Finite Group Databasebeta

← #10944108

Subgroups — #10944108

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5572C211yesmaximal, characteristic
2C2×#5031C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4809C212yesmaximal
5#74193C211yesmaximal, characteristic
6C3×#1805C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic
8#70239C212yesmaximal, complemented
9#69559C211yesmaximal, characteristic, complemented
10#68908C212yesmaximal, complemented
11#68879C212yesmaximal
12#32915—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#216C2212yes—
17C22×Dic12C2212yes—
18#5572—24no—
19#5572—24no—
20C2×#897C2212yes—
21C2×#892—21no—
22C2×#892C2211yescharacteristic
23C2×#887—21no—
24C2×#887C2211yescharacteristic
25C2×#872—21no—
26C2×#872C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×C4×Dic6—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C3×#382C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4973C2212yes—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4849—24no—
41#4809—28no—
42#4674—21no—
43#4674—21no—
44#4661—21no—
45#4661—21no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#519—31no—
50C2×#462—31no—
51C2×#462—31no—
52C2×#451—32no—
53#2621—31no—
54#1878—32no—
55#1878—32no—
56#1873—32no—
57#1805—31no—
58#1805D311yescharacteristic, complemented
59#1798—31no—
60C22×C3×Q8C2311yescharacteristic
61C2×D3×Q8—22no—
62C2×D3×Q8—22no—
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8D414yes—
67C22×C4×D3—21no—
68C22×C4×D3C2311yescharacteristic
69C22×Dic6—21no—
70C22×Dic6C2312yes—
71C22×Dic6C2311yescharacteristic
72C22×C3×C8C2312yes—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80Q8×Dic3—22no—
81Q8×Dic3—22no—
82C2×#216—24no—
83C2×#216—24no—
84C2×#216—24no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D411yescharacteristic
88C2×(C2×D3)⋊C4D411yescharacteristic
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C2×Dic12—24no—
105C2×Dic12—24no—
106C2×Dic12—24no—
107#897—24no—
108#897—24no—
109#892—28no—
110#892—22no—
111#892—22no—
112#887—28no—
113#887—22no—
114#887—22no—
115#872—28no—
116#872—22no—
117#872—22no—
118#869—22no—
119#869—22no—
120C4×Dic6—28no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
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#810—24no—
133C4⋊(C3⋊C8)—22no—
134C23×Q8—31no—
135C22×Q16—32no—
136C22×Q16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139#519—64no—
140#519—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×Q8—61no—
146C2×C4×Q8C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C2×#117—61no—
150C2×#117—61no—
151#462—64no—
152#462—64no—
153#451—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#103C2×D312yes—
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×D3C2×D411yescharacteristic
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8#4112yes—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189D3×Q8—28no—
190D3×Q8—28no—
191D3×Q8—28no—
192D3×Q8—28no—
193C2×C4×D3—24no—
194C2×C4×D3—24no—
195C2×C4×D3—24no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3C2×D412yes—
201C2×C4×D3#4112yes—
202C2×Dic6—24no—
203C2×Dic6—24no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—24no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6C2×D412yes—
215C2×Dic6C2×D412yes—
216C2×Dic6#4112yes—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4C2×D412yes—
225C3×C42—24no—
226C3×C42C2×D412yes—
227C3×C42C2×D412yes—
228#216—48no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—44no—
233C4⋊Dic3—24no—
234C4⋊Dic3—21no—
235C4⋊Dic3—42no—
236C4⋊Dic3—42no—
237C4⋊Dic3—44no—
238C4⋊Dic3C2×D412yes—
239Dic3⋊C4—24no—
240Dic3⋊C4—22no—
241Dic3⋊C4—24no—
242Dic3⋊C4—22no—
243C4×Dic3—24no—
244C4×Dic3—24no—
245C4×Dic3—24no—
246C4×Dic3—24no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252Dic12—48no—
253C22×Q8—34no—
254C22×Q8—31no—
255C22×Q8—32no—
256C22×Q8—61no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8C22×D311yescharacteristic
263C23×C4—31no—
264C23×C4—61no—
265C2×Q16—64no—
266C2×Q16—64no—
267C2×Q16—64no—
268C2×Q16—64no—
269C2×Q16—64no—
270C2×Q16—64no—
271C22×C8—32no—
272C22×C8C22×D312yes—
273C4⋊Q8—62no—
274C4⋊Q8—62no—
275C22⋊Q8—64no—
276C22⋊Q8—64no—
277C4×Q8—22no—
278C4×Q8—34no—
279C4×Q8—68no—
280C4×Q8—62no—
281C4×Q8—68no—
282C4×Q8—62no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8D1214yes—
287#117—68no—
288#117—62no—
289#117—68no—
290#117—62no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D312yes—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—62no—
309C2×C42—21no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42C22×D311yescharacteristic
316C4⋊C8—22no—
317C4⋊C8—62no—
318#103—24no—
319#103—64no—
320#103—64no—
321#103—64no—
322C22⋊C8—64no—
323#95—21no—
324#95—21no—
325#95—62no—
326#95—62no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3#12112yes—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8#12114yes—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4#12012yes—
348C2×C3×C4#12012yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3#12112yes—
370C4×D3—216no—
371C4×D3—22no—
372C4×D3—22no—
373C4×D3—22no—
374C4×D3—22no—
375Dic6—216no—
376Dic6—22no—
377Dic6—22no—
378Dic6—22no—
379Dic6—42no—
380Dic6—42no—
381Dic6—42no—
382Dic6—42no—
383Dic6#12114yes—
384C3×C8—42no—
385C3⋊C8—42no—
386C24—31no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—32no—
391C2×Q8—32no—
392C2×Q8—32no—
393C2×Q8—64no—
394C2×Q8—64no—
395C2×Q8—64no—
396C2×Q8—68no—
397C2×Q8—68no—
398C2×Q8—68no—
399C2×Q8—68no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
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—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4C23×D311yescharacteristic
433Q16—128no—
434Q16—128no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—62no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—32no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—124no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4C2×D1212yes—
456C22⋊C4—64no—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—124no—
460C42—24no—
461C42—64no—
462C42—64no—
463C42—64no—
464C42—64no—
465C42—61no—
466C42—61no—
467C42C2×D1212yes—
468C42D3×D412yes—
469C22×C3C2×#12011yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#57812yes—
473C22×C3#57812yes—
474C2×D3—22no—
475C2×D3—24no—
476C2×D3—42no—
477C2×D3—41no—
478C2×D3—41no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—82no—
484C3×C4C2×#12111yescharacteristic
485C3×C4C2×#12111yescharacteristic
486C3×C4#52511yescharacteristic
487C3×C4#52511yescharacteristic
488Dic3—22no—
489Dic3—24no—
490Dic3—41no—
491Dic3—41no—
492Dic3—42no—
493Dic3—82no—
494C23—32no—
495C23—61no—
496C23—62no—
497C23—62no—
498C23—61no—
499C23C2×D3×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—34no—
504Q8—616no—
505Q8—62no—
506Q8—62no—
507Q8—62no—
508Q8—122no—
509Q8—122no—
510Q8—122no—
511Q8—122no—
512Q8#89514yes—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—616no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4#88212yes—
544C2×C4#88212yes—
545C2×C4C22×D1211yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4C2×#23911yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C8—42no—
552C8—122no—
553C2×C3#284711yescharacteristic
554C2×C3#301111yescharacteristic
555C2×C3#301111yescharacteristic
556C2×C3#304412yes—
557C2×C3#312712yes—
558D3—42no—
559C22—62no—
560C22—64no—
561C22—122no—
562C22—121no—
563C22—121no—
564C22C2×#88211yescharacteristic
565C22D3×#12111yescharacteristic
566C22#562711yescharacteristic
567C22#577112yes—
568C22#577912yes—
569C4—24no—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—62no—
574C4—64no—
575C4—82no—
576C4—121no—
577C4—122no—
578C4—121no—
579C4—242no—
580C4C2×#89511yescharacteristic
581C4#552411yescharacteristic
582C4D3×#12111yescharacteristic
583C4#562711yescharacteristic
584C3#3291511yescharacteristic, complemented, Sylow
585C2—122no—
586C2#7404811yescharacteristic
587C2#7883711yescharacteristic
588C2#7896111yescharacteristic
589C2#8060112yes—
590C2#8121112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1805D3
hypocenterC3#32915
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×#4849C2semidirect
6. C3×#1805C2semidirect
8. #70239C2semidirect
9. #69559C2semidirect
10. #68908C2semidirect
58. #1805D3semidirect
584. C3#32915semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×#50312
C2×#48492
C2×#48092
#741932
C3×#18052
#704762
#702392
#695592
#689082
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