Finite Group Databasebeta

← #10972891

Subgroups — #10972891

Order 768.

2230 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic
3C3×#2855C211yesmaximal, characteristic
4#82810C211yesmaximal, characteristic
5#82075C211yesmaximal, characteristic
6#76547C212yesmaximal, complemented
7#76545C212yesmaximal, complemented
8#76076C212yesmaximal, complemented
9#76060C212yesmaximal, complemented
10#75135C214yesmaximal, complemented
11#74948C214yesmaximal, complemented
12#74456C212yesmaximal
13#74412C212yesmaximal
14#74079C212yesmaximal
15#74073C212yesmaximal
16#74000C211yesmaximal, characteristic
17#73863C211yesmaximal, characteristic
18#35248—31nomaximal, Sylow
19C23×Dic6C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×#944C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C22×C4×Dic3C2211yescharacteristic
29C22×C4×Dic3C2211yescharacteristic
30#5622—28no—
31C2×#889C2212yes—
32C2×#888C2211yescharacteristic
33C2×#888C2214yes—
34C2×#887C2211yescharacteristic
35C2×#887C2212yes—
36C2×#878C2212yes—
37C2×#877C2212yes—
38C2×C4×Dic6—28no—
39C2×C4×Dic6C2212yes—
40C2×C4×Dic6C2212yes—
41C2×C4×Dic6C2212yes—
42C2×C4×Dic6C2212yes—
43C3×#391C2212yes—
44C3×#389C2212yes—
45C3×#387C2214yes—
46C3×#385C2212yes—
47C3×#381C2212yes—
48C3×#380C2211yescharacteristic
49#5229C2212yes—
50#5212C2211yescharacteristic
51#5211C2212yes—
52#5210C2212yes—
53#4984C2212yes—
54#4982C2214yes—
55#4980C2214yes—
56#4979C2214yes—
57#4978C2214yes—
58#4977C2212yes—
59Dic3×C4⋊C4C2212yes—
60#4973C2212yes—
61#4972C2212yes—
62#4950C2212yes—
63#4946C2212yes—
64#4943C2212yes—
65#4943C2212yes—
66#4943C2212yes—
67Dic3×C22⋊C4C2212yes—
68#4933C2214yes—
69#4929C2212yes—
70#4928C2212yes—
71#4660C2214yes—
72#4660C2214yes—
73#4658C2214yes—
74#4658C2214yes—
75#4657C2214yes—
76#4656C2214yes—
77#4655C2214yes—
78#4654C2214yes—
79#4653C2214yes—
80#4652C2214yes—
81#4651C2212yes—
82#4650C2214yes—
83#4648C2212yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×#533—31no—
87C22×C4×Q8—31no—
88#2855—31no—
89#2855—31no—
90#2855—31no—
91#2855D311yescharacteristic
92#2634—34no—
93#2581—32no—
94#2581—32no—
95#2568—32no—
96#2546—32no—
97#2503—32no—
98#2484—32no—
99#2484—32no—
100#2434—32no—
101#2327—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C23×C3×C4C2311yescharacteristic
107C23×Dic3C2311yescharacteristic
108C23×Dic3C2311yescharacteristic
109C23×Dic3C2311yescharacteristic
110C23×Dic3C2311yescharacteristic
111C22×Dic6—24no—
112C22×Dic6—28no—
113C22×Dic6C2312yes—
114C22×Dic6C2312yes—
115C22×Dic6C2311yescharacteristic
116C22×Dic6C2312yes—
117C3×C22⋊Q8—24no—
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2314yes—
122C2×C3×C22⋊C4C2312yes—
123C2×C3×C22⋊C4C2312yes—
124C2×C3×C42—24no—
125C2×C3×C42C2312yes—
126C2×C3×C42C2312yes—
127C2×#217C2312yes—
128C2×#217C2312yes—
129Q8×Dic3—24no—
130#944—24no—
131C2×C4⋊Dic3—24no—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2312yes—
138#924—24no—
139C2×Dic3⋊C4—24no—
140C2×Dic3⋊C4—24no—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×C4×Dic3—24no—
153C2×C4×Dic3D418yes—
154C2×C4×Dic3C2314yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2312yes—
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2311yescharacteristic
160C2×C4×Dic3C2311yescharacteristic
161C2×C4×Dic3C2311yescharacteristic
162C2×C4×Dic3C2311yescharacteristic
163#889—28no—
164#888—216no—
165#888—24no—
166#887—24no—
167#887—28no—
168#878—28no—
169#877—28no—
170C4×Dic6—28no—
171C4×Dic6—28no—
172C4×Dic6—28no—
173C4×Dic6—28no—
174C3×#95C2314yes—
175C3×#95C2314yes—
176C3×#95C2312yes—
177C3×#95C2314yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2314yes—
189C3⋊#95C2314yes—
190C3⋊#95C2314yes—
191C23×Q8—31no—
192#533—68no—
193C2×C4⋊Q8—31no—
194C2×C4⋊Q8—34no—
195C2×C4⋊Q8—31no—
196C2×#125—32no—
197C2×C22⋊Q8—31no—
198C2×C22⋊Q8—32no—
199C2×C22⋊Q8C2×D311yescharacteristic
200C2×C4×Q8—31no—
201C2×C4×Q8—31no—
202C2×C4×Q8—32no—
203C2×C4×Q8—32no—
204C2×C4×Q8—32no—
205C2×C4×Q8—32no—
206C2×C4×Q8—32no—
207C2×C4×Q8—68no—
208C2×#117—32no—
209C22×C4⋊C4—31no—
210C22×C4⋊C4—31no—
211C22×C4⋊C4—31no—
212C22×C42—31no—
213C22×C42—31no—
214C22×C42C2×D311yescharacteristic
215#400—34no—
216#400—34no—
217#400—34no—
218#400—34no—
219#398—34no—
220#398—34no—
221#395—34no—
222#395—34no—
223#391—32no—
224#391—32no—
225#391—32no—
226#391—34no—
227#391—34no—
228#391C2×D312yes—
229#389—32no—
230#389—32no—
231#389—32no—
232#389C2×D312yes—
233#387—34no—
234#387—34no—
235#387—34no—
236#387—34no—
237#387—34no—
238#387—34no—
239#387—34no—
240#387C2×D314yes—
241#385—32no—
242#385—32no—
243#385—32no—
244#385C2×D312yes—
245#383—32no—
246#382—32no—
247#382—34no—
248#381—32no—
249#381—32no—
250#381—32no—
251#381—32no—
252#381C2×D312yes—
253#380—31no—
254#380C2×D311yescharacteristic
255C4×C4⋊C4—32no—
256C4×C22⋊C4—32no—
257#376—32no—
258C24×C3C2411yescharacteristic
259C2×C3×Q8—22no—
260C2×C3×Q8—24no—
261C22×C3×C4—22no—
262C22×C3×C4—24no—
263C22×C3×C4—24no—
264C22×C3×C4—22no—
265C22×C3×C4—22no—
266C22×C3×C4—22no—
267C22×C3×C4C2×D414yes—
268C22×C3×C4C2414yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2412yes—
271C22×C3×C4C2412yes—
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2412yes—
274C22×C3×C4C2411yescharacteristic
275C22×Dic3—22no—
276C22×Dic3—22no—
277C22×Dic3—22no—
278C22×Dic3—24no—
279C22×Dic3—24no—
280C22×Dic3—22no—
281C22×Dic3—22no—
282C22×Dic3—22no—
283C22×Dic3C2×D414yes—
284C22×Dic3C2×D414yes—
285C22×Dic3C2414yes—
286C22×Dic3C2412yes—
287C22×Dic3C2412yes—
288C22×Dic3C2412yes—
289C22×Dic3C2411yescharacteristic
290C22×Dic3C2411yescharacteristic
291C22×Dic3C2411yescharacteristic
292C22×Dic3C2412yes—
293C22×Dic3C2411yescharacteristic
294C2×Dic6—232no—
295C2×Dic6—24no—
296C2×Dic6—24no—
297C2×Dic6—24no—
298C2×Dic6—24no—
299C2×Dic6—24no—
300C2×Dic6—22no—
301C2×Dic6—24no—
302C2×Dic6—24no—
303C2×Dic6—28no—
304C2×Dic6#4118yes—
305C3×C4⋊C4—22no—
306C3×C4⋊C4—24no—
307C3×C4⋊C4—28no—
308C3×C4⋊C4—24no—
309C3×C22⋊C4—24no—
310C3×C22⋊C4—24no—
311C3×C42—24no—
312C3×C42—24no—
313#217—24no—
314#217—24no—
315C4⋊Dic3—22no—
316C4⋊Dic3—24no—
317C4⋊Dic3—24no—
318C4⋊Dic3—24no—
319C4⋊Dic3—24no—
320C4⋊Dic3—28no—
321Dic3⋊C4—24no—
322Dic3⋊C4—28no—
323Dic3⋊C4—24no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—24no—
331Dic3⋊C4—28no—
332C4×Dic3—24no—
333C4×Dic3—24no—
334C4×Dic3—24no—
335C4×Dic3—24no—
336C4×Dic3—28no—
337C4×Dic3C2×D414yes—
338C4×Dic3C2×D414yes—
339C4×Dic3C2×D414yes—
340C4×Dic3C2×D414yes—
341C22×Q8—32no—
342C22×Q8—32no—
343C22×Q8—32no—
344C22×Q8—31no—
345C22×Q8—64no—
346C22×Q8—68no—
347C22×Q8C22×D311yescharacteristic
348C23×C4—31no—
349C23×C4—31no—
350C23×C4—31no—
351C23×C4—31no—
352C23×C4C22×D311yescharacteristic
353C23×C4C22×D311yescharacteristic
354C23×C4C22×D311yescharacteristic
355C4⋊Q8—64no—
356C4⋊Q8—64no—
357C4⋊Q8—616no—
358#125—68no—
359C22⋊Q8—24no—
360C22⋊Q8—68no—
361C22⋊Q8—64no—
362C4×Q8—68no—
363C4×Q8—68no—
364C4×Q8—68no—
365C4×Q8—68no—
366C4×Q8—68no—
367C4×Q8—64no—
368C4×Q8—64no—
369#117—68no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—32no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—31no—
383C2×C4⋊C4—34no—
384C2×C4⋊C4—34no—
385C2×C4⋊C4—34no—
386C2×C4⋊C4—32no—
387C2×C4⋊C4—64no—
388C2×C4⋊C4—64no—
389C2×C4⋊C4—64no—
390C2×C4⋊C4C22×D312yes—
391C2×C4⋊C4C22×D311yescharacteristic
392C2×C4⋊C4C22×D314yes—
393C2×C4⋊C4C22×D312yes—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—32no—
396C2×C22⋊C4C22×D312yes—
397C2×C22⋊C4C22×D312yes—
398C2×C42—24no—
399C2×C42—38no—
400C2×C42—31no—
401C2×C42—31no—
402C2×C42—31no—
403C2×C42—31no—
404C2×C42—32no—
405C2×C42—32no—
406C2×C42—32no—
407C2×C42—32no—
408C2×C42—34no—
409C2×C42—64no—
410C2×C42C22×D312yes—
411C2×C42C22×D312yes—
412#95—32no—
413#95—34no—
414#95—34no—
415#95—34no—
416#95—34no—
417#95—34no—
418#95—34no—
419#95—34no—
420#95—34no—
421#95—34no—
422#95—34no—
423#95—34no—
424#95—34no—
425#95C22×D312yes—
426#95C22×D314yes—
427#95C22×D314yes—
428#95C22×D314yes—
429C23×C3—22no—
430C23×C3—22no—
431C23×C3C22×D412yes—
432C23×C32−1+412yes—
433C23×C32−1+412yes—
434C23×C3C2511yescharacteristic
435C3×Q8—44no—
436C2×C3×C4—28no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—28no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—24no—
445C2×C3×C4—24no—
446C2×C3×C4—24no—
447C2×C3×C4—24no—
448C2×C3×C4—22no—
449C2×C3×C4—22no—
450C2×C3×C4—24no—
451C2×C3×C4—44no—
452C2×C3×C4C22×D412yes—
453C2×C3×C4C22×D412yes—
454C2×C3×C4C2×#4114yes—
455C2×C3×C42−1+414yes—
456C2×Dic3—28no—
457C2×Dic3—28no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—21no—
462C2×Dic3—21no—
463C2×Dic3—21no—
464C2×Dic3—21no—
465C2×Dic3—28no—
466C2×Dic3—22no—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—24no—
470C2×Dic3—24no—
471C2×Dic3—24no—
472C2×Dic3—24no—
473C2×Dic3—24no—
474C2×Dic3—22no—
475C2×Dic3—22no—
476C2×Dic3—44no—
477C2×Dic3C22×D412yes—
478C2×Dic3C22×D412yes—
479C2×Dic3C22×D412yes—
480C2×Dic3C22×D412yes—
481C2×Dic3C2×#4114yes—
482C2×Dic3C2×#4114yes—
483Dic6—216no—
484Dic6—216no—
485Dic6—44no—
486Dic6—48no—
487C24C23×D311yescharacteristic
488C2×Q8—22no—
489C2×Q8—24no—
490C2×Q8—38no—
491C2×Q8—64no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—64no—
496C2×Q8—68no—
497C2×Q8—632no—
498C2×Q8—62no—
499C2×Q8—64no—
500C2×Q8—64no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—31no—
508C22×C4—34no—
509C22×C4—34no—
510C22×C4—31no—
511C22×C4—31no—
512C22×C4—31no—
513C22×C4—32no—
514C22×C4—34no—
515C22×C4—32no—
516C22×C4—32no—
517C22×C4—32no—
518C22×C4—64no—
519C22×C4—64no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4—62no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—62no—
526C22×C4D3×D414yes—
527C22×C4C23×D314yes—
528C22×C4C23×D312yes—
529C22×C4C23×D312yes—
530C22×C4C23×D312yes—
531C22×C4C23×D311yescharacteristic
532C22×C4C23×D312yes—
533C22×C4C23×D311yescharacteristic
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—64no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—64no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—64no—
549C4⋊C4—68no—
550C4⋊C4—68no—
551C4⋊C4—64no—
552C4⋊C4—62no—
553C4⋊C4—68no—
554C4⋊C4—64no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—64no—
558C22⋊C4—64no—
559C42—24no—
560C42—24no—
561C42—34no—
562C42—34no—
563C42—34no—
564C42—34no—
565C42—64no—
566C42—64no—
567C42—68no—
568C42—64no—
569C42—64no—
570C22×C3—22no—
571C22×C3—22no—
572C22×C3—24no—
573C22×C3—24no—
574C22×C3#53614yes—
575C22×C3C23×D411yescharacteristic
576C22×C3C22×#4112yes—
577C22×C3C2×2−1+411yescharacteristic
578C22×C3C2×2−1+411yescharacteristic
579C22×C3C2×2−1+412yes—
580C3×C4—24no—
581C3×C4—22no—
582C3×C4—42no—
583C3×C4—42no—
584C3×C4—42no—
585C3×C4—44no—
586C3×C4#53614yes—
587Dic3—22no—
588Dic3—24no—
589Dic3—22no—
590Dic3—22no—
591Dic3—42no—
592Dic3—42no—
593Dic3—42no—
594Dic3—44no—
595Dic3#52514yes—
596C23—22no—
597C23—22no—
598C23C2×D3×D412yes—
599C23#100712yes—
600C23#101012yes—
601C23C24×D311yescharacteristic
602Q8—44no—
603Q8—616no—
604Q8—616no—
605Q8—128no—
606Q8—124no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—32no—
623C2×C4—34no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—34no—
628C2×C4—44no—
629C2×C4—68no—
630C2×C4—68no—
631C2×C4—62no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—68no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—64no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—62no—
648C2×C4—64no—
649C2×C4—124no—
650C2×C4C2×D3×D412yes—
651C2×C4C2×D3×D412yes—
652C2×C4D3×#4114yes—
653C2×C4#101014yes—
654C2×C3—24no—
655C2×C3C2×#52511yescharacteristic
656C2×C3C2×#53611yescharacteristic
657C2×C3C2×#53611yescharacteristic
658C2×C3#352312yes—
659C2×C3#352312yes—
660C22—22no—
661C22—24no—
662C22—22no—
663C22—24no—
664C22#562814yes—
665C22C22×D3×D411yescharacteristic
666C22C2×#100711yescharacteristic
667C22C2×D3×#4112yes—
668C22C2×#101011yescharacteristic
669C22C2×#101012yes—
670C4—24no—
671C4—22no—
672C4—34no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—62no—
678C4—64no—
679C4—62no—
680C4—62no—
681C4—122no—
682C4—122no—
683C4—122no—
684C4—124no—
685C4#564814yes—
686C3#3524811yescharacteristic, complemented, Sylow
687C2—24no—
688C2C2×#562811yescharacteristic
689C2C2×#564811yescharacteristic
690C2D3×#52511yescharacteristic
691C2#8612912yes—
692C2#8626612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2855C2
prime radicalC24×C3C24
hypercenter#2855D3
hypocenterC3#35248
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2855

Complements

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

NH ≅ G/Ntype
6. #76547C2semidirect
7. #76545C2semidirect
8. #76076C2semidirect
9. #76060C2semidirect
10. #75135C2semidirect
11. #74948C2semidirect
686. C3#35248semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C22×C4×Dic62
C3×#28552
#828102
#820752
#765472
#765452
#760762
#760602
#751352
#749482
#744562
#744122
#740792
#740732
#740002
#738632
#352483