Finite Group Databasebeta

← #10824650

Subgroups — #10824650

Order 768.

978 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×C3×#431C211yesmaximal, characteristic, complemented
2C4×#924C211yesmaximal, characteristic, complemented
3C4×#930C211yesmaximal, characteristic
4C2×C4×#204C211yesmaximal, characteristic, complemented
5#70489C211yesmaximal, characteristic, complemented
6#70484C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#69962C211yesmaximal, characteristic
10#69940C211yesmaximal, characteristic, complemented
11#69937C211yesmaximal, characteristic, complemented
12#69934C211yesmaximal, characteristic
13C4×#818C211yesmaximal, characteristic, complemented
14C4×#816C211yesmaximal, characteristic, complemented
15C4×#821C211yesmaximal, characteristic, complemented
16#23701—31nomaximal, Sylow
17C22×C3×C42C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×C4×#67—21no—
20C2×#924C2211yescharacteristic
21C22×#204C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#431—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31#5213—21no—
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5210—21no—
35C4×#217—21no—
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45#4940—21no—
46#4939C2211yescharacteristic
47C4×(C2×D3)⋊C4—21no—
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4933—21no—
51C4×Dic3⋊C4—21no—
52#4929C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55C4×#204—24no—
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431D311yescharacteristic, complemented
64C4×C22⋊Q8—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2067—31no—
70#2055—31no—
71#2036—31no—
72#2033—31no—
73#2000—31no—
74C4×#107—31no—
75C4×#103—31no—
76C4×#102—31no—
77C23×C3×C4D411yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97#930—22no—
98#930—22no—
99C4×#67—44no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—22no—
111#924—22no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×#204—22no—
120C2×#204—22no—
121C2×#204—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×C4⋊C8—22no—
127C3×C4⋊C8—22no—
128C3×C22⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×C4×C8—22no—
131C3×C4×C8D414yescomplemented
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#821—22no—
143#818—22no—
144#818—22no—
145#817—22no—
146#816—22no—
147#816—22no—
148#816—22no—
149C22×SD16—31no—
150C2×C22⋊Q8—31no—
151C2×#121—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42C2×D311yescharacteristic
156C4×SD16—64no—
157#431—24no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×C4⋊C8C2×D311yescharacteristic
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×C22⋊C8C2×D311yescharacteristic
171C2×C4×C8C2×D311yescharacteristic
172C2×C4×C8C2×D311yescharacteristic
173#391—31no—
174#390—31no—
175#389—31no—
176#389—31no—
177#388—61no—
178#387—61no—
179#386—61no—
180#385—61no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186C4×C22⋊C4—61no—
187C4×C22⋊C4—61no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—24no—
215C2×#67—42no—
216C2×#67—42no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12#4112yes—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8C2×D412yes—
250C2×C3×C8C2×D412yes—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42C2×D412yes—
256C3×C42#4112yes—
257#217—24no—
258#217—42no—
259#217—42no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3#4112yes—
275C4⋊Dic3#4112yes—
276Dic3⋊C4—24no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—42no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C4×Dic3—42no—
283#204—28no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4—61no—
287C23×C4—61no—
288C23×C4D1211yescharacteristic
289C23×C4D1211yescharacteristic
290C23×C4C22×D311yescharacteristic
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300#121—62no—
301#121—62no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42D1211yescharacteristic
330C2×C42D1211yescharacteristic
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—62no—
335#106—62no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C4×C8—22no—
347C4×C8#6714yescomplemented
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3#13512yes—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C2×C3×C4—28no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—44no—
387C2×C3×C4#12112yes—
388C2×C3×C4#12112yes—
389C2×C3×C4#13512yes—
390C2×C3×C4AGL(1,ℤ/8)12yes—
391C2×C3×C4#13712yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398#67—48no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424C4×D3—84no—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—22no—
431C3×C8—41no—
432C3×C8—41no—
433C24—61no—
434C24C2×D1211yescharacteristic
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—122no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4C2×D1211yescharacteristic
473C22×C4C2×D1211yescharacteristic
474C22×C4C2×D1211yescharacteristic
475C22×C4C2×D1211yescharacteristic
476C22×C4C2×D1211yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8C2×#6712yes—
490C2×C8C2×#6712yes—
491C4⋊C4—32no—
492C4⋊C4—32no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—122no—
519C42—122no—
520C42—122no—
521C42#23512yes—
522C42C2×#6712yes—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3C2×#12111yescharacteristic
531C22×C3#52511yescharacteristic
532C22×C3#52511yescharacteristic
533C22×C3C2×#13511yescharacteristic
534C22×C3C2×#13511yescharacteristic
535C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
536C22×C3C2×#13711yescharacteristic
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4#46814yes—
549C3×C4#52511yescharacteristic
550C3×C4#52911yescharacteristic
551C3×C4#52911yescharacteristic
552C3×C4#53211yescharacteristic
553Dic3—42no—
554Dic3—42no—
555Dic3—42no—
556Dic3—81no—
557Dic3—81no—
558Dic3—81no—
559Dic3—81no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—62no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23#90412yes—
574C23C22×D1211yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—62no—
580D4—62no—
581D4—128no—
582D4—121no—
583D4—121no—
584C2×C4—28no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—244no—
622C2×C4#90412yes—
623C2×C4#90812yes—
624C2×C4#90912yes—
625C2×C4#93012yes—
626C2×C4#93012yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4C22×#6711yescharacteristic
633C8—22no—
634C8—22no—
635C8—41no—
636C8—41no—
637C2×C3—42no—
638C2×C3#239211yescharacteristic
639C2×C3#305011yescharacteristic
640C2×C3C2×#46811yescharacteristic
641C2×C3#310511yescharacteristic
642C2×C3#310611yescharacteristic
643C2×C3#310711yescharacteristic
644C2×C3#310811yescharacteristic
645D3—82no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22#552411yescharacteristic
661C22#552411yescharacteristic
662C22C2×#90411yescharacteristic
663C22C2×#90411yescharacteristic
664C22C2×#90811yescharacteristic
665C22C2×#90911yescharacteristic
666C22C2×#93011yescharacteristic
667C4—22no—
668C4—22no—
669C4—41no—
670C4—41no—
671C4—122no—
672C4—122no—
673C4—122no—
674C4—241no—
675C4—241no—
676C4—241no—
677C4—241no—
678C4#513414yes—
679C4#552811yescharacteristic
680C4#552811yescharacteristic
681C4#582811yescharacteristic
682C4#583211yescharacteristic
683C3#2370111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—242no—
686C2#7930011yescharacteristic
687C2#7930111yescharacteristic
688C2#7930211yescharacteristic
689C2#7931111yescharacteristic
690C2#8227311yescharacteristic
691C2#8374211yescharacteristic
692C2C2×#513411yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#431C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#431D3
hypocenterC3#23701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
1. C2×C3×#431C2semidirect
2. C4×#924C2semidirect
4. C2×C4×#204C2semidirect
5. #70489C2semidirect
6. #70484C2semidirect
7. #70482C2semidirect
8. #70464C2semidirect
10. #69940C2semidirect
11. #69937C2semidirect
13. C4×#818C2semidirect
14. C4×#816C2semidirect
15. C4×#821C2semidirect
63. C2×#431D3semidirect
131. C3×C4×C8D4semidirect
347. C4×C8#67semidirect
683. C3#23701semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4312
C4×#9242
C4×#9302
C2×C4×#2042
#704892
#704842
#704822
#704642
#699622
#699402
#699372
#699342
C4×#8182
C4×#8162
C4×#8212
#237013