Finite Group Databasebeta

← #10824658

Subgroups — #10824658

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
1C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C4×#930C211yesmaximal, characteristic
4C2×#4925C211yesmaximal, characteristic, complemented
5#71699C211yesmaximal, characteristic
6#71695C211yesmaximal, characteristic, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8#69683C211yesmaximal, characteristic, complemented
9#69680C211yesmaximal, characteristic, complemented
10#69630C211yesmaximal, characteristic, complemented
11#69624C211yesmaximal, characteristic, complemented
12#69523C211yesmaximal, characteristic
13#69520C211yesmaximal, characteristic, complemented
14C4×#807C211yesmaximal, characteristic, complemented
15C4×#809C211yesmaximal, characteristic, complemented
16#23701—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×C4×#67—21no—
23C2×C4×D12C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#387—21no—
27C3×#385—21no—
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5213—21no—
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C4×#215—24no—
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4940—21no—
47#4939C2211yescharacteristic
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51#4925—24no—
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431—31no—
64C4×C22⋊Q8D311yescharacteristic, complemented
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—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4D411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—22no—
97C2×C3×C42—22no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#215—22no—
106C2×#215—22no—
107C2×#215—22no—
108#930—22no—
109#930—22no—
110C4×#67—44no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C4×D12—22no—
125C4×D12—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#835—22no—
133#835—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830—22no—
138#830—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—22no—
147C4×C3⋊C8—22no—
148C4×C3⋊C8D414yescomplemented
149C22×SD16—31no—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×#121—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42C2×D311yescharacteristic
156C4×SD16—64no—
157#431—64no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—31no—
171C2×C4×C8—31no—
172C2×C4×C8—31no—
173#391C2×D311yescharacteristic
174#390—31no—
175#389—31no—
176#389C2×D311yescharacteristic
177#388—61no—
178#387—21no—
179#386—61no—
180#385—21no—
181#380—31no—
182#380C2×D311yescharacteristic
183C4×C4⋊C4—21no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—21no—
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—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C2×#67—24no—
225C2×#67—42no—
226C2×#67—42no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12#4112yes—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4#4112yes—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—42no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—42no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#217—42no—
259#215—28no—
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#4112yes—
270Dic3⋊C4—42no—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8C2×D412yes—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—61no—
287C23×C4—61no—
288C23×C4#6711yescharacteristic
289C23×C4#6711yescharacteristic
290C23×C4C22×D311yescharacteristic
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—31no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300#121—62no—
301#121—62no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
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—21no—
327C2×C42—21no—
328C2×C42—61no—
329C2×C42#6711yescharacteristic
330C2×C42#6711yescharacteristic
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—62no—
335#106—62no—
336C4⋊C8—62no—
337C4⋊C8—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344C22⋊C8—62no—
345C22⋊C8—62no—
346C4×C8—34no—
347C4×C8—62no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
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—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—28no—
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—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—44no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4#12112yes—
404C2×C3×C4#12112yes—
405C2×C3×C4#13512yes—
406C2×C3×C4AGL(1,ℤ/8)12yes—
407C2×C3×C4#13712yes—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414#67—48no—
415C2×Dic3—22no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428C4×D3—84no—
429C3⋊C8—22no—
430C3⋊C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434C24C2×#6711yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23512yes—
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—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479C2×C8—32no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—32no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—122no—
506C4⋊C4#23512yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—64no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—122no—
521C42D3×D412yes—
522C42#23712yes—
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—42no—
547C3×C4—41no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4#46814yes—
553C3×C4#52511yescharacteristic
554C3×C4#52911yescharacteristic
555C3×C4#52911yescharacteristic
556C3×C4#53211yescharacteristic
557Dic3—42no—
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#95012yes—
574C23C22×#6711yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
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—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—62no—
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#93912yes—
623C2×C4#93912yes—
624C2×C4#94212yes—
625C2×C4#94912yes—
626C2×C4#95012yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—62no—
634C8—62no—
635C8—121no—
636C8—121no—
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#579111yescharacteristic
661C22C2×#93911yescharacteristic
662C22C2×#94211yescharacteristic
663C22C2×#94911yescharacteristic
664C22C2×#95011yescharacteristic
665C22C2×#95011yescharacteristic
666C22#582811yescharacteristic
667C4—22no—
668C4—22no—
669C4—41no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—122no—
676C4—241no—
677C4—241no—
678C4#504914yes—
679C4#553811yescharacteristic
680C4#554711yescharacteristic
681C4#554811yescharacteristic
682C4#555711yescharacteristic
683C3#2370111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—242no—
686C2#8234511yescharacteristic
687C2#8244311yescharacteristic
688C2#8252911yescharacteristic
689C2C2×#504911yescharacteristic
690C2#8292211yescharacteristic
691C2#8314111yescharacteristic
692C2#8335211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC4×C22⋊Q8D3
hypocenterC3#23701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×C22⋊Q8C2semidirect
2. C2×C4×#215C2semidirect
4. C2×#4925C2semidirect
6. #71695C2semidirect
7. C4×#835C2semidirect
8. #69683C2semidirect
9. #69680C2semidirect
10. #69630C2semidirect
11. #69624C2semidirect
13. #69520C2semidirect
14. C4×#807C2semidirect
15. C4×#809C2semidirect
64. C4×C22⋊Q8D3semidirect
148. C4×C3⋊C8D4semidirect
683. C3#23701semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×C4×#2152
C4×#9302
C2×#49252
#716992
#716952
C4×#8352
#696832
#696802
#696302
#696242
#695232
#695202
C4×#8072
C4×#8092
#237013