Finite Group Databasebeta

← #10944119

Subgroups — #10944119

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4841C211yesmaximal, characteristic, complemented
5C2×#4733C212yesmaximal
6C3×#1798C211yesmaximal, characteristic, complemented
7#70486C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#69930C212yesmaximal, complemented
10#68685C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#32916—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C2×#924C2212yes—
17C22×Dic12C2212yes—
18C22×#204C2212yes—
19#5523—24no—
20#5523—24no—
21C2×#892—21no—
22C2×D3×C4⋊C4—21no—
23C2×#67⋊C4—21no—
24C2×#67⋊C4—21no—
25C2×C4×D12C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×#821C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#4981—21no—
38#4981—21no—
39#4953—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42#4929C2212yes—
43#4849—24no—
44#4841—24no—
45#4733—28no—
46C3×#336C2212yes—
47#4551C2212yes—
48#4551C2212yes—
49C2×#520—31no—
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2621—31no—
54#1878—32no—
55#1877—32no—
56#1872—32no—
57#1805—31no—
58#1804—31no—
59#1798D311yescharacteristic, complemented
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235D414yes—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×Dic6C2312yes—
74C22×Dic6C2311yescharacteristic
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×#117—22no—
78C3×#117—22no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—24no—
97#924—24no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×Dic12—24no—
104C2×Dic12—24no—
105C2×Dic12—24no—
106C2×#204—24no—
107C2×#204—24no—
108C2×#204—24no—
109#892—28no—
110D3×C4⋊C4—28no—
111#67⋊C4—28no—
112#67⋊C4—28no—
113C4×D12—22no—
114C4×D12—22no—
115C4×D12—22no—
116C4×D12D414yes—
117#869—22no—
118#869—22no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×C22⋊C8—24no—
126C3⋊#95—22no—
127C3⋊#95—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—24no—
131#816—24no—
132#816—24no—
133#816—24no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—61no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8C2×D311yescharacteristic
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8C2×D312yes—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336C2×D312yes—
170C24×C3C2×D411yescharacteristic
171C23×D3—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—24no—
185C2×#67—24no—
186C2×#67—24no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—82no—
194#235—28no—
195#235—28no—
196#235—28no—
197#235—28no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C2×D412yes—
202C2×D12C2×D412yes—
203C2×D12#4112yes—
204C2×C4×D3—24no—
205C2×C4×D3—24no—
206C2×C4×D3—24no—
207C2×C4×D3—24no—
208C2×C4×D3—24no—
209C2×C4×D3—82no—
210C2×C4×D3—82no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6C2×D412yes—
221C2×Dic6C2×D412yes—
222C2×Dic6#4112yes—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237#217—44no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4⋊Dic3—24no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3—42no—
246C4⋊Dic3—42no—
247C4⋊Dic3C2×D412yes—
248C4⋊Dic3C2×D412yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—24no—
251Dic3⋊C4—44no—
252C4×Dic3—24no—
253C4×Dic3—24no—
254Dic12—48no—
255#204—48no—
256C2×#41—34no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C22×Q8—31no—
263C22×Q8—32no—
264C22×D4—31no—
265C22×D4—61no—
266C23×C4—61no—
267C23×C4—61no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×Q16—64no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8C22×D312yes—
278C22×C8C22×D312yes—
279C4⋊Q8—62no—
280C4⋊Q8—62no—
281C22⋊Q8—64no—
282C22⋊Q8—64no—
283C4×Q8—34no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×D4—34no—
288C4×D4—62no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—68no—
292C4×D4—62no—
293#117—22no—
294#117—22no—
295#117—68no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—68no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4D1211yescharacteristic
309C2×C4⋊C4D1211yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4D1211yescharacteristic
314C2×C22⋊C4D1211yescharacteristic
315C2×C42—61no—
316C2×C42—61no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—22no—
321#103—64no—
322#103—64no—
323#103—64no—
324#102—64no—
325C22⋊C8—24no—
326#95—61no—
327#95—61no—
328#95—62no—
329#95—62no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3#12112yes—
335C23×C3C22×D411yescharacteristic
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—42no—
340C22×D3—42no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4#12012yes—
354C2×C3×C4#12012yes—
355C2×C3×C4#12112yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362#67—216no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—84no—
377D12—22no—
378D12—22no—
379D12—22no—
380D12#12114yes—
381C4×D3—216no—
382C4×D3—84no—
383C4×D3—84no—
384Dic6—22no—
385Dic6—22no—
386Dic6—22no—
387Dic6—42no—
388Dic6—42no—
389Dic6—42no—
390Dic6—42no—
391Dic6#12114yes—
392C3×C8—42no—
393C3×C8—42no—
394C24—61no—
395C24C2×D1211yescharacteristic
396#41—68no—
397#41—68no—
398#41—68no—
399#41—68no—
400C2×Q8—32no—
401C2×Q8—32no—
402C2×Q8—32no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8—62no—
410C2×Q8—62no—
411C2×Q8—62no—
412C2×D4—32no—
413C2×D4—32no—
414C2×D4—32no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—61no—
429C22×C4—64no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—242no—
437C22×C4—242no—
438C22×C4—242no—
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×D1211yescharacteristic
442C22×C4C2×D1211yescharacteristic
443C22×C4C2×D1211yescharacteristic
444C22×C4D3×D412yes—
445C22×C4#23912yes—
446C22×C4C23×D311yescharacteristic
447Q16—128no—
448SD16—128no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C4⋊C4—64no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—124no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—124no—
474C42—21no—
475C42—21no—
476C42—64no—
477C42—64no—
478C42D3×D412yes—
479C42D3×D412yes—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3C2×#12011yescharacteristic
486C22×C3C2×#12111yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3#57712yes—
489C22×C3#57812yes—
490C2×D3—24no—
491C2×D3—44no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—44no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4C2×#12111yescharacteristic
501C3×C4C2×#12111yescharacteristic
502C3×C4#52511yescharacteristic
503C3×C4#52511yescharacteristic
504Dic3—24no—
505Dic3—42no—
506Dic3—42no—
507Dic3—82no—
508Dic3—82no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—122no—
517C23—122no—
518C23#89512yes—
519C23C22×D1211yescharacteristic
520Q8—34no—
521Q8—62no—
522Q8—62no—
523Q8—62no—
524Q8—122no—
525Q8—122no—
526Q8—122no—
527Q8—122no—
528D4—34no—
529D4—616no—
530D4—62no—
531D4—62no—
532D4—62no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—616no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—244no—
560C2×C4—244no—
561C2×C4—244no—
562C2×C4#88212yes—
563C2×C4#88212yes—
564C2×C4#89512yes—
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23911yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—42no—
572C8—42no—
573C2×C3—42no—
574C2×C3#284711yescharacteristic
575C2×C3#300911yescharacteristic
576C2×C3#301111yescharacteristic
577C2×C3#304612yes—
578C2×C3#312012yes—
579D3—44no—
580C22—22no—
581C22—24no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—64no—
586C22—124no—
587C22—122no—
588C22—122no—
589C22—124no—
590C22C2×#88211yescharacteristic
591C22C2×#89511yescharacteristic
592C22#552411yescharacteristic
593C22#575112yes—
594C22#575312yes—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—64no—
601C4—122no—
602C4—122no—
603C4—242no—
604C4—242no—
605C4D3×#12111yescharacteristic
606C4D3×#12111yescharacteristic
607C4#562711yescharacteristic
608C4#562711yescharacteristic
609C3#3291611yescharacteristic, complemented, Sylow
610C2—42no—
611C2—124no—
612C2#7404811yescharacteristic
613C2#7867111yescharacteristic
614C2#7867911yescharacteristic
615C2#7908912yes—
616C2#7938512yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1798D3
hypocenterC3#32916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. #82262C2semidirect
3. C2×#4849C2semidirect
4. C2×#4841C2semidirect
6. C3×#1798C2semidirect
7. #70486C2semidirect
8. #70480C2semidirect
9. #69930C2semidirect
10. #68685C2semidirect
11. #68672C2semidirect
59. #1798D3semidirect
609. C3#32916semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
#822622
C2×#48492
C2×#48412
C2×#47332
C3×#17982
#704862
#704802
#699302
#686852
#686722
#329163