Finite Group Databasebeta

← #10822289

Subgroups — #10822289

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic, complemented
2C23×#215C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5#82269C211yesmaximal, characteristic, complemented
6C2×#5002C211yesmaximal, characteristic, complemented
7#71980C211yesmaximal, characteristic, complemented
8#70282C214yesmaximal, complemented
9#70277C214yesmaximal, complemented
10#23424—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×D12C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5816—24no—
17#5816—24no—
18C2×#947C2212yescomplemented
19C2×#946—22no—
20C2×#944C2212yescomplemented
21C22×#215—24no—
22C22×#215—24no—
23C22×#215—24no—
24C22×#215C2214yes—
25C2×#929—22no—
26C2×#926C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×#908—24no—
29C2×#908—24no—
30C2×#908—24no—
31C22×D24C2212yes—
32C22×#204C2212yes—
33C3×#419—24no—
34C2×C3×#103C2214yescomplemented
35#5002—24no—
36#4988C2214yes—
37C2×#810C2212yescomplemented
38C2×#809C2212yescomplemented
39#4955—22no—
40#4939—22no—
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×SD16—31no—
44C2×#536—31no—
45C2×#419D311yescharacteristic, complemented
46C2×#416—31no—
47#2309—31no—
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C24×D3—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8D414yes—
59C22×C3×Q8C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235D414yes—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12D414yes—
76C22×D12C2312yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×#217D412yes—
90#947—24no—
91#947—24no—
92#946—216no—
93#944—24no—
94#944—24no—
95C2×#215—232no—
96C2×#215—28no—
97C2×#215—28no—
98C2×#215—28no—
99#929—216no—
100C2×(C2×D3)⋊C4—24no—
101C2×(C2×D3)⋊C4—28no—
102#926—22no—
103#926—22no—
104C2×C4⋊Dic3D412yes—
105C2×Dic3⋊C4—24no—
106C2×C4×Dic3C2312yes—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8D414yes—
110C22×C3⋊C8C2312yes—
111#908—232no—
112C2×D24—24no—
113C2×D24—24no—
114C2×D24—24no—
115C2×#204—24no—
116C2×#204—24no—
117C2×#204—24no—
118C3×#103—28no—
119#810—24no—
120#809—24no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×D4—31no—
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—34no—
128C22×SD16—32no—
129C22×SD16—64no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×D8—32no—
133C22×M16C2×D311yescharacteristic
134C23×C8—31no—
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#419—24no—
144#416—64no—
145C2×#103—32no—
146C2×#103C2×D314yescomplemented
147C2×#102—32no—
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—42no—
157C23×D3—42no—
158C23×D3—84no—
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—44no—
168C2×C3×Q8—42no—
169C2×C3×Q8—44no—
170C2×C3×Q8—42no—
171C2×C3×Q8C2×D414yes—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4112yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—21no—
189C22×Dic3C2×D414yes—
190#235—28no—
191#235—28no—
192#235—44no—
193#235—44no—
194C2×D12—28no—
195C2×D12—28no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—21no—
204C2×D12—24no—
205C2×D12—21no—
206C2×D12—22no—
207C2×D12—42no—
208C2×D12—44no—
209C2×D12—44no—
210C2×D12—42no—
211C2×D12C2×D414yes—
212C2×D12C2×D412yes—
213C2×C4×D3—24no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C2×D412yes—
222C3×M16—28no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8C2×D414yes—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C22⋊C4—42no—
229C3×C42—42no—
230#217—24no—
231#215—232no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—88no—
234C4⋊Dic3—24no—
235Dic3⋊C4—216no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238C2×C3⋊C8—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×D414yes—
242D24—216no—
243#204—216no—
244C25—61no—
245C2×#41—34no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—31no—
258C22×Q8D1214yes—
259C22×Q8C22×D312yes—
260C22×D4—34no—
261C22×D4—31no—
262C22×D4—32no—
263C22×D4—61no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—61no—
269C22×D4—62no—
270C23×C4—21no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—632no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—68no—
277C2×SD16—68no—
278C2×SD16—632no—
279C2×SD16—68no—
280C2×SD16—64no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—64no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16#6714yes—
287C22×C8—34no—
288C22×C8—32no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8C22×D312yes—
292C4⋊Q8—64no—
293C4⋊Q8—64no—
294#124—64no—
295#124—64no—
296#123—616no—
297C22⋊Q8—616no—
298#117—22no—
299#117—22no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—64no—
304C2×C4⋊C4C22×D312yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—32no—
307C2×C22⋊C4—64no—
308C2×C22⋊C4—68no—
309C2×C42—21no—
310C2×C42—32no—
311#103—28no—
312#103—64no—
313#102—64no—
314C23×C3—24no—
315C23×C3#12012yes—
316C23×C3#12112yes—
317C23×C3#12712yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—24no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—48no—
326C22×D3—44no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—88no—
330C22×D3—88no—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—48no—
336C2×C3×C4—24no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—48no—
340C2×C3×C4—42no—
341C2×C3×C4—44no—
342C2×C3×C4#12014yes—
343C2×C3×C4#12012yes—
344C2×C3×C4#12014yes—
345C2×C3×C4#12114yes—
346C2×C3×C4#12112yes—
347C2×C3×C4#12114yes—
348C2×C3×C4#12712yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355#67—48no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—28no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361D12—24no—
362D12—22no—
363D12—24no—
364D12—22no—
365D12—41no—
366D12—42no—
367D12—42no—
368D12—41no—
369D12—48no—
370C4×D3—48no—
371Dic6—22no—
372Dic6—22no—
373Dic6—41no—
374Dic6—41no—
375C3×C8—24no—
376C3⋊C8—24no—
377C24—61no—
378C24—62no—
379C24—64no—
380C24—122no—
381C24—122no—
382C24—244no—
383C24D3×D411yescharacteristic
384#41—68no—
385#41—68no—
386#41—124no—
387#41—124no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—32no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8C2×D1214yes—
407C2×D4—32no—
408C2×D4—34no—
409C2×D4—64no—
410C2×D4—68no—
411C2×D4—64no—
412C2×D4—68no—
413C2×D4—62no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—62no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—61no—
422C2×D4—62no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4—122no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—34no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—61no—
436C22×C4—64no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4C2×D1212yes—
440C22×C4#23512yes—
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D412yes—
443C22×C4D3×D412yes—
444C22×C4D3×D412yes—
445C22×C4C2×#6712yes—
446C22×C4C23×D311yescharacteristic
447SD16—616no—
448SD16—632no—
449D8—616no—
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—34no—
454C2×C8—68no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8C2×#6714yes—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—616no—
461C4⋊C4—64no—
462C22⋊C4—42no—
463C22⋊C4—616no—
464C22⋊C4—64no—
465C22⋊C4—248no—
466C42—42no—
467C42—62no—
468C42—62no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3#39214yes—
473C22×C3C2×#12011yescharacteristic
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×#12711yescharacteristic
476C22×C3#57714yes—
477C2×D3—42no—
478C2×D3—44no—
479C2×D3—44no—
480C2×D3—48no—
481C2×D3—816no—
482C2×D3—82no—
483C2×D3—84no—
484C2×D3—84no—
485C2×D3—82no—
486C3×C4—44no—
487C3×C4—82no—
488C3×C4#39214yes—
489C3×C4C2×#12011yescharacteristic
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12111yescharacteristic
492C3×C4C2×#12111yescharacteristic
493Dic3—42no—
494Dic3—44no—
495C23—24no—
496C23—62no—
497C23—64no—
498C23—128no—
499C23—122no—
500C23—121no—
501C23—128no—
502C23—124no—
503C23—121no—
504C23—122no—
505C23—248no—
506C23—248no—
507C23—248no—
508C23#88212yes—
509C23#88412yes—
510C23#94012yes—
511C23C2×D3×D411yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—48no—
515Q8—42no—
516Q8—42no—
517Q8—62no—
518Q8—62no—
519Q8—121no—
520Q8—121no—
521D4—64no—
522D4—62no—
523D4—62no—
524D4—64no—
525D4—128no—
526D4—128no—
527D4—121no—
528D4—122no—
529D4—122no—
530D4—121no—
531C2×C4—24no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—64no—
538C2×C4—62no—
539C2×C4—68no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—128no—
543C2×C4#88212yes—
544C2×C4#88214yes—
545C2×C4#88412yes—
546C2×C4#88414yes—
547C2×C4#93014yes—
548C2×C4#93714yes—
549C2×C4#94012yes—
550C2×C4C22×D1211yescharacteristic
551C2×C4C2×#23511yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—24no—
557C8—64no—
558C2×C3—24no—
559C2×C3C2×#39211yescharacteristic
560C2×C3#301012yes—
561C2×C3#304212yes—
562C2×C3#315412yes—
563D3—82no—
564D3—84no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—124no—
569C22—128no—
570C22—122no—
571C22—124no—
572C22—2416no—
573C22—244no—
574C22—242no—
575C22—244no—
576C22—242no—
577C22#496014yes—
578C22C2×#88211yescharacteristic
579C22C2×#88411yescharacteristic
580C22#577814yes—
581C22C2×#94011yescharacteristic
582C4—44no—
583C4—82no—
584C4—124no—
585C4—122no—
586C4#496014yes—
587C4C2×#88211yescharacteristic
588C4C2×#88411yescharacteristic
589C4C2×#93011yescharacteristic
590C4C2×#93711yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—244no—
594C2—242no—
595C2#7896012yes—
596C2#7897612yes—
597C2C2×#496011yescharacteristic
598C2#8397412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#908C2
hypercenterC2×#419D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C2×#5816C2semidirect
2. C23×#215C2semidirect
4. C2×C3×#419C2semidirect
5. #82269C2semidirect
6. C2×#5002C2semidirect
7. #71980C2semidirect
8. #70282C2semidirect
9. #70277C2semidirect
18. C2×#947C22semidirect
20. C2×#944C22semidirect
34. C2×C3×#103C22semidirect
37. C2×#810C22semidirect
38. C2×#809C22semidirect
41. #4550C22semidirect
45. C2×#419D3semidirect
146. C2×#103C2×D3semidirect
591. C3#23424semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C23×#2152
C22×#9082
C2×C3×#4192
#822692
C2×#50022
#719802
#702822
#702772
#234243