Finite Group Databasebeta

← #10944120

Subgroups — #10944120

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×#5574C211yesmaximal, characteristic
2C2×#5031C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic, complemented
4C2×#4808C212yesmaximal
5#74193C211yesmaximal, characteristic, complemented
6C3×#1805C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic, complemented
8#70239C212yesmaximal, complemented
9#69556C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68878C212yesmaximal, complemented
12#32916—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#216C2212yes—
17C22×#204C2212yes—
18#5574—24no—
19#5574—24no—
20C2×#897C2212yes—
21C2×#893—21no—
22C2×#893C2211yescharacteristic
23C2×D3×C4⋊C4—21no—
24C2×D3×C4⋊C4C2211yescharacteristic
25C2×C4×D12—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#872—21no—
28C2×#872C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C3×#382C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4973C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4841—24no—
41#4808—28no—
42#4674—21no—
43#4674—21no—
44#4661—21no—
45#4661—21no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
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#1805D311yescharacteristic, complemented
58#1804—31no—
59#1798—31no—
60C22×C3×Q8C2311yescharacteristic
61C2×#239—22no—
62C2×#239—22no—
63C2×#239—22no—
64C2×#239—22no—
65C2×#239—22no—
66C2×#239D414yes—
67C22×D12—21no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3—21no—
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3C2311yescharacteristic
74C22×Dic6C2312yes—
75C22×C3×C8C2312yes—
76C3×C4×Q8—22no—
77C3×C4×Q8D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83Q8×Dic3—22no—
84Q8×Dic3—22no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—24no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C2×#204—24no—
105C2×#204—24no—
106C2×#204—24no—
107#897—24no—
108#897—24no—
109#893—28no—
110#893—22no—
111#893—22no—
112D3×C4⋊C4—28no—
113D3×C4⋊C4—22no—
114D3×C4⋊C4—22no—
115C4×D12—28no—
116C4×D12—22no—
117C4×D12D414yes—
118#872—28no—
119#872—22no—
120#872—22no—
121#869—22no—
122#869—22no—
123C3×C4⋊C8—22no—
124C3×#103—24no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—22no—
128C3⋊#95—22no—
129(C2×D3)⋊C8—24no—
130#816—24no—
131#810—24no—
132#809—24no—
133C4⋊(C3⋊C8)—22no—
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×Q8C2×D311yescharacteristic
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—31no—
148C2×#117—61no—
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103C2×D312yes—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382C2×D311yescharacteristic
167#340—32no—
168#340C2×D312yes—
169#336—32no—
170C23×D3—21no—
171C23×D3C2×D411yescharacteristic
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8#4112yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189#239—28no—
190#239—28no—
191#239—28no—
192#239—28no—
193C2×D12—24no—
194C2×D12—24no—
195C2×D12—24no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12C2×D412yes—
200C2×D12C2×D412yes—
201C2×D12#4112yes—
202C2×C4×D3—24no—
203C2×C4×D3—24no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—24no—
209C2×C4×D3—24no—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—82no—
213C2×C4×D3—82no—
214C2×C4×D3—82no—
215C2×C4×D3C2×D412yes—
216C2×C4×D3#4112yes—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4C2×D412yes—
231C3×C42—24no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234#216—48no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—44no—
240C4⋊Dic3—24no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3—42no—
244C4⋊Dic3—44no—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249C4×Dic3—24no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255#204—48no—
256C2×#41—34no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C22×Q8—32no—
263C22×Q8C22×D311yescharacteristic
264C22×D4—31no—
265C22×D4—61no—
266C23×C4—31no—
267C23×C4—61no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4—61no—
271C2×Q16—64no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279C4⋊Q8—62no—
280C4⋊Q8—62no—
281C22⋊Q8—64no—
282C22⋊Q8—64no—
283C4×Q8—22no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8D1214yes—
287C4×D4—34no—
288C4×D4—68no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—62no—
292C4×D4—62no—
293#117—68no—
294#117—62no—
295#117—62no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—68no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C42—21no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—62no—
321#103—24no—
322#103—64no—
323#103—64no—
324#102—64no—
325C22⋊C8—64no—
326#95—21no—
327#95—21no—
328#95—62no—
329#95—62no—
330C23×C3C22×D411yescharacteristic
331C22×D3—21no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—21no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3#12112yes—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8#12114yes—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4#12012yes—
356C2×C3×C4#12012yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3#12112yes—
375D12—216no—
376D12—22no—
377D12—22no—
378D12—22no—
379D12#12114yes—
380C4×D3—216no—
381C4×D3—22no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—84no—
386C4×D3—84no—
387C4×D3—84no—
388Dic6—42no—
389Dic6—42no—
390Dic6—42no—
391Dic6—42no—
392C3×C8—42no—
393C3⋊C8—42no—
394C24—31no—
395C24—61no—
396#41—68no—
397#41—68no—
398#41—68no—
399#41—68no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8C2×D1212yes—
410C2×Q8D3×D412yes—
411C2×Q8#23912yes—
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—22no—
424C22×C4—31no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—242no—
442C22×C4—242no—
443C22×C4—242no—
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4C23×D311yescharacteristic
447Q16—128no—
448SD16—128no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—32no—
460C4⋊C4—64no—
461C4⋊C4—64no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—61no—
465C4⋊C4—124no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4C2×D1212yes—
469C22⋊C4—64no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C22⋊C4—64no—
473C22⋊C4—124no—
474C42—24no—
475C42—64no—
476C42—61no—
477C42—61no—
478C42C2×D1212yes—
479C42D3×D412yes—
480C22×C3C2×#12011yescharacteristic
481C22×C3C2×#12111yescharacteristic
482C22×C3#52511yescharacteristic
483C22×C3#57712yes—
484C22×C3#57812yes—
485C2×D3—22no—
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—42no—
489C2×D3—44no—
490C2×D3—41no—
491C2×D3—42no—
492C2×D3—41no—
493C2×D3—42no—
494C2×D3—44no—
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—42no—
499C3×C4—82no—
500C3×C4C2×#12111yescharacteristic
501C3×C4C2×#12111yescharacteristic
502C3×C4#52511yescharacteristic
503C3×C4#52511yescharacteristic
504Dic3—22no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—82no—
509C23—32no—
510C23—62no—
511C23—61no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—61no—
517C23—122no—
518C23—122no—
519C23C2×D3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—122no—
524Q8—122no—
525Q8—122no—
526Q8—122no—
527Q8#89514yes—
528D4—34no—
529D4—616no—
530D4—62no—
531D4—62no—
532D4—62no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—32no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—616no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—244no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4#88212yes—
564C2×C4#88212yes—
565C2×C4C22×D1211yescharacteristic
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23911yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—42no—
572C8—122no—
573C2×C3#284711yescharacteristic
574C2×C3#300911yescharacteristic
575C2×C3#301111yescharacteristic
576C2×C3#304612yes—
577C2×C3#312012yes—
578D3—42no—
579D3—44no—
580C22—64no—
581C22—62no—
582C22—64no—
583C22—122no—
584C22—124no—
585C22—121no—
586C22—122no—
587C22—124no—
588C22—121no—
589C22—122no—
590C22C2×#88211yescharacteristic
591C22D3×#12111yescharacteristic
592C22#562711yescharacteristic
593C22#577012yes—
594C22#577912yes—
595C4—24no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—62no—
600C4—82no—
601C4—121no—
602C4—121no—
603C4—122no—
604C4—242no—
605C4C2×#89511yescharacteristic
606C4#552411yescharacteristic
607C4D3×#12111yescharacteristic
608C4#562711yescharacteristic
609C3#3291611yescharacteristic, complemented, Sylow
610C2—122no—
611C2—124no—
612C2#7404811yescharacteristic
613C2#7883511yescharacteristic
614C2#7896111yescharacteristic
615C2#8060312yes—
616C2#8120912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#1805D3
hypocenterC3#32916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#5031C2semidirect
3. C2×#4841C2semidirect
5. #74193C2semidirect
6. C3×#1805C2semidirect
7. #70476C2semidirect
8. #70239C2semidirect
9. #69556C2semidirect
10. #68896C2semidirect
11. #68878C2semidirect
57. #1805D3semidirect
609. C3#32916semidirect

Maximal subgroups

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

subgroupindex
C2×#55742
C2×#50312
C2×#48412
C2×#48082
#741932
C3×#18052
#704762
#702392
#695562
#688962
#688782
#329163