Finite Group Databasebeta

← #10822288

Subgroups — #10822288

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×#5825C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic
3C23×#204C211yesmaximal, characteristic, complemented
4C2×C3×#416C211yesmaximal, characteristic, complemented
5#82269C211yesmaximal, characteristic, complemented
6C2×#5004C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#70282C214yesmaximal, complemented
9#70214C214yesmaximal, complemented
10#23424—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5825—24no—
17#5825—24no—
18C2×#949—24no—
19C2×#949—24no—
20C2×#949—24no—
21C2×#944C2212yescomplemented
22C22×#215C2212yes—
23C2×#936C2212yescomplemented
24C2×#935—22no—
25C22×#213C2212yes—
26C2×#926C2211yescharacteristic
27C2×#924—22no—
28C22×#208C2211yescharacteristic
29C22×#204—24no—
30C22×#204—24no—
31C22×#204—24no—
32C22×#204C2214yes—
33C3×#416—24no—
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yescomplemented
36#5004—24no—
37#4988C2214yes—
38C2×#810C2214yescomplemented
39#4955—22no—
40#4939—22no—
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×SD16—31no—
44C2×#536—31no—
45C2×#419—31no—
46C2×#416D311yescharacteristic, complemented
47#2309—31no—
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C24×D3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—22no—
70C22×D12D414yes—
71C22×D12C2312yes—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—22no—
78C22×Dic6D414yes—
79C22×Dic6C2312yes—
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×#217—24no—
90C2×#217D412yes—
91#949—232no—
92#944—24no—
93#944—24no—
94C2×#215—24no—
95C2×#215—24no—
96C2×#215—24no—
97#936—24no—
98#936—24no—
99#935—216no—
100C2×#213—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×(C2×D3)⋊C4—28no—
104#926—22no—
105#926—22no—
106C2×C4⋊Dic3D412yes—
107#924—216no—
108C2×Dic3⋊C4—24no—
109C2×C4×Dic3C2312yes—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208D414yes—
113C22×C3⋊C8C2312yes—
114C2×#204—232no—
115C2×#204—28no—
116C2×#204—28no—
117C2×#204—28no—
118C3×#103—24no—
119C3×#102—24no—
120#810—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4—31no—
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—32no—
128C22×SD16—34no—
129C22×SD16—64no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×D8—32no—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
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—64no—
144#416—24no—
145C2×#103—34no—
146C2×#103C2×D312yescomplemented
147C2×#102C2×D312yescomplemented
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—42no—
157C23×D3—42no—
158C23×D3—84no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—42no—
194C22×Dic3—42no—
195C22×Dic3C2×D414yes—
196C2×D12—28no—
197C2×D12—28no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—24no—
203C2×D12—22no—
204C2×D12—42no—
205C2×D12—44no—
206C2×D12—44no—
207C2×D12—42no—
208C2×D12C2×D414yes—
209C2×Dic6—28no—
210C2×Dic6—28no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—24no—
216C2×Dic6—22no—
217C2×Dic6—44no—
218C2×Dic6—42no—
219C2×Dic6—44no—
220C2×Dic6—42no—
221C2×Dic6C2×D414yes—
222C2×C3×C8—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—216no—
231#217—24no—
232#215—216no—
233#213—216no—
234(C2×D3)⋊C4—88no—
235C4⋊Dic3—24no—
236Dic3⋊C4—216no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239#208—28no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243#204—232no—
244C25—61no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41D1214yes—
251C22×Q8—34no—
252C22×Q8—32no—
253C22×Q8—62no—
254C22×Q8—61no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8C22×D311yescharacteristic
260C22×D4—21no—
261C22×D4—34no—
262C22×D4—32no—
263C22×D4—62no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—61no—
268C22×D4—62no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—632no—
274C2×SD16—68no—
275C2×SD16—68no—
276C2×SD16—632no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—64no—
280C2×SD16—68no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—64no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—32no—
290C22×C8#6714yes—
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—24no—
312#103—68no—
313#102—24no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—44no—
319C23×C3#12012yes—
320C23×C3#12112yes—
321C23×C3#12712yes—
322C23×C3C22×D411yescharacteristic
323C22×D3—24no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—88no—
330C22×D3—88no—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—48no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—48no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4#12014yes—
347C2×C3×C4#12012yes—
348C2×C3×C4#12014yes—
349C2×C3×C4#12114yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12114yes—
352C2×C3×C4#12712yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—28no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—48no—
365D12—24no—
366D12—24no—
367D12—42no—
368D12—42no—
369D12—48no—
370Dic6—24no—
371Dic6—24no—
372Dic6—42no—
373Dic6—42no—
374Dic6—48no—
375C3×C8—24no—
376C3⋊C8—24no—
377C24—21no—
378C24—62no—
379C24—64no—
380C24—122no—
381C24—122no—
382C24—244no—
383C24D3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—34no—
394C2×Q8—64no—
395C2×Q8—62no—
396C2×Q8—68no—
397C2×Q8—68no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—124no—
403C2×Q8—122no—
404C2×Q8—124no—
405C2×Q8—122no—
406C2×Q8C2×D1212yes—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—34no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—68no—
417C2×D4—64no—
418C2×D4—68no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—122no—
425C2×D4—124no—
426C2×D4—124no—
427C2×D4—122no—
428C2×D4C2×D1212yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—34no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—64no—
436C22×C4—62no—
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—632no—
448SD16—616no—
449D8—616no—
450M16—68no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—34no—
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—42no—
473C22×C3—44no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3#39214yes—
477C22×C3C2×#12011yescharacteristic
478C22×C3C2×#12111yescharacteristic
479C22×C3C2×#12711yescharacteristic
480C22×C3#57714yes—
481C2×D3—44no—
482C2×D3—48no—
483C2×D3—816no—
484C2×D3—84no—
485C2×D3—84no—
486C3×C4—42no—
487C3×C4—82no—
488C3×C4#39214yes—
489C3×C4C2×#12011yescharacteristic
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12111yescharacteristic
492C3×C4C2×#12111yescharacteristic
493Dic3—44no—
494Dic3—44no—
495C23—24no—
496C23—22no—
497C23—41no—
498C23—44no—
499C23—41no—
500C23—64no—
501C23—128no—
502C23—122no—
503C23—128no—
504C23—122no—
505C23—248no—
506C23—248no—
507C23—248no—
508C23#88212yes—
509C23#88412yes—
510C23#94012yes—
511C23C2×D3×D411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—64no—
517Q8—64no—
518Q8—128no—
519Q8—122no—
520Q8—122no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526D4—64no—
527D4—64no—
528D4—128no—
529D4—122no—
530D4—122no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—68no—
540C2×C4—128no—
541C2×C4—122no—
542C2×C4—122no—
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×C3—82no—
560C2×C3C2×#39211yescharacteristic
561C2×C3#301012yes—
562C2×C3#304212yes—
563C2×C3#315412yes—
564D3—84no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—128no—
573C22—124no—
574C22—2416no—
575C22—244no—
576C22—244no—
577C22#496014yes—
578C22C2×#88211yescharacteristic
579C22C2×#88411yescharacteristic
580C22#577014yes—
581C22C2×#94011yescharacteristic
582C4—42no—
583C4—82no—
584C4—124no—
585C4—124no—
586C4#496014yes—
587C4C2×#88211yescharacteristic
588C4C2×#88411yescharacteristic
589C4C2×#93011yescharacteristic
590C4C2×#93711yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—82no—
594C2—244no—
595C2#7883612yes—
596C2#7886312yes—
597C2C2×#496011yescharacteristic
598C2#8390512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC22×#949C2
hypercenterC2×#416D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#5825C2semidirect
3. C23×#204C2semidirect
4. C2×C3×#416C2semidirect
5. #82269C2semidirect
6. C2×#5004C2semidirect
7. #71965C2semidirect
8. #70282C2semidirect
9. #70214C2semidirect
21. C2×#944C22semidirect
23. C2×#936C22semidirect
34. C2×C3×#103C22semidirect
35. C2×C3×#102C22semidirect
38. C2×#810C22semidirect
41. #4550C22semidirect
46. C2×#416D3semidirect
146. C2×#103C2×D3semidirect
147. C2×#102C2×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×#58252
C22×#9492
C23×#2042
C2×C3×#4162
#822692
C2×#50042
#719652
#702822
#702142
#234243