Finite Group Databasebeta

← #11660815

Subgroups — #11660815

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3480C211yesmaximal, characteristic
2#83280C213yesmaximal, complemented
3#83278C213yesmaximal, complemented
4#83132C213yesmaximal, complemented
5#83131C213yesmaximal, complemented
6#83129C213yesmaximal, complemented
7#83126C213yesmaximal, complemented
8#82523C211yesmaximal, characteristic, complemented
9#82520C211yesmaximal, characteristic, complemented
10#82519C211yesmaximal, characteristic, complemented
11#41×C3⋊C8C211yesmaximal, characteristic, complemented
12#82515C213yesmaximal, complemented
13#82513C213yesmaximal, complemented
14#82437C211yesmaximal, characteristic, complemented
15#82435C211yesmaximal, characteristic, complemented
16#60301—31nomaximal, Sylow
17C3×#559C413yes—
18C3×#557C411yescharacteristic
19C3×#556C413yes—
20C3×#553C413yes—
21C3×#553C413yes—
22C3×#550C411yescharacteristic
23C3×#548C411yescharacteristic
24C3×#548C411yescharacteristic
25C3×#532C2213yes—
26C3×#529C2213yes—
27C3×#529C2213yes—
28C3×C4×#41C2211yescharacteristic
29C3×C4×#41C2211yescharacteristic
30C3×C4×#41C2213yes—
31C2×C3×C4×Q8C2211yescharacteristic
32#5234C2213yes—
33#5234C2216yes—
34#5234C2213yes—
35#5024C2213yes—
36#5024C2213yes—
37Q8×C3⋊C8C2211yescharacteristic
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2213yes—
40Q8×C3⋊C8C2213yes—
41Q8×C3⋊C8C2211yescharacteristic
42Q8×C3⋊C8C2211yescharacteristic
43#5012C2213yes—
44#5012C2213yes—
45#5012C2216yes—
46#5012C2213yes—
47#5012C2213yes—
48#5011C2216yes—
49#5011C2216yes—
50D4×C3⋊C8C2213yes—
51D4×C3⋊C8C2213yes—
52D4×C3⋊C8C2216yes—
53D4×C3⋊C8C2213yes—
54D4×C3⋊C8C2213yes—
55#5000C2211yescharacteristic
56#5000C2213yes—
57#5000C2213yes—
58#5000C2213yes—
59#5000C2216yes—
60#4999C2213yes—
61#4999C2213yes—
62#4997C2216yes—
63#4926C2213yes—
64#4926C2213yes—
65#4925C2213yes—
66#4925C2213yes—
67#4925C2213yes—
68#4925C2213yes—
69#4925C2213yes—
70#4925C2213yes—
71#4924C2213yes—
72C2×C4⋊(C3⋊C8)C2213yes—
73C4×#208C2213yes—
74C2×C4×C3⋊C8C2213yes—
75#3480D311yescharacteristic
76#2994—33no—
77#2993—33no—
78#2991—31no—
79#2988—31no—
80#2984—31no—
81#2982—31no—
82#2981—33no—
83#2979—33no—
84#2971—31no—
85#2969—33no—
86#2966—33no—
87C8×#41—31no—
88#2925—33no—
89#2922—33no—
90C2×C3×#41C2×C411yescharacteristic
91C2×C3×#41C2×C411yescharacteristic
92C2×C3×#41C2313yes—
93C22×C3×Q8C2×C411yescharacteristic
94C3×C4⋊Q8C2×C413yes—
95C3×#127C2×C413yes—
96C3×#126C2×C416yes—
97C3×#126C2×C416yes—
98C3×#125C2×C413yes—
99C3×#125C2×C413yes—
100C3×#124C2×C413yes—
101C3×#124C2×C413yes—
102C3×#124C2×C416yes—
103C3×#123C2×C416yes—
104C3×#123C2×C416yes—
105C3×C22⋊Q8C2×C413yes—
106C3×C22⋊Q8C2×C413yes—
107C3×#121C2×C413yes—
108C3×#121C2×C413yes—
109C3×#121C2×C413yes—
110C3×#121C2×C413yes—
111C3×#121C2×C416yes—
112C3×C4×Q8C2×C413yes—
113C3×C4×Q8C2×C413yes—
114C3×C4×Q8C2×C411yescharacteristic
115C3×C4×Q8C2×C411yescharacteristic
116C3×C4×Q8C2×C411yescharacteristic
117C3×C4×Q8C2×C413yes—
118C3×C4×Q8C2311yescharacteristic
119C3×C4×Q8C2313yes—
120C3×C4×D4C2×C413yes—
121C3×C4×D4C2×C413yes—
122C3×C4×D4C2×C413yes—
123C3×C4×D4C2×C413yes—
124C3×C4×D4C2×C413yes—
125C3×C4×D4C2×C413yes—
126C3×C4×D4C2×C413yes—
127C3×C4×D4C2×C413yes—
128C3×C4×D4C2313yes—
129C3×C4×D4C2313yes—
130C3×C4×D4C2316yes—
131C3×#117C2×C413yes—
132C3×#117C2×C413yes—
133C3×#117C2313yes—
134C3×#117C2313yes—
135C3×#117C2311yescharacteristic
136C2×C3×C4⋊C4C2×C413yes—
137C2×C3×C42C2313yes—
138C2×C3×C42C2313yes—
139C2×C3×C42C2313yes—
140C2×#208C2313yes—
141C2×#208C2313yes—
142C2×#208C2316yes—
143C22×C3⋊C8C2313yes—
144C22×C3⋊C8C2313yes—
145C22×C3⋊C8C2316yes—
146#830C2313yes—
147#830C2311yescharacteristic
148#830C2311yescharacteristic
149#830C2313yes—
150#830C2313yes—
151#830C2316yes—
152#830C2313yes—
153#830C2313yes—
154#830C2313yes—
155#830C2316yes—
156C4⋊(C3⋊C8)C2313yes—
157C4⋊(C3⋊C8)C2313yes—
158C4⋊(C3⋊C8)C2313yes—
159C4⋊(C3⋊C8)C2313yes—
160C4⋊(C3⋊C8)C2313yes—
161C4⋊(C3⋊C8)C2313yes—
162C4⋊(C3⋊C8)C2311yescharacteristic
163C4⋊(C3⋊C8)C2311yescharacteristic
164C4⋊(C3⋊C8)C2313yes—
165C4⋊(C3⋊C8)C2313yes—
166C4⋊(C3⋊C8)C2313yes—
167C4⋊(C3⋊C8)C2313yes—
168#803C2311yescharacteristic
169#803C2313yes—
170#803C2316yes—
171C4×C3⋊C8C2311yescharacteristic
172C4×C3⋊C8C2313yes—
173C4×C3⋊C8C2313yes—
174C4×C3⋊C8C2313yes—
175C4×C3⋊C8C2313yes—
176C4×C3⋊C8C2313yes—
177C4×C3⋊C8C2316yes—
178#559Dic313yes—
179#557Dic311yescharacteristic
180#556Dic313yes—
181#553Dic313yes—
182#553Dic313yes—
183#550Dic311yescharacteristic
184#548Dic311yescharacteristic
185#548Dic311yescharacteristic
186#532C2×D313yes—
187#529C2×D313yes—
188#529C2×D313yes—
189C4×#41C2×D311yescharacteristic
190C4×#41C2×D311yescharacteristic
191C4×#41C2×D313yes—
192C2×C4×Q8C2×D311yescharacteristic
193#446—33no—
194#446—33no—
195C8×Q8—31no—
196C8×Q8—31no—
197C8×Q8—33no—
198C8×Q8—33no—
199C8×Q8—31no—
200C8×Q8—31no—
201#436—33no—
202#436—36no—
203#436—33no—
204#436—33no—
205#436—33no—
206#435—36no—
207#435—36no—
208C8×D4—33no—
209C8×D4—33no—
210C8×D4—36no—
211C8×D4—33no—
212C8×D4—33no—
213#433—31no—
214#433—33no—
215#433—33no—
216#433—33no—
217#433—36no—
218#432—33no—
219#432—33no—
220#431—33no—
221#431—33no—
222#431—33no—
223#431—33no—
224#431—33no—
225#431—33no—
226#424—33no—
227#424—33no—
228#423—33no—
229C2×C4⋊C8—33no—
230#408—33no—
231#408—33no—
232#408—36no—
233#405—36no—
234C4×M16—33no—
235C2×C4×C8—33no—
236C3×#41—24no—
237C3×#41—24no—
238C3×#41—43no—
239C3×#41—43no—
240C2×C3×Q8—24no—
241C2×C3×Q8C22×C413yes—
242C2×C3×Q8C22×C413yes—
243C2×C3×Q8C22×C411yescharacteristic
244C2×C3×D4C22×C413yes—
245C2×C3×D4C22×C413yes—
246C2×C3×D4C22×C413yes—
247C2×C3×D4C22×C413yes—
248C2×C3×D4C22×C413yes—
249C2×C3×D4C22×C416yes—
250C22×C3×C4C22×C413yes—
251C22×C3×C4C22×C413yes—
252C22×C3×C4C22×C413yes—
253C22×C3×C4C22×C413yes—
254C22×C3×C4C22×C413yes—
255C22×C3×C4C22×C413yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2413yes—
258C22×C3×C4C2413yes—
259C3×C4⋊C4C22×C413yes—
260C3×C4⋊C4C22×C413yes—
261C3×C4⋊C4C22×C416yes—
262C3×C4⋊C4C22×C413yes—
263C3×C4⋊C4C22×C413yes—
264C3×C4⋊C4C22×C411yescharacteristic
265C3×C4⋊C4C22×C411yescharacteristic
266C3×C4⋊C4C22×C413yes—
267C3×C4⋊C4C22×C413yes—
268C3×C4⋊C4C22×C413yes—
269C3×C4⋊C4C22×C413yes—
270C3×C4⋊C4C22×C416yes—
271C3×C22⋊C4C22×C416yes—
272C3×C22⋊C4C22×C416yes—
273C3×C22⋊C4C22×C411yescharacteristic
274C3×C22⋊C4C22×C411yescharacteristic
275C3×C22⋊C4C22×C413yes—
276C3×C22⋊C4C22×C413yes—
277C3×C22⋊C4C22×C413yes—
278C3×C22⋊C4C22×C413yes—
279C3×C22⋊C4C22×C416yes—
280C3×C22⋊C4C22×C416yes—
281C3×C42C22×C413yes—
282C3×C42C22×C413yes—
283C3×C42C22×C413yes—
284C3×C42C22×C413yes—
285C3×C42C22×C413yes—
286C3×C42C22×C413yes—
287C3×C42C2411yescharacteristic
288C3×C42C2413yes—
289C3×C42C2413yes—
290C3×C42C2411yescharacteristic
291#208—43no—
292#208—46no—
293#208—43no—
294C2×C3⋊C8—43no—
295C2×C3⋊C8—46no—
296C2×C3⋊C8—43no—
297C2×C3⋊C8C2411yescharacteristic
298C2×C3⋊C8C2411yescharacteristic
299C2×C3⋊C8C2413yes—
300C2×C3⋊C8C2411yescharacteristic
301C2×C3⋊C8C2411yescharacteristic
302C2×C3⋊C8C2413yes—
303C2×C3⋊C8C2413yes—
304C2×C3⋊C8C2413yes—
305C2×#41C2×Dic311yescharacteristic
306C2×#41C2×Dic311yescharacteristic
307C2×#41C22×D313yes—
308C22×Q8C2×Dic311yescharacteristic
309C2×M16—33no—
310C2×M16—33no—
311C2×M16—36no—
312C22×C8—33no—
313C22×C8—33no—
314C22×C8—36no—
315C4⋊Q8C2×Dic313yes—
316#127C2×Dic313yes—
317#126C2×Dic316yes—
318#126C2×Dic316yes—
319#125C2×Dic313yes—
320#125C2×Dic313yes—
321#124C2×Dic313yes—
322#124C2×Dic313yes—
323#124C2×Dic316yes—
324#123C2×Dic316yes—
325#123C2×Dic316yes—
326C22⋊Q8C2×Dic313yes—
327C22⋊Q8C2×Dic313yes—
328#121C2×Dic313yes—
329#121C2×Dic313yes—
330#121C2×Dic313yes—
331#121C2×Dic313yes—
332#121C2×Dic316yes—
333C4×Q8C2×Dic311yescharacteristic
334C4×Q8C2×Dic313yes—
335C4×Q8C2×Dic311yescharacteristic
336C4×Q8C2×Dic311yescharacteristic
337C4×Q8C2×Dic313yes—
338C4×Q8C2×Dic313yes—
339C4×Q8C22×D311yescharacteristic
340C4×Q8C22×D313yes—
341C4×D4C2×Dic313yes—
342C4×D4C2×Dic313yes—
343C4×D4C2×Dic313yes—
344C4×D4C2×Dic313yes—
345C4×D4C2×Dic313yes—
346C4×D4C2×Dic313yes—
347C4×D4C2×Dic313yes—
348C4×D4C2×Dic313yes—
349C4×D4C22×D313yes—
350C4×D4C22×D313yes—
351C4×D4C22×D316yes—
352#117C2×Dic313yes—
353#117C2×Dic313yes—
354#117C22×D311yescharacteristic
355#117C22×D313yes—
356#117C22×D313yes—
357C2×C4⋊C4C2×Dic313yes—
358C2×C42C22×D313yes—
359C2×C42C22×D313yes—
360C2×C42C22×D313yes—
361C4⋊C8—33no—
362C4⋊C8—33no—
363C4⋊C8—33no—
364C4⋊C8—33no—
365C4⋊C8—33no—
366C4⋊C8—33no—
367C4⋊C8—31no—
368C4⋊C8—31no—
369C4⋊C8—33no—
370C4⋊C8—33no—
371C4⋊C8—33no—
372C4⋊C8—33no—
373C22⋊C8—31no—
374C22⋊C8—33no—
375C22⋊C8—31no—
376C22⋊C8—33no—
377C22⋊C8—33no—
378C22⋊C8—36no—
379C22⋊C8—33no—
380C22⋊C8—33no—
381C22⋊C8—33no—
382C22⋊C8—36no—
383#97—31no—
384#97—33no—
385#97—36no—
386C4×C8—31no—
387C4×C8—33no—
388C4×C8—33no—
389C4×C8—33no—
390C4×C8—33no—
391C4×C8—33no—
392C4×C8—36no—
393C23×C3C23×C413yes—
394C23×C3C23×C413yes—
395C23×C3C23×C411yescharacteristic
396C3×Q8—26no—
397C3×Q8—43no—
398C3×Q8#13114yes—
399C3×D4—26no—
400C3×D4—26no—
401C3×D4—46no—
402C3×D4—43no—
403C3×D4—43no—
404C3×D4—43no—
405C2×C3×C4—26no—
406C2×C3×C4—26no—
407C2×C3×C4—26no—
408C2×C3×C4—23no—
409C2×C3×C4—23no—
410C2×C3×C4—23no—
411C2×C3×C4—23no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—43no—
415C2×C3×C4—43no—
416C2×C3×C4—43no—
417C2×C3×C4C23×C411yescharacteristic
418C2×C3×C4C23×C411yescharacteristic
419C2×C3×C4C23×C411yescharacteristic
420C2×C3×C4C23×C413yes—
421C2×C3×C4C23×C411yescharacteristic
422C2×C3×C4C23×C411yescharacteristic
423C2×C3×C4C23×C413yes—
424C2×C3×C4C23×C413yes—
425C2×C3×C4C23×C413yes—
426C2×C3×C4C23×C413yes—
427C2×C3×C4C23×C413yes—
428C2×C3×C4C2511yescharacteristic
429C3⋊C8—21no—
430C3⋊C8—23no—
431C3⋊C8—21no—
432C3⋊C8—23no—
433C3⋊C8—23no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C3⋊C8—23no—
437#41—24no—
438#41—24no—
439#41—43no—
440#41—43no—
441C2×Q8—24no—
442C2×Q8C22×Dic313yes—
443C2×Q8C22×Dic313yes—
444C2×Q8C22×Dic311yescharacteristic
445C2×D4C22×Dic313yes—
446C2×D4C22×Dic313yes—
447C2×D4C22×Dic313yes—
448C2×D4C22×Dic313yes—
449C2×D4C22×Dic313yes—
450C2×D4C22×Dic316yes—
451C22×C4C22×Dic313yes—
452C22×C4C22×Dic313yes—
453C22×C4C22×Dic313yes—
454C22×C4C22×Dic313yes—
455C22×C4C22×Dic313yes—
456C22×C4C22×Dic313yes—
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D313yes—
459C22×C4C23×D313yes—
460M16—123no—
461M16—126no—
462M16—123no—
463C2×C8—31no—
464C2×C8—31no—
465C2×C8—33no—
466C2×C8—31no—
467C2×C8—31no—
468C2×C8—33no—
469C2×C8—33no—
470C2×C8—33no—
471C2×C8—123no—
472C2×C8—126no—
473C2×C8—123no—
474C4⋊C4C22×Dic313yes—
475C4⋊C4C22×Dic313yes—
476C4⋊C4C22×Dic316yes—
477C4⋊C4C22×Dic313yes—
478C4⋊C4C22×Dic313yes—
479C4⋊C4C22×Dic311yescharacteristic
480C4⋊C4C22×Dic311yescharacteristic
481C4⋊C4C22×Dic313yes—
482C4⋊C4C22×Dic313yes—
483C4⋊C4C22×Dic313yes—
484C4⋊C4C22×Dic313yes—
485C4⋊C4C22×Dic316yes—
486C22⋊C4C22×Dic316yes—
487C22⋊C4C22×Dic316yes—
488C22⋊C4C22×Dic311yescharacteristic
489C22⋊C4C22×Dic311yescharacteristic
490C22⋊C4C22×Dic313yes—
491C22⋊C4C22×Dic313yes—
492C22⋊C4C22×Dic313yes—
493C22⋊C4C22×Dic313yes—
494C22⋊C4C22×Dic316yes—
495C22⋊C4C22×Dic316yes—
496C42C22×Dic313yes—
497C42C22×Dic313yes—
498C42C22×Dic313yes—
499C42C22×Dic313yes—
500C42C22×Dic313yes—
501C42C22×Dic313yes—
502C42C23×D311yescharacteristic
503C42C23×D313yes—
504C42C23×D313yes—
505C42C23×D311yescharacteristic
506C22×C3—21no—
507C22×C3—23no—
508C22×C3—23no—
509C22×C3—21no—
510C22×C3—23no—
511C22×C3—23no—
512C22×C3—21no—
513C22×C3—23no—
514C22×C3—23no—
515C22×C3C24×C411yescharacteristic
516C3×C4—21no—
517C3×C4—23no—
518C3×C4—23no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—23no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—23no—
525C3×C4—23no—
526C3×C4C2×#13116yes—
527C3×C4#58511yescharacteristic
528C3×C4#58511yescharacteristic
529C23C23×Dic313yes—
530C23C23×Dic313yes—
531C23C23×Dic311yescharacteristic
532Q8—26no—
533Q8—43no—
534Q8#94814yes—
535D4—26no—
536D4—26no—
537D4—43no—
538D4—43no—
539D4—43no—
540D4—46no—
541C2×C4—23no—
542C2×C4—23no—
543C2×C4—23no—
544C2×C4—23no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4C23×Dic311yescharacteristic
554C2×C4C23×Dic313yes—
555C2×C4C23×Dic311yescharacteristic
556C2×C4C23×Dic311yescharacteristic
557C2×C4C23×Dic311yescharacteristic
558C2×C4C23×Dic311yescharacteristic
559C2×C4C23×Dic313yes—
560C2×C4C23×Dic313yes—
561C2×C4C23×Dic313yes—
562C2×C4C23×Dic313yes—
563C2×C4C23×Dic313yes—
564C2×C4C24×D311yescharacteristic
565C8—61no—
566C8—63no—
567C8—61no—
568C8—63no—
569C8—63no—
570C8—61no—
571C8—61no—
572C8—63no—
573C2×C3—41no—
574C2×C3—43no—
575C2×C3—43no—
576C2×C3#342811yescharacteristic
577C2×C3C22×#13111yescharacteristic
578C2×C3#357311yescharacteristic
579C22—21no—
580C22—23no—
581C22—23no—
582C22—21no—
583C22—23no—
584C22—23no—
585C22—21no—
586C22—23no—
587C22—23no—
588C22C24×Dic311yescharacteristic
589C4—21no—
590C4—21no—
591C4—23no—
592C4—23no—
593C4—21no—
594C4—21no—
595C4—23no—
596C4—21no—
597C4—23no—
598C4—23no—
599C4C2×#94816yes—
600C4#596411yescharacteristic
601C4#596411yescharacteristic
602C3#6030111yescharacteristic, complemented, Sylow
603C2—41no—
604C2—43no—
605C2—43no—
606C2C22×#94811yescharacteristic
607C2#8726011yescharacteristic
608C2#8729611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
Fitting, prime radicalC3×#3480C2
cosocleC2×C3×C4C25
hypercenter#3480D3
hypocenterC3#60301
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #3480

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. #83278C2semidirect
4. #83132C2semidirect
5. #83131C2semidirect
6. #83129C2semidirect
7. #83126C2semidirect
8. #82523C2semidirect
9. #82520C2semidirect
10. #82519C2semidirect
11. #41×C3⋊C8C2semidirect
12. #82515C2semidirect
13. #82513C2semidirect
14. #82437C2semidirect
15. #82435C2semidirect
602. C3#60301semidirect

Maximal subgroups

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

subgroupindex
C3×#34802
#832802
#832782
#831322
#831312
#831292
#831262
#825232
#825202
#825192
#41×C3⋊C82
#825152
#825132
#824372
#824352
#603013