Finite Group Databasebeta

← #10724385

Subgroups — #10724385

Order 768.

1406 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×C3×#381C211yesmaximal, characteristic
2#72127C212yesmaximal
3#71546C211yesmaximal, characteristic
4#71546C211yesmaximal, characteristic
5#71545C212yesmaximal
6#69416C212yesmaximal, complemented
7#69412C212yesmaximal, complemented
8#69407C212yesmaximal, complemented
9#69401C212yesmaximal, complemented
10#14508—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4×Dic3C2212yes—
19C3×#381C418yescomplemented
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×#381D311yescharacteristic
34#1443—32no—
35#1442—32no—
36#1439—31no—
37#1439—31no—
38#1437—32no—
39#1436—32no—
40#1434—32no—
41C22⋊#95—32no—
42C25×C3C2311yescharacteristic
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—41no—
46C23×C3×C4—41no—
47C23×C3×C4—41no—
48C23×C3×C4—41no—
49C23×C3×C4C2×C414yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C2×C3×C22⋊C4—41no—
59C2×C3×C22⋊C4—41no—
60C2×C3×C22⋊C4—41no—
61C2×C3×C22⋊C4—41no—
62C2×C3×C22⋊C4C2×C414yes—
63C2×C3×C22⋊C4C2×C414yes—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—42no—
69C2×#217—42no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—41no—
72C2×C4⋊Dic3—41no—
73C2×C4⋊Dic3—42no—
74C2×C4×Dic3—24no—
75C2×C4×Dic3—42no—
76C2×C4×Dic3—42no—
77C3×#95C2×C418yes—
78C3×#95C2×C418yes—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—41no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95—42no—
88C3⋊#95—41no—
89C3⋊#95—42no—
90C24×C4C2×D311yescharacteristic
91C22×C4⋊C4—31no—
92C22×C4⋊C4—31no—
93C22×C22⋊C4—32no—
94C22×C22⋊C4—32no—
95C22×C22⋊C4C2×D311yescharacteristic
96C22×C22⋊C4C2×D311yescharacteristic
97C22×C42—32no—
98#381Dic318yescomplemented
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—31no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—31no—
110C2×#95C2×D312yes—
111C2×#95C2×D312yes—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—22no—
120C24×C3C22×C412yes—
121C24×C3#4112yes—
122C24×C3#4112yes—
123C24×C3C2411yescharacteristic
124C22×C3×C4—28no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—44no—
151C22×C3×C4C22×C412yes—
152C22×C3×C4C22×C414yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C22×C414yes—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3#4114yes—
182C22×Dic3#4114yes—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—44no—
185#217—48no—
186#217—48no—
187C4⋊Dic3—44no—
188C4⋊Dic3—44no—
189C4×Dic3—48no—
190C25C22×D311yescharacteristic
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—32no—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4C2×Dic314yes—
202C23×C4C22×D311yescharacteristic
203C23×C4C22×D311yescharacteristic
204C23×C4C22×D312yes—
205C23×C4C22×D312yes—
206C2×C4⋊C4—64no—
207C2×C4⋊C4—121no—
208C2×C4⋊C4—121no—
209C2×C4⋊C4—122no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—64no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—122no—
219C2×C22⋊C4—122no—
220C2×C22⋊C4C2×Dic314yes—
221C2×C22⋊C4C2×Dic314yes—
222C2×C42—64no—
223C2×C42—122no—
224C2×C42—122no—
225#95—64no—
226#95—64no—
227#95—121no—
228#95—122no—
229#95—121no—
230#95—122no—
231#95—122no—
232#95—122no—
233#95—122no—
234#95—122no—
235#95—122no—
236#95C2×Dic318yes—
237#95C2×Dic318yes—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—42no—
259C23×C3—41no—
260C23×C3#11714yes—
261C23×C3C23×C411yescharacteristic
262C23×C3C2×#4111yescharacteristic
263C23×C3C2×#4111yescharacteristic
264C23×C3C2×#4112yes—
265C23×C3C2×#4112yes—
266C23×C3H(5,𝔽2)11yescharacteristic
267C23×C3H(5,𝔽2)11yescharacteristic
268C23×C3H(5,𝔽2)11yescharacteristic
269C23×C3H(5,𝔽2)11yescharacteristic
270C23×C3H(5,𝔽2)12yes—
271C23×C32−1+412yes—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—416no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24#23512yes—
334C24#23512yes—
335C24C22×Dic312yes—
336C24C23×D311yescharacteristic
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—34no—
359C22×C4—34no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4C22×Dic312yes—
393C22×C4C22×Dic314yes—
394C22×C4C22×Dic312yes—
395C22×C4C22×Dic314yes—
396C4⋊C4—124no—
397C4⋊C4—124no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—128no—
401C22⋊C4—128no—
402C42—128no—
403C22×C3—24no—
404C22×C3—28no—
405C22×C3—44no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—44no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3C2×#11711yescharacteristic
415C22×C3C4×#4112yes—
416C22×C3#51811yescharacteristic
417C22×C3#51811yescharacteristic
418C22×C3#52012yes—
419C22×C3#53812yes—
420C22×C3#53811yescharacteristic
421C22×C3#53811yescharacteristic
422C22×C3#53811yescharacteristic
423C22×C3#53811yescharacteristic
424C22×C3#53912yes—
425C22×C3#54012yes—
426C22×C3#54212yes—
427C22×C3#55111yescharacteristic
428C22×C3#55112yes—
429C22×C3#55112yes—
430C22×C3#55212yes—
431C22×C3#55212yes—
432C22×C3#55312yes—
433C22×C3#55911yescharacteristic
434C22×C3#56011yescharacteristic
435C22×C3#56212yes—
436C22×C3#56311yescharacteristic
437C3×C4—44no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—42no—
441Dic3—44no—
442Dic3—44no—
443Dic3—82no—
444Dic3—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—41no—
458C23—42no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23#92614yes—
468C23C2×#23511yescharacteristic
469C23C2×#23511yescharacteristic
470C23#100411yescharacteristic
471C23#100411yescharacteristic
472C23#100411yescharacteristic
473C23#100411yescharacteristic
474C23D3×#4112yes—
475C23D3×#4112yes—
476C23#100912yes—
477C23#101012yes—
478C23C23×Dic311yescharacteristic
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—416no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C3—44no—
533C2×C3#231211yescharacteristic
534C2×C3#233512yes—
535C2×C3#234312yes—
536C2×C3#237411yescharacteristic
537C2×C3#237611yescharacteristic
538C2×C3#275012yes—
539C2×C3#275211yescharacteristic
540C2×C3#276011yescharacteristic
541C2×C3#276112yes—
542C2×C3#276411yescharacteristic
543C2×C3#277111yescharacteristic
544C22—28no—
545C22—24no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22#549411yescharacteristic
556C22#549411yescharacteristic
557C22#549411yescharacteristic
558C22#549411yescharacteristic
559C22#552712yes—
560C22#552712yes—
561C22#553112yes—
562C22#553112yes—
563C22#565111yescharacteristic
564C22#565712yes—
565C22#565911yescharacteristic
566C22#566012yes—
567C22#566112yes—
568C22#566212yes—
569C22#566312yes—
570C22#566411yescharacteristic
571C22#566611yescharacteristic
572C22#566712yes—
573C22C2×#92611yescharacteristic
574C22#579811yescharacteristic
575C22#579811yescharacteristic
576C22Dic3×#4112yes—
577C22#582712yes—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—124no—
583C4—124no—
584C4—242no—
585C4—242no—
586C3#1450811yescharacteristic, complemented, Sylow
587C2—44no—
588C2#7410911yescharacteristic
589C2#7411012yes—
590C2#7411111yescharacteristic
591C2#7411311yescharacteristic
592C2#7414212yes—
593C2#7414311yescharacteristic
594C2#8209311yescharacteristic
595C2#8223712yes—
596C2#8283712yes—
597C2#8283911yescharacteristic
598C2#8284011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#381C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#381D3
hypocenterC3#14508
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
6. #69416C2semidirect
7. #69412C2semidirect
8. #69407C2semidirect
9. #69401C2semidirect
19. C3×#381C4semidirect
98. #381Dic3semidirect
586. C3#14508semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3812
#721272
#715462
#715452
#694162
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