Finite Group Databasebeta

← #10826285

Subgroups — #10826285

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C23×#216C211yesmaximal, characteristic, complemented
3C2×#5228C212yesmaximal, complemented
4C2×#5050C211yesmaximal, characteristic, complemented
5#82349C211yesmaximal, characteristic
6#72132C211yesmaximal, characteristic, complemented
7#71793C212yesmaximal, complemented
8#71700C212yesmaximal, complemented
9#71696C211yesmaximal, characteristic, complemented
10#71632C212yesmaximal, complemented
11#71610C211yesmaximal, characteristic
12#23856—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C3×D4×Q8—24no—
17C3×D4×Q8—24no—
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C22×#216—24no—
28C22×#216—28no—
29C22×#216C2212yes—
30C22×#216C2212yes—
31C22×#216C2212yes—
32C2×#924C2212yes—
33C23×C3⋊C8C2211yescharacteristic
34#5228—28no—
35C2×#835C2212yes—
36C2×#835C2212yes—
37C2×#835C2212yes—
38#5220—21no—
39#5212C2211yescharacteristic
40C2×#830C2212yes—
41C2×#830C2211yescharacteristic
42#5050—24no—
43#4979—21no—
44C2×#810C2212yes—
45#4947—21no—
46#4946—21no—
47#4929C2212yes—
48#4529C2212yes—
49#4524C2212yes—
50C23×Q16—31no—
51C2×D4×Q8D311yescharacteristic
52C2×#451—32no—
53C2×#451—31no—
54#2409—31no—
55#2016—32no—
56#2001—32no—
57#1892—31no—
58#1878—32no—
59#1869—31no—
60#1787—31no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—21no—
64C22×C3×Q8—21no—
65C22×C3×Q8—22no—
66C22×C3×Q8—24no—
67C22×C3×Q8—24no—
68C22×C3×Q8—42no—
69C22×C3×Q8—42no—
70C22×C3×Q8D414yes—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×Q8C2311yescharacteristic
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4D411yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6—22no—
81C22×Dic6—24no—
82C22×Dic6—22no—
83C22×Dic6—21no—
84C22×Dic6—42no—
85C22×Dic6C2312yes—
86C3×C4⋊Q8—28no—
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8—22no—
89C3×C22⋊Q8—28no—
90C3×C22⋊Q8—28no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8D414yes—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×D4—28no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×#217—21no—
114C2×#217—22no—
115C2×#216—232no—
116C2×#216—24no—
117C2×#216—24no—
118C2×#216—24no—
119C2×#216—24no—
120C2×#216—24no—
121C2×#216—24no—
122C2×#216—24no—
123C2×#216—42no—
124C2×#216D418yescomplemented
125C2×C4⋊Dic3C2312yes—
126#924—24no—
127#924—42no—
128C2×Dic3⋊C4—22no—
129C22×C3⋊C8—22no—
130C22×C3⋊C8—24no—
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2312yes—
133#835—24no—
134#835—24no—
135#835—42no—
136C3⋊#95—22no—
137C3⋊#95—22no—
138C3⋊#95—21no—
139#830—22no—
140#830—24no—
141#810—24no—
142C23×Q8—31no—
143C23×Q8C2×D311yescharacteristic
144C23×Q8C2×D311yescharacteristic
145C22×Q16—32no—
146C22×Q16—32no—
147C22×Q16—32no—
148C22×Q16—68no—
149C22×Q16—64no—
150C23×C8—31no—
151D4×Q8—24no—
152D4×Q8—24no—
153C2×C4⋊Q8—21no—
154C2×C4⋊Q8C2×D311yescharacteristic
155C2×C22⋊Q8—21no—
156C2×C22⋊Q8—21no—
157C2×C22⋊Q8—32no—
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×C22⋊Q8C2×D311yescharacteristic
160C2×C4×Q8C2×D311yescharacteristic
161C2×C4×D4—21no—
162C2×C4×D4C2×D311yescharacteristic
163#451—68no—
164#451—64no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#103—32no—
169C2×C22⋊C8—32no—
170C2×C22⋊C8—31no—
171#395—61no—
172#393—61no—
173#391—32no—
174#386—61no—
175#385—61no—
176#380—31no—
177#340—32no—
178#336—32no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D411yescharacteristic
181C2×C3×Q8—24no—
182C2×C3×Q8—24no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—22no—
187C2×C3×Q8—28no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—44no—
193C2×C3×Q8—44no—
194C2×C3×Q8—44no—
195C2×C3×Q8—48no—
196C2×C3×Q8—48no—
197C2×C3×Q8—44no—
198C2×C3×Q8—44no—
199C2×C3×Q8C2×D414yes—
200C2×C3×Q8C2×D412yes—
201C2×C3×Q8C2×D412yes—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4Q1614yescomplemented
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—44no—
219C22×C3×C4—44no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—22no—
228C22×Dic3—22no—
229C22×Dic3—44no—
230C22×Dic3—42no—
231C2×Dic6—28no—
232C2×Dic6—22no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—48no—
238C2×Dic6—44no—
239C2×Dic6—44no—
240C2×Dic6—42no—
241C2×Dic6C2×D414yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D412yes—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C42—24no—
255C3×C42—21no—
256C3×C42—21no—
257#217—44no—
258#217—42no—
259#216—216no—
260#216—216no—
261#216—48no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264Dic3⋊C4—44no—
265C2×C3⋊C8—28no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8C2×D414yes—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—32no—
280C22×Q8—42no—
281C22×Q8—42no—
282C22×Q8—64no—
283C22×Q8—61no—
284C22×Q8—62no—
285C22×Q8—62no—
286C22×Q8—122no—
287C22×Q8#6714yes—
288C22×Q8C22×D311yescharacteristic
289C22×Q8C22×D311yescharacteristic
290C22×Q8C22×D311yescharacteristic
291C22×D4#6711yescharacteristic
292C23×C4—21no—
293C23×C4—21no—
294C23×C4—61no—
295C23×C4C22×D311yescharacteristic
296C23×C4C22×D311yescharacteristic
297C2×Q16—38no—
298C2×Q16—64no—
299C2×Q16—64no—
300C2×Q16—632no—
301C2×Q16—64no—
302C2×Q16—64no—
303C2×Q16—64no—
304C2×Q16—64no—
305C2×Q16—64no—
306C2×Q16—122no—
307C22×C8—32no—
308C22×C8—32no—
309C22×C8—64no—
310C22×C8—62no—
311C4⋊Q8—28no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C22⋊Q8—28no—
315C22⋊Q8—28no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—64no—
320C22⋊Q8—122no—
321C22⋊Q8#6714yes—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×D4—28no—
325C4×D4—22no—
326C4×D4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—62no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4#6711yescharacteristic
341C2×C22⋊C4#6711yescharacteristic
342C2×C42—21no—
343C2×C42C22×D311yescharacteristic
344#103—64no—
345#103—64no—
346#103—64no—
347#103—122no—
348C22⋊C8—64no—
349C22⋊C8—62no—
350#95—61no—
351#95—62no—
352#95—62no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3#12012yes—
360C23×C3C2×Q1612yes—
361C23×C3C2×Q1612yescomplemented
362C23×C3#13712yes—
363C23×C3C22×D411yescharacteristic
364C3×Q8—22no—
365C3×Q8—24no—
366C3×Q8—22no—
367C3×Q8—22no—
368C3×Q8—24no—
369C3×Q8—48no—
370C3×Q8—48no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—42no—
374C3×Q8#12014yes—
375C3×D4—28no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—48no—
392C2×C3×C4—48no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4#12012yes—
400C2×C3×C4C2×Q1612yescomplemented
401C2×C3×C4C22×D412yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4H(5,𝔽2)12yes—
405C2×Dic3—24no—
406C2×Dic3—48no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412Dic6—24no—
413Dic6—24no—
414Dic6—48no—
415Dic6—42no—
416Dic6—42no—
417C3⋊C8—24no—
418C3⋊C8—42no—
419C24C2×#6711yescharacteristic
420C24C2×#6711yescharacteristic
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—34no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—68no—
441C2×Q8—62no—
442C2×Q8—62no—
443C2×Q8—62no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—124no—
447C2×Q8—128no—
448C2×Q8—124no—
449C2×Q8—122no—
450C2×Q8D3×D414yes—
451C2×Q8C2×#6712yes—
452C2×Q8C2×#6712yes—
453C2×Q8C2×#6712yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4#21614yescomplemented
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—124no—
474C22×C4—122no—
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6712yes—
480C22×C4C2×#6712yes—
481C22×C4C23×D311yescharacteristic
482Q16—616no—
483Q16—616no—
484Q16—128no—
485C2×C8—34no—
486C2×C8—68no—
487C2×C8—62no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8—62no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—124no—
502C4⋊C4C2×#6712yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—124no—
508C22⋊C4—122no—
509C42—24no—
510C42—21no—
511C42—21no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3—22no—
517C22×C3—24no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3#45114yescomplemented
522C22×C3C2×#12011yescharacteristic
523C22×C3D4212yes—
524C22×C3C22×Q1611yescharacteristic
525C22×C3C2×#13711yescharacteristic
526C22×C3#57812yes—
527C3×C4—22no—
528C3×C4—24no—
529C3×C4—22no—
530C3×C4—41no—
531C3×C4—42no—
532C3×C4—42no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4C2×#12011yescharacteristic
536C3×C4#53411yescharacteristic
537C3×C4D4212yes—
538Dic3—44no—
539Dic3—82no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23#94212yes—
547C23C2×#21612yes—
548C23C2×#21612yescomplemented
549C23#95312yes—
550C23C22×#6711yescharacteristic
551Q8—24no—
552Q8—22no—
553Q8—22no—
554Q8—24no—
555Q8—22no—
556Q8—48no—
557Q8—48no—
558Q8—41no—
559Q8—42no—
560Q8—41no—
561Q8—64no—
562Q8—64no—
563Q8—128no—
564Q8—122no—
565Q8—122no—
566Q8#93714yes—
567D4—28no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—64no—
592C2×C4—128no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4C2×#21612yescomplemented
599C2×C4#95312yes—
600C2×C4C2×D3×D412yes—
601C2×C4#100412yes—
602C2×C4C22×#6711yescharacteristic
603C2×C4C22×#6711yescharacteristic
604C8—64no—
605C8—122no—
606C2×C3—24no—
607C2×C3—42no—
608C2×C3#240311yescharacteristic
609C2×C3C2×#45111yescharacteristic
610C2×C3#301111yescharacteristic
611C2×C3#327012yes—
612C2×C3D4×Q1612yes—
613C22—22no—
614C22—24no—
615C22—24no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22#522814yescomplemented
623C22D4×#6712yes—
624C22C2×#94211yescharacteristic
625C22C22×#21611yescharacteristic
626C22#583812yes—
627C22C2×#95311yescharacteristic
628C4—22no—
629C4—24no—
630C4—22no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—124no—
637C4—242no—
638C4C2×#93711yescharacteristic
639C4D4×#6712yes—
640C4#580411yescharacteristic
641C3#2385611yescharacteristic, complemented, Sylow
642C2—24no—
643C2—42no—
644C2#8356512yes—
645C2D4×#21612yes—
646C2#8405211yescharacteristic
647C2C2×#522811yescharacteristic
648C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×D4×Q8D3
hypocenterC3#23856
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C23×#216C2semidirect
3. C2×#5228C2semidirect
4. C2×#5050C2semidirect
6. #72132C2semidirect
7. #71793C2semidirect
8. #71700C2semidirect
9. #71696C2semidirect
10. #71632C2semidirect
124. C2×#216D4semidirect
205. C2×C3×D4Q16semidirect
361. C23×C3C2×Q16semidirect
400. C2×C3×C4C2×Q16semidirect
456. C2×D4#216semidirect
521. C22×C3#451semidirect
548. C23C2×#216semidirect
598. C2×C4C2×#216semidirect
622. C22#5228semidirect
641. C3#23856semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C23×#2162
C2×#52282
C2×#50502
#823492
#721322
#717932
#717002
#716962
#716322
#716102
#238563