—
163#389—21no—
164#384C2×D311yescharacteristic
165#380—21no—
166#380—21no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3C2×D411yescharacteristic
171C23×D3—21no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—41no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—24no—
200C2×#67—24no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204#235—28no—
205#235—28no—
206#235—28no—
207#235—28no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C2×D412yes—
212C2×D12C2×D412yes—
213C2×D12#4112yes—
214C2×C4×D3—24no—
215C2×C4×D3—24no—
216C2×C4×D3—24no—
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×D412yes—
227C2×Dic6C2×D412yes—
228C2×Dic6#4112yes—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—44no—
235C3×C4⋊C4—42no—
236C3×C22⋊C4—44no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243#217—24no—
244#216—48no—
245#215—48no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3C2×D412yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254C4×Dic3—24no—
255C4×Dic3—24no—
256C4×Dic3—24no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C22×Q8—31no—
270C22×Q8C22×D312yes—
271C22×D4—31no—
272C22×D4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4—61no—
276C23×C4#6711yescharacteristic
277C23×C4#6711yescharacteristic
278C23×C4C22×D311yescharacteristic
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C22×C8—32no—
286C22×C8—32no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—34no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—68no—
299C4×D4—68no—
300C4×D4—62no—
301#117—68no—
302#117—62no—
303#117—62no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D312yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—68no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42#6711yescharacteristic
325C2×C42#6711yescharacteristic
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C4⋊C8—62no—
329C4⋊C8—62no—
330#103—64no—
331#103—64no—
332#103—64no—
333#102—64no—
334C22⋊C8—64no—
335#95—22no—
336#95—22no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3#12112yes—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—42no—
353C3×Q8—42no—
354C3×Q8—42no—
355C3×Q8—42no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—44no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4#12012yes—
378C2×C3×C4#12012yes—
379C2×C3×C4#12112yes—
380C2×C3×C4#13512yes—
381C2×C3×C4#13712yes—
382C2×C3×C4#13712yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389#67—216no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399D12—22no—
400D12—22no—
401D12—22no—
402D12#12114yes—
403C4×D3—216no—
404C4×D3—48no—
405C4×D3—48no—
406Dic6—22no—
407Dic6—22no—
408Dic6—22no—
409Dic6#12114yes—
410C3⋊C8—42no—
411C3⋊C8—42no—
412C24—61no—
413C24C2×#6711yescharacteristic
414#41—68no—
415#41—68no—
416#41—68no—
417#41—68no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—32no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—32no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—64no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—64no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4D3×D412yes—
464C22×C4#23712yes—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C23×D311yescharacteristic
471Q16—128no—
472SD16—128no—
473C2×C8—62no—
474C2×C8—62no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—32no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C22⋊C4—44no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—21no—
500C42—64no—
501C42—64no—
502C42—64no—
503C42D3×D412yes—
504C42D3×D412yes—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3C2×#13511yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C2×D3—24no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C3×C4—22no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—82no—
531C3×C4—82no—
532C3×C4#45214yescomplemented
533C3×C4C2×#12111yescharacteristic
534C3×C4C2×#12111yescharacteristic
535C3×C4#52511yescharacteristic
536C3×C4#52511yescharacteristic
537Dic3—24no—
538Dic3—42no—
539Dic3—42no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23#93812yes—
554C23C22×#6711yescharacteristic
555Q8—34no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563D4—34no—
564D4—616no—
565D4—62no—
566D4—62no—
567D4—62no—
568C2×C4—21no—
569C2×C4—21no—
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—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—616no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—128no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—128no—
601C2×C4#93712yes—
602C2×C4#93712yes—
603C2×C4#93812yes—
604C2×C4#94212yes—
605C2×C4#94212yes—
606C2×C4#95012yes—
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—122no—
614C8—122no—
615C2×C3—42no—
616C2×C3#284711yescharacteristic
617C2×C3C2×#45211yescharacteristic
618C2×C3#300711yescharacteristic
619C2×C3#303511yescharacteristic
620C2×C3#303911yescharacteristic
621C2×C3#310711yescharacteristic
622C2×C3#311611yescharacteristic
623D3—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—64no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22C2×#93711yescharacteristic
639C22C2×#93811yescharacteristic
640C22C2×#94211yescharacteristic
641C22C2×#94211yescharacteristic
642C22C2×#95011yescharacteristic
643C22#582211yescharacteristic
644C22#582811yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—64no—
650C4—82no—
651C4—82no—
652C4—122no—
653C4—122no—
654C4#504714yescomplemented
655C4D3×#12111yescharacteristic
656C4D3×#12111yescharacteristic
657C4#560611yescharacteristic
658C4#560611yescharacteristic
659C3#3290711yescharacteristic, complemented, Sylow
660C2—42no—
661C2—124no—
662C2#8272311yescharacteristic
663C2C2×#504711yescharacteristic
664C2#8273511yescharacteristic
665C2#8320911yescharacteristic
666C2#8329611yescharacteristic
667C2#8335211yescharacteristic
668C2#8335411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2857D3
hypocenterC3#32907
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. C2×#5090C2semidirect
4. C2×#5083C2semidirect
6. #71693C2semidirect
7. #71530C2semidirect
8. #69690C2semidirect
9. #69687C2semidirect
10. #69588C2semidirect
11. #69584C2semidirect
53. #2857D3semidirect
532. C3×C4#452semidirect
654. C4#5047semidirect
659. C3#32907semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C3×#28572
C2×#50902
C2×#50832
C2×#50472
#716932
#715302
#696902
#696872
#695882
#695842
#329073