Finite Group Databasebeta

← #10957277

Subgroups — #10957277

Order 768.

3276 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic, complemented
2C2×#5834C211yesmaximal, characteristic
3C2×#5574C211yesmaximal, characteristic, complemented
4#84089C211yesmaximal, characteristic, complemented
5Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
6#82457C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#82399C213yesmaximal, complemented
9#82396C213yesmaximal, complemented
10#82343C213yesmaximal, complemented
11C4×#940C213yesmaximal, complemented
12#82236C213yesmaximal, complemented
13#82225C213yesmaximal, complemented
14C4⋊C4×#67C213yesmaximal, complemented
15#82163C213yesmaximal, complemented
16#34313—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C22×#239C2211yescharacteristic
19C3×#520—28no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2213yes—
23C2×C3×C4×D4C2213yes—
24C2×C3×#117C2213yes—
25C22×C3×C4⋊C4C2213yes—
26#5834C4116yescomplemented
27C2×Q8×Dic3C2211yescharacteristic
28C2×Q8×Dic3C2211yescharacteristic
29C2×Q8×Dic3C2211yescharacteristic
30C2×#940C2213yes—
31C2×#938C2213yes—
32C2×#930C2213yes—
33C2×C4×#67C2216yes—
34C2×C4×#67C2216yes—
35C2×C4×#67C2213yes—
36#5574—28no—
37C2×#893C2213yes—
38C2×D3×C4⋊C4C2213yes—
39C2×C4×D12C2213yes—
40C2×#872C2213yes—
41#5221C2213yes—
42#5221C2213yes—
43#5213C2216yes—
44#5212C2213yes—
45#5210C2216yes—
46C4×#217C2213yes—
47#4990C2213yes—
48#4990C2213yes—
49#4989C2216yes—
50#4988C2213yes—
51#4987C2213yes—
52#4976C2213yes—
53#4976C2211yescharacteristic
54#4975C2213yes—
55Dic3×C4⋊C4C2213yes—
56Dic3×C4⋊C4C2213yes—
57#4972C2213yes—
58#4957C2213yes—
59#4957C2213yes—
60#4957C2216yes—
61#4956C2216yes—
62#4945C2213yes—
63#4945C2213yes—
64#4939C2213yes—
65C4×(C2×D3)⋊C4C2213yes—
66C4×C4⋊Dic3C2213yes—
67C4×Dic3⋊C4C2213yes—
68C42×Dic3C2213yes—
69#4928C2213yes—
70C2×#550—31no—
71C2×#520—31no—
72C2×#520D311yescharacteristic, complemented
73#2366—33no—
74#2361—33no—
75#2361—33no—
76Q8×C4⋊C4—31no—
77D4×C4⋊C4—33no—
78#2341—31no—
79#2341—31no—
80#2324—33no—
81#2324—33no—
82C4×#127—33no—
83#2293—33no—
84C42×Q8—31no—
85C2×C3×#41—24no—
86C2×C3×#41—24no—
87C2×C3×#41C2×C418yescomplemented
88C22×C3×Q8C2311yescharacteristic
89C22×C3×D4C2313yes—
90C23×C3×C4C2313yes—
91C23×C3×C4C2316yes—
92C22×#67C2313yes—
93C22×#67C2316yes—
94C2×#239—24no—
95C2×#239—24no—
96C2×#239C2×C418yescomplemented
97C22×D12C2313yes—
98C22×C4×D3C2313yes—
99C22×C4×D3C2316yes—
100C3×C4×Q8—24no—
101C3×C4×Q8D418yescomplemented
102C3×C4×D4—212no—
103C3×C4×D4—212no—
104C3×#117—212no—
105C2×C3×C4⋊C4—212no—
106C2×C3×C4⋊C4C2313yes—
107C2×C3×C4⋊C4C2313yes—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2313yes—
110C2×C3×C22⋊C4C2313yes—
111C2×C3×C22⋊C4C2313yes—
112C2×C3×C42C2313yes—
113C2×C3×C42C2313yes—
114C2×#217C2313yes—
115C2×#217C2313yes—
116C2×#217C2316yes—
117C2×#217C2313yes—
118Q8×Dic3—24no—
119Q8×Dic3—24no—
120Q8×Dic3C2×C418yescomplemented
121#940C2×C4124yescomplemented
122#938C2×C4124yescomplemented
123#930C2×C4124yescomplemented
124C4×#67—412no—
125C4×#67—412no—
126C4×#67C2×C4124yescomplemented
127C2×(C2×D3)⋊C4C2313yes—
128C2×(C2×D3)⋊C4C2313yes—
129C2×(C2×D3)⋊C4C2313yes—
130C2×(C2×D3)⋊C4C2313yes—
131C2×(C2×D3)⋊C4C2316yes—
132C2×(C2×D3)⋊C4C2313yes—
133C2×C4⋊Dic3C2313yes—
134C2×C4⋊Dic3C2313yes—
135C2×C4⋊Dic3C2313yes—
136C2×Dic3⋊C4C2311yescharacteristic
137C2×Dic3⋊C4C2313yes—
138C2×Dic3⋊C4C2311yescharacteristic
139C2×Dic3⋊C4C2313yes—
140C2×Dic3⋊C4C2316yes—
141C2×Dic3⋊C4C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145C2×C4×Dic3C2313yes—
146C2×C4×Dic3C2313yes—
147C2×C4×Dic3C2313yes—
148C2×C4×Dic3C2313yes—
149#893—212no—
150D3×C4⋊C4—212no—
151C4×D12—212no—
152#872—212no—
153C3⋊#95C2313yes—
154C3⋊#95C2313yes—
155C3⋊#95C2316yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2313yes—
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2313yes—
160C22×#41—31no—
161C22×#41C2×D311yescharacteristic
162#550—316no—
163C2×#127—33no—
164C2×#121—33no—
165C2×#121—33no—
166#520—28no—
167#520—68no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×Q8C2×D311yescharacteristic
173C2×C4×D4—33no—
174C2×C4×D4—36no—
175C2×C4×D4—36no—
176C2×C4×D4—33no—
177C2×C4×D4—33no—
178C2×C4×D4C2×D313yes—
179C2×C4×D4C2×D313yes—
180C2×#117—33no—
181C2×#117C2×D313yes—
182C22×C4⋊C4—33no—
183C22×C4⋊C4C2×D313yes—
184#390—33no—
185#390—33no—
186#390—33no—
187#390—33no—
188#390—36no—
189#390—33no—
190#390—33no—
191#389—33no—
192#389—33no—
193#389—31no—
194#389—33no—
195#388—33no—
196#388—33no—
197#388—33no—
198#388—33no—
199#386—36no—
200#386—36no—
201#385—36no—
202#385—36no—
203#384—33no—
204#380—33no—
205#380—33no—
206C4×C4⋊C4—33no—
207C4×C4⋊C4—33no—
208C4×C4⋊C4—33no—
209C4×C4⋊C4—33no—
210C4×C22⋊C4—33no—
211C4×C22⋊C4—33no—
212C43—33no—
213C24×C3C2413yes—
214C23×D3C2413yes—
215C3×#41—232no—
216C2×C3×Q8C22×C414yes—
217C2×C3×Q8C2×D414yes—
218C2×C3×Q8#4114yes—
219C2×C3×D4—26no—
220C2×C3×D4—26no—
221C2×C3×D4C22×C4112yes—
222C22×C3×C4—26no—
223C22×C3×C4—26no—
224C22×C3×C4—212no—
225C22×C3×C4—46no—
226C22×C3×C4—46no—
227C22×C3×C4C22×C4112yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2413yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2413yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2413yes—
234C2×#67—26no—
235C2×#67—26no—
236C2×#67—46no—
237C2×#67—46no—
238C2×#67C22×C4112yes—
239C2×#67C22×C4124yes—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2413yes—
242C22×Dic3C2413yes—
243C22×Dic3C2413yes—
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2411yescharacteristic
247#239—232no—
248C2×D12—26no—
249C2×D12—26no—
250C2×D12C22×C4112yes—
251C2×C4×D3—26no—
252C2×C4×D3—26no—
253C2×C4×D3—212no—
254C2×C4×D3—46no—
255C2×C4×D3—46no—
256C2×C4×D3C22×C4112yes—
257C3×C4⋊C4—26no—
258C3×C4⋊C4—26no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4C2×D4112yes—
261C3×C22⋊C4—26no—
262C3×C22⋊C4—26no—
263C3×C42—26no—
264C3×C42C2×D4112yes—
265#217—26no—
266#217—46no—
267#217—43no—
268#217C22×C4112yes—
269(C2×D3)⋊C4—26no—
270(C2×D3)⋊C4—26no—
271(C2×D3)⋊C4—26no—
272(C2×D3)⋊C4—43no—
273(C2×D3)⋊C4—46no—
274(C2×D3)⋊C4C22×C4112yes—
275C4⋊Dic3—26no—
276C4⋊Dic3—26no—
277C4⋊Dic3C22×C4112yes—
278Dic3⋊C4—22no—
279Dic3⋊C4—26no—
280Dic3⋊C4—26no—
281Dic3⋊C4—22no—
282Dic3⋊C4—46no—
283Dic3⋊C4C22×C414yes—
284C4×Dic3—26no—
285C4×Dic3—26no—
286C4×Dic3—26no—
287C4×Dic3—26no—
288C4×Dic3—22no—
289C4×Dic3—43no—
290C4×Dic3C22×C4112yes—
291C2×#41—24no—
292C2×#41—24no—
293C2×#41—38no—
294C2×#41—64no—
295C2×#41—64no—
296C2×#41C4×D318yescomplemented
297C22×Q8C22×D311yescharacteristic
298C22×D4—33no—
299C22×D4—36no—
300C22×D4—33no—
301C22×D4C22×D313yes—
302C23×C4—33no—
303C23×C4—36no—
304C23×C4C22×D313yes—
305C23×C4C22×D316yes—
306#127—324no—
307#121—324no—
308#121—324no—
309C4×Q8—24no—
310C4×Q8—38no—
311C4×Q8—64no—
312C4×Q8—64no—
313C4×Q8#6718yescomplemented
314C4×D4—212no—
315C4×D4—212no—
316C4×D4—324no—
317C4×D4—612no—
318C4×D4—612no—
319C4×D4—1212no—
320C4×D4—1212no—
321#117—212no—
322#117—612no—
323C2×C4⋊C4—212no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—33no—
326C2×C4⋊C4—33no—
327C2×C4⋊C4—33no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—33no—
330C2×C4⋊C4—33no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—36no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4C22×D313yes—
335C2×C4⋊C4C22×D313yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D313yes—
338C2×C22⋊C4—33no—
339C2×C22⋊C4—33no—
340C2×C22⋊C4—33no—
341C2×C22⋊C4—33no—
342C2×C22⋊C4—33no—
343C2×C22⋊C4—33no—
344C2×C22⋊C4—36no—
345C2×C22⋊C4—33no—
346C2×C22⋊C4—36no—
347C2×C22⋊C4—33no—
348C2×C22⋊C4C22×D313yes—
349C2×C22⋊C4C22×D313yes—
350C2×C42—33no—
351C2×C42—33no—
352C2×C42—33no—
353C2×C42—33no—
354C2×C42—33no—
355C2×C42—31no—
356C2×C42—33no—
357C2×C42C22×D313yes—
358C2×C42C22×D313yes—
359#95—31no—
360#95—31no—
361#95—33no—
362#95—33no—
363#95—36no—
364#95—33no—
365#95—33no—
366C23×C3—23no—
367C23×C3—23no—
368C23×C3—23no—
369C23×C3—23no—
370C23×C3—23no—
371C23×C3—23no—
372C23×C3C23×C416yes—
373C23×C3C2511yescharacteristic
374C22×D3—23no—
375C22×D3—23no—
376C22×D3—23no—
377C22×D3—23no—
378C22×D3—23no—
379C22×D3—23no—
380C22×D3C23×C416yes—
381C3×Q8C4×D4116yescomplemented
382C3×D4—224no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—23no—
386C2×C3×C4—23no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—224no—
390C2×C3×C4—21no—
391C2×C3×C4—23no—
392C2×C3×C4—23no—
393C2×C3×C4—23no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—812no—
397C2×C3×C4C23×C412yes—
398C2×C3×C4C23×C416yes—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C22×D416yes—
401C2×C3×C4C22×D416yes—
402C2×C3×C4C2×#4116yes—
403#67—224no—
404#67—424no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—23no—
409C2×Dic3—21no—
410C2×Dic3—23no—
411C2×Dic3—21no—
412C2×Dic3—23no—
413C2×Dic3—23no—
414C2×Dic3—23no—
415C2×Dic3—23no—
416C2×Dic3—23no—
417C2×Dic3—21no—
418C2×Dic3—23no—
419C2×Dic3—21no—
420C2×Dic3C23×C416yes—
421C2×Dic3C23×C412yes—
422C2×Dic3C23×C412yes—
423C2×Dic3H(5,𝔽2)12yes—
424C2×Dic3H(5,𝔽2)12yes—
425C2×Dic32−1+412yes—
426D12—224no—
427C4×D3—224no—
428C4×D3—812no—
429C24—33no—
430C24C23×D313yes—
431#41—232no—
432#41—632no—
433C2×Q8C2×C4×D314yes—
434C2×Q8#23514yes—
435C2×Q8C2×#6714yes—
436C2×D4—26no—
437C2×D4—26no—
438C2×D4—312no—
439C2×D4—324no—
440C2×D4—312no—
441C2×D4—66no—
442C2×D4—66no—
443C2×D4—66no—
444C2×D4—66no—
445C2×D4—126no—
446C2×D4—126no—
447C2×D4C2×C4×D3112yes—
448C22×C4—26no—
449C22×C4—26no—
450C22×C4—212no—
451C22×C4—31no—
452C22×C4—33no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—312no—
456C22×C4—33no—
457C22×C4—31no—
458C22×C4—33no—
459C22×C4—46no—
460C22×C4—46no—
461C22×C4—66no—
462C22×C4—66no—
463C22×C4—612no—
464C22×C4—126no—
465C22×C4—126no—
466C22×C4C2×C4×D3112yes—
467C22×C4C23×D311yescharacteristic
468C22×C4C23×D313yes—
469C22×C4C23×D311yescharacteristic
470C22×C4C23×D313yes—
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D313yes—
473C4⋊C4—26no—
474C4⋊C4—26no—
475C4⋊C4—22no—
476C4⋊C4—34no—
477C4⋊C4—312no—
478C4⋊C4—66no—
479C4⋊C4—62no—
480C4⋊C4—66no—
481C4⋊C4—66no—
482C4⋊C4—66no—
483C4⋊C4—62no—
484C4⋊C4—126no—
485C4⋊C4C2×#67112yes—
486C22⋊C4—26no—
487C22⋊C4—26no—
488C22⋊C4—312no—
489C22⋊C4—312no—
490C22⋊C4—66no—
491C22⋊C4—66no—
492C22⋊C4—66no—
493C22⋊C4—66no—
494C22⋊C4—126no—
495C22⋊C4—123no—
496C22⋊C4—126no—
497C22⋊C4—123no—
498C42—26no—
499C42—312no—
500C42—66no—
501C42—66no—
502C42—62no—
503C42—66no—
504C42—66no—
505C42—123no—
506C42C2×#67112yes—
507C22×C3—26no—
508C22×C3—26no—
509C22×C3—26no—
510C22×C3—43no—
511C22×C3—43no—
512C22×C3—43no—
513C22×C3—43no—
514C22×C3C24×C411yescharacteristic
515C22×C3C23×D411yescharacteristic
516C22×C3C22×#4111yescharacteristic
517C22×C3C2×H(5,𝔽2)11yescharacteristic
518C22×C3C2×H(5,𝔽2)11yescharacteristic
519C22×C3C2×2−1+411yescharacteristic
520C22×C3#58511yescharacteristic
521C2×D3—26no—
522C2×D3—26no—
523C2×D3—26no—
524C2×D3—43no—
525C2×D3—43no—
526C2×D3—43no—
527C2×D3—43no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—26no—
531C3×C4—41no—
532C3×C4—43no—
533C3×C4C2×C4×D4112yes—
534Dic3—22no—
535Dic3—26no—
536Dic3—22no—
537Dic3—41no—
538Dic3—43no—
539Dic3—43no—
540Dic3#52014yescomplemented
541C23—23no—
542C23—23no—
543C23—23no—
544C23—23no—
545C23—23no—
546C23—23no—
547C23—36no—
548C23—63no—
549C23—63no—
550C23—63no—
551C23—63no—
552C23—63no—
553C23—63no—
554C23C22×C4×D316yes—
555C23C24×D311yescharacteristic
556Q8C4×#67116yescomplemented
557D4—224no—
558D4—624no—
559D4—624no—
560D4—1224no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—224no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—36no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—63no—
583C2×C4—61no—
584C2×C4—63no—
585C2×C4—61no—
586C2×C4—624no—
587C2×C4—63no—
588C2×C4—63no—
589C2×C4—63no—
590C2×C4—63no—
591C2×C4—61no—
592C2×C4—63no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—63no—
596C2×C4—812no—
597C2×C4—2412no—
598C2×C4C22×C4×D312yes—
599C2×C4C22×C4×D316yes—
600C2×C4C2×#23516yes—
601C2×C4C22×#6712yes—
602C2×C4C22×#6716yes—
603C2×C4C22×#6716yes—
604C2×C3—46no—
605C2×C3C22×C4×D411yescharacteristic
606C2×C3C2×#52011yescharacteristic
607C2×C3C4×H(5,𝔽2)11yescharacteristic
608C2×C3#349711yescharacteristic
609C2×C3#349811yescharacteristic
610C2×C3#351711yescharacteristic
611C2×C3#352211yescharacteristic
612D3—46no—
613C22—26no—
614C22—26no—
615C22—26no—
616C22—43no—
617C22—43no—
618C22—43no—
619C22—43no—
620C22—66no—
621C22—66no—
622C22—66no—
623C22—123no—
624C22—123no—
625C22—123no—
626C22—123no—
627C22C23×C4×D311yescharacteristic
628C22C22×#23511yescharacteristic
629C22D3×H(5,𝔽2)11yescharacteristic
630C22D3×H(5,𝔽2)11yescharacteristic
631C22D3×2−1+411yescharacteristic
632C22#596811yescharacteristic
633C22C23×#6711yescharacteristic
634C4—22no—
635C4—26no—
636C4—22no—
637C4—34no—
638C4—43no—
639C4—41no—
640C4—66no—
641C4—62no—
642C4—62no—
643C4—123no—
644C4—123no—
645C4—121no—
646C4C2×C4×#67112yes—
647C3#3431311yescharacteristic, complemented, Sylow
648C2—46no—
649C2—126no—
650C2#8588611yescharacteristic
651C2#8589611yescharacteristic
652C2D3×#52011yescharacteristic
653C2#8590811yescharacteristic
654C2C22×C4×#6711yescharacteristic
655C2#8728211yescharacteristic
656C2#8729111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC2×C3×#520C2
socle, cosocleC23×C3C25
prime radicalC2×#5834C2
hypercenterC2×#520D3
hypocenterC3#34313
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×C3×#520C2semidirect
3. C2×#5574C2semidirect
4. #84089C2semidirect
5. Q8×Dic3⋊C4C2semidirect
6. #82457C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #82399C2semidirect
9. #82396C2semidirect
10. #82343C2semidirect
11. C4×#940C2semidirect
12. #82236C2semidirect
13. #82225C2semidirect
14. C4⋊C4×#67C2semidirect
15. #82163C2semidirect
26. #5834C4semidirect
72. C2×#520D3semidirect
87. C2×C3×#41C2×C4semidirect
96. C2×#239C2×C4semidirect
101. C3×C4×Q8D4semidirect
120. Q8×Dic3C2×C4semidirect
121. #940C2×C4semidirect
122. #938C2×C4semidirect
123. #930C2×C4semidirect
126. C4×#67C2×C4semidirect
296. C2×#41C4×D3semidirect
313. C4×Q8#67semidirect
381. C3×Q8C4×D4semidirect
540. Dic3#520semidirect
556. Q8C4×#67semidirect
647. C3#34313semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×#58342
C2×#55742
#840892
Q8×Dic3⋊C42
#824572
C4×Q8×Dic32
#823992
#823962
#823432
C4×#9402
#822362
#822252
C4⋊C4×#672
#821632
#343133