Finite Group Databasebeta

← #10957278

Subgroups — #10957278

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×#5834C211yesmaximal, characteristic
2C2×#5827C211yesmaximal, characteristic, complemented
3C2×#5574C211yesmaximal, characteristic, complemented
4C3×#2341C211yesmaximal, characteristic, complemented
5Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
6#82457C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#76433C213yesmaximal, complemented
9#76430C213yesmaximal, complemented
10#76306C213yesmaximal, complemented
11#76293C213yesmaximal, complemented
12#74010C213yesmaximal, complemented
13#73872C213yesmaximal, complemented
14#73467C213yesmaximal, complemented
15#73439C213yesmaximal, complemented
16#34313—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C22×#239C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21#5834C4116yescomplemented
22#5827—28no—
23C2×Q8×Dic3C2211yescharacteristic
24C2×Q8×Dic3C2211yescharacteristic
25C2×Q8×Dic3C2211yescharacteristic
26C2×#940C2213yes—
27C2×#938C2213yes—
28C2×D4×Dic3C2213yes—
29C2×D4×Dic3C2213yes—
30C2×#930C2213yes—
31C2×C4×#67C2213yes—
32C2×#926C2213yes—
33C22×C4⋊Dic3C2213yes—
34#5574—28no—
35C2×#893C2213yes—
36C2×D3×C4⋊C4C2213yes—
37C2×#67⋊C4C2216yes—
38C2×#67⋊C4C2216yes—
39C2×C4×D12C2213yes—
40C2×#872C2213yes—
41C3×#390C2213yes—
42C3×#390C2213yes—
43C3×#380C2213yes—
44C3×C4×C22⋊C4C2213yes—
45#4988C2213yes—
46#4987C2213yes—
47#4975C2213yes—
48Dic3×C4⋊C4C2213yes—
49Dic3×C4⋊C4C2213yes—
50#4972C2213yes—
51#4958C2216yes—
52#4957C2213yes—
53#4957C2216yes—
54#4957C2213yes—
55#4956C2216yes—
56#4946C2216yes—
57#4944C2213yes—
58#4944C2213yes—
59#4939C2213yes—
60C4×(C2×D3)⋊C4C2213yes—
61C4×C4⋊Dic3C2213yes—
62C4×Dic3⋊C4C2213yes—
63C42×Dic3C2213yes—
64#4928C2213yes—
65#4670C2213yes—
66#4670C2213yes—
67#4668C2216yes—
68#4647C2213yes—
69#4647C2211yescharacteristic
70C2×#550—31no—
71C2×#520—31no—
72C2×#520—31no—
73#2366—33no—
74#2361—33no—
75#2361—33no—
76Q8×C4⋊C4—31no—
77D4×C4⋊C4—33no—
78#2341—31no—
79#2341D311yescharacteristic, complemented
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—
91C22×#67C2313yes—
92C22×#67C2316yes—
93C23×Dic3C2316yes—
94C2×#239—24no—
95C2×#239—24no—
96C2×#239C2×C418yescomplemented
97C22×D12C2313yes—
98C22×C4×D3C2313yes—
99C22×C4×D3C2316yes—
100C3×C4×Q8—24no—
101C3×C4×Q8—24no—
102C2×C3×C4⋊C4C2313yes—
103C2×C3×C4⋊C4C2313yes—
104C2×C3×C22⋊C4C2313yes—
105C2×C3×C22⋊C4C2313yes—
106C2×C3×C22⋊C4C2316yes—
107C2×C3×C42C2313yes—
108C2×C3×C42C2313yes—
109C2×#217C2313yes—
110C2×#217C2313yes—
111C2×#217C2313yes—
112Q8×Dic3—24no—
113Q8×Dic3C2×C418yescomplemented
114Q8×Dic3D418yescomplemented
115#940C2×C4124yescomplemented
116#938C2×C4124yescomplemented
117D4×Dic3—212no—
118D4×Dic3—212no—
119#930C2×C4124yescomplemented
120C4×#67C2×C4124yescomplemented
121C2×(C2×D3)⋊C4C2313yes—
122C2×(C2×D3)⋊C4C2313yes—
123C2×(C2×D3)⋊C4C2313yes—
124C2×(C2×D3)⋊C4C2316yes—
125C2×(C2×D3)⋊C4C2313yes—
126C2×(C2×D3)⋊C4C2313yes—
127#926—212no—
128C2×C4⋊Dic3—212no—
129C2×C4⋊Dic3C2313yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2313yes—
132C2×C4⋊Dic3C2313yes—
133C2×C4⋊Dic3C2313yes—
134C2×Dic3⋊C4C2311yescharacteristic
135C2×Dic3⋊C4C2313yes—
136C2×Dic3⋊C4C2311yescharacteristic
137C2×Dic3⋊C4C2313yes—
138C2×Dic3⋊C4C2316yes—
139C2×Dic3⋊C4C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145C2×C4×Dic3C2313yes—
146C2×C4×Dic3C2313yes—
147#893—212no—
148D3×C4⋊C4—212no—
149#67⋊C4—412no—
150#67⋊C4—412no—
151C4×D12—212no—
152#872—212no—
153C3×#95C2313yes—
154C3×#95C2311yescharacteristic
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2313yes—
157C3⋊#95C2316yes—
158C3⋊#95C2313yes—
159C3⋊#95C2313yes—
160C22×#41—31no—
161C22×#41C2×D311yescharacteristic
162#550—316no—
163C2×#127—33no—
164C2×#121—33no—
165C2×#121—33no—
166#520—68no—
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—33no—
175C2×C4×D4—33no—
176C2×C4×D4—33no—
177C2×C4×D4—36no—
178C2×C4×D4—36no—
179C2×C4×D4—33no—
180C2×#117—33no—
181C2×#117—33no—
182C22×C4⋊C4—33no—
183C22×C4⋊C4—33no—
184#390—33no—
185#390—33no—
186#390—36no—
187#390—33no—
188#390—33no—
189#390C2×D313yes—
190#390C2×D313yes—
191#389—33no—
192#389—33no—
193#389—33no—
194#389—31no—
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#380C2×D313yes—
206C4×C4⋊C4—33no—
207C4×C4⋊C4—33no—
208C4×C4⋊C4—33no—
209C4×C4⋊C4—33no—
210C4×C22⋊C4—33no—
211C4×C22⋊C4C2×D313yes—
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×C4C22×C4112yes—
225C22×C3×C4C2413yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2413yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2413yes—
231C2×#67—26no—
232C2×#67—26no—
233C2×#67—46no—
234C2×#67—46no—
235C2×#67C22×C4112yes—
236C2×#67C22×C4124yes—
237C22×Dic3—212no—
238C22×Dic3—46no—
239C22×Dic3—46no—
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2413yes—
242C22×Dic3C2413yes—
243C22×Dic3C2411yescharacteristic
244C22×Dic3C2413yes—
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×C22⋊C4—26no—
260C3×C22⋊C4—46no—
261C3×C22⋊C4—43no—
262C3×C42—26no—
263C3×C42—26no—
264#217—26no—
265#217—26no—
266#217C22×C4112yes—
267(C2×D3)⋊C4—26no—
268(C2×D3)⋊C4—26no—
269(C2×D3)⋊C4—26no—
270(C2×D3)⋊C4—46no—
271(C2×D3)⋊C4—43no—
272(C2×D3)⋊C4C22×C4112yes—
273C4⋊Dic3—26no—
274C4⋊Dic3—26no—
275C4⋊Dic3—22no—
276C4⋊Dic3C22×C4112yes—
277C4⋊Dic3C2×D4112yes—
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—22no—
288C4×Dic3—43no—
289C4×Dic3C22×C4112yes—
290C4×Dic3C2×D4112yes—
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×C4—36no—
305C23×C4C22×D313yes—
306#127—324no—
307#121—324no—
308#121—324no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—38no—
312C4×Q8—38no—
313C4×Q8—64no—
314C4×D4—324no—
315C4×D4—612no—
316C4×D4—612no—
317C4×D4—612no—
318C4×D4—612no—
319C4×D4—1212no—
320C4×D4—1212no—
321#117—612no—
322#117—612no—
323C2×C4⋊C4—33no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—33no—
326C2×C4⋊C4—33no—
327C2×C4⋊C4—33no—
328C2×C4⋊C4—33no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—33no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—36no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4C22×D313yes—
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—33no—
345C2×C22⋊C4—36no—
346C2×C22⋊C4—33no—
347C2×C22⋊C4C22×D313yes—
348C2×C22⋊C4C22×D316yes—
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—33no—
361#95—36no—
362#95—33no—
363#95—33no—
364#95C22×D311yescharacteristic
365#95C22×D313yes—
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—21no—
386C2×C3×C4—21no—
387C2×C3×C4—224no—
388C2×C3×C4—21no—
389C2×C3×C4—23no—
390C2×C3×C4—23no—
391C2×C3×C4—23no—
392C2×C3×C4—23no—
393C2×C3×C4—23no—
394C2×C3×C4—21no—
395C2×C3×C4—23no—
396C2×C3×C4—21no—
397C2×C3×C4C23×C412yes—
398C2×C3×C4C23×C416yes—
399C2×C3×C4C22×D416yes—
400C2×C3×C4C2×#4116yes—
401C2×C3×C4H(5,𝔽2)12yes—
402#67—224no—
403#67—424no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—23no—
407C2×Dic3—23no—
408C2×Dic3—23no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—23no—
412C2×Dic3—21no—
413C2×Dic3—23no—
414C2×Dic3—23no—
415C2×Dic3—23no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—812no—
419C2×Dic3C23×C416yes—
420C2×Dic3C23×C412yes—
421C2×Dic3C23×C412yes—
422C2×Dic3C22×D412yes—
423C2×Dic3C22×D416yes—
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×Q8D3×D414yes—
435C2×Q8#23714yes—
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—31no—
451C22×C4—33no—
452C22×C4—31no—
453C22×C4—33no—
454C22×C4—31no—
455C22×C4—312no—
456C22×C4—31no—
457C22×C4—33no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—66no—
461C22×C4—66no—
462C22×C4—126no—
463C22×C4—126no—
464C22×C4—126no—
465C22×C4—126no—
466C22×C4C2×C4×D3112yes—
467C22×C4C23×D313yes—
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D313yes—
470C22×C4C23×D311yescharacteristic
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D313yes—
473C4⋊C4—26no—
474C4⋊C4—26no—
475C4⋊C4—34no—
476C4⋊C4—312no—
477C4⋊C4—312no—
478C4⋊C4—66no—
479C4⋊C4—66no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—66no—
483C4⋊C4—62no—
484C4⋊C4—66no—
485C4⋊C4—126no—
486C22⋊C4—26no—
487C22⋊C4—312no—
488C22⋊C4—312no—
489C22⋊C4—46no—
490C22⋊C4—43no—
491C22⋊C4—66no—
492C22⋊C4—66no—
493C22⋊C4—66no—
494C22⋊C4—66no—
495C22⋊C4—66no—
496C22⋊C4—126no—
497C22⋊C4—123no—
498C42—26no—
499C42—26no—
500C42—312no—
501C42—312no—
502C42—66no—
503C42—62no—
504C42—66no—
505C42—66no—
506C42—123no—
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—41no—
531C3×C4—43no—
532C3×C4—43no—
533C3×C4C2×C4×D4112yes—
534Dic3—22no—
535Dic3—26no—
536Dic3—22no—
537Dic3—26no—
538Dic3—41no—
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
556Q8#67⋊C4116yescomplemented
557D4—224no—
558D4—624no—
559D4—624no—
560D4—1224no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—224no—
566C2×C4—21no—
567C2×C4—23no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—21no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—21no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—36no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—36no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—63no—
585C2×C4—63no—
586C2×C4—61no—
587C2×C4—63no—
588C2×C4—63no—
589C2×C4—61no—
590C2×C4—624no—
591C2×C4—61no—
592C2×C4—63no—
593C2×C4—63no—
594C2×C4—61no—
595C2×C4—63no—
596C2×C4—61no—
597C2×C4—2412no—
598C2×C4—2412no—
599C2×C4C22×C4×D312yes—
600C2×C4C22×C4×D316yes—
601C2×C4C2×D3×D416yes—
602C2×C4C2×#23716yes—
603C2×C4#100912yes—
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×D3×D411yescharacteristic
629C22C22×#23711yescharacteristic
630C22C2×#100911yescharacteristic
631C22#596411yescharacteristic
632C22D3×H(5,𝔽2)11yescharacteristic
633C22D3×2−1+411yescharacteristic
634C4—22no—
635C4—22no—
636C4—34no—
637C4—41no—
638C4—43no—
639C4—43no—
640C4—66no—
641C4—62no—
642C4—62no—
643C4—66no—
644C4—121no—
645C4—123no—
646C4C2×#67⋊C4112yes—
647C3#3431311yescharacteristic, complemented, Sylow
648C2—46no—
649C2—126no—
650C2C22×#67⋊C411yescharacteristic
651C2C4×#100911yescharacteristic
652C2D3×#52011yescharacteristic
653C2#8672211yescharacteristic
654C2#8673711yescharacteristic
655C2#8677011yescharacteristic
656C2#8677211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2341C2
socle, cosocleC23×C3C25
prime radicalC2×#5834C2
hypercenter#2341D3
hypocenterC3#34313
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. C2×#5827C2semidirect
3. C2×#5574C2semidirect
4. C3×#2341C2semidirect
5. Q8×Dic3⋊C4C2semidirect
6. #82457C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #76433C2semidirect
9. #76430C2semidirect
10. #76306C2semidirect
11. #76293C2semidirect
12. #74010C2semidirect
13. #73872C2semidirect
14. #73467C2semidirect
15. #73439C2semidirect
21. #5834C4semidirect
79. #2341D3semidirect
87. C2×C3×#41C2×C4semidirect
96. C2×#239C2×C4semidirect
113. Q8×Dic3C2×C4semidirect
114. Q8×Dic3D4semidirect
115. #940C2×C4semidirect
116. #938C2×C4semidirect
119. #930C2×C4semidirect
120. C4×#67C2×C4semidirect
296. C2×#41C4×D3semidirect
381. C3×Q8C4×D4semidirect
540. Dic3#520semidirect
556. Q8#67⋊C4semidirect
647. C3#34313semidirect

Maximal subgroups

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

subgroupindex
C2×#58342
C2×#58272
C2×#55742
C3×#23412
Q8×Dic3⋊C42
#824572
C4×Q8×Dic32
#764332
#764302
#763062
#762932
#740102
#738722
#734672
#734392
#343133