Finite Group Databasebeta

← #10804087

Subgroups — #10804087

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C2×#4726C211yesmaximal, characteristic, complemented
3C2×D3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×#1779C211yesmaximal, characteristic, complemented
5#72126C211yesmaximal, characteristic, complemented
6#69958C214yesmaximal, complemented
7#69957C214yesmaximal, complemented
8#69910C211yesmaximal, characteristic, complemented
9#69910C211yesmaximal, characteristic, complemented
10#21740—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67C411yescharacteristic
13C23×C4×D3C2211yescharacteristic
14C22×#217C411yescharacteristic
15C2×C4×#67—24no—
16C2×C4×#67—24no—
17C2×C4×#67C412yes—
18C2×C4×#67C412yes—
19C2×C4×#67C2212yes—
20C2×C4×#67C2212yes—
21C22×(C2×D3)⋊C4C411yescharacteristic
22C22×Dic3⋊C4C411yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×ΓL(1,𝔽25)C2212yescomplemented
25C22×D3×C8C2212yescomplemented
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2214yescomplemented
29C2×#830C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#830C2214yescomplemented
32C2×(C2×D3)⋊C8C2212yescomplemented
33C2×(C2×D3)⋊C8C2212yescomplemented
34#4726—24no—
35D3×C22⋊C8—24no—
36#4550C2214yescomplemented
37#4550C2214yescomplemented
38C22×C4×D4—31no—
39C2×#407—31no—
40C22×C22⋊C8—31no—
41#1870—34no—
42#1869—34no—
43#1817—31no—
44#1817—31no—
45#1779—31no—
46#1779D311yescharacteristic, complemented
47C25×C3C2×C411yescharacteristic
48C24×D3C2×C411yescharacteristic
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—44no—
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2×C412yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67—24no—
57C22×#67—28no—
58C22×#67C2×C412yes—
59C22×#67C2×C412yes—
60C23×Dic3C2×C411yescharacteristic
61C23×Dic3C2×C411yescharacteristic
62C23×Dic3C2×C411yescharacteristic
63C22×C4×D3—44no—
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3C2×C411yescharacteristic
66C22×C4×D3D414yes—
67C22×C4×D3D414yes—
68C22×C4×D3C2311yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C2×#217—24no—
73C2×#217C2×C412yes—
74C2×#217C2×C412yes—
75C4×#67—28no—
76C4×#67—28no—
77C4×#67—24no—
78C4×#67—24no—
79C4×#67—24no—
80C4×#67—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4C2×C412yes—
83C2×(C2×D3)⋊C4C2×C412yes—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4C2×C412yes—
86C2×Dic3⋊C4C2×C412yes—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3C2×C412yes—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2312yes—
92C22×C3⋊C8C2312yes—
93C2×ΓL(1,𝔽25)—24no—
94C2×ΓL(1,𝔽25)—44no—
95C2×D3×C8—24no—
96C2×D3×C8—44no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—44no—
100#830—22no—
101#830—22no—
102#830—44no—
103(C2×D3)⋊C8—24no—
104(C2×D3)⋊C8—24no—
105C23×D4—31no—
106C24×C4—31no—
107C24×C4C2×D311yescharacteristic
108C22×M16—32no—
109C23×C8—32no—
110C2×C4×D4—32no—
111C2×C4×D4—32no—
112C2×C4×D4—32no—
113C2×C4×D4—32no—
114C2×C4×D4—64no—
115C2×C4×D4—64no—
116C22×C4⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C42—31no—
120#407—64no—
121C2×C22⋊C8—31no—
122C2×C22⋊C8—31no—
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—34no—
126C2×C22⋊C8—64no—
127C2×C22⋊C8C2×D311yescharacteristic
128C2×C22⋊C8C2×D311yescharacteristic
129C2×C22⋊C8C2×D314yescomplemented
130#336—34no—
131#336—34no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—42no—
137C24×C3C22×C412yes—
138C24×C3C22×C411yescharacteristic
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—24no—
142C23×D3—42no—
143C23×D3C22⋊C414yes—
144C23×D3C22×C411yescharacteristic
145C23×D3C22×C411yescharacteristic
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—48no—
157C22×C3×C4—48no—
158C22×C3×C4C22×C412yes—
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C2×#67—232no—
164C2×#67—24no—
165C2×#67—24no—
166C2×#67—24no—
167C2×#67—24no—
168C2×#67—24no—
169C2×#67M1618yescomplemented
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—42no—
176C22×Dic3C22⋊C414yes—
177C22×Dic3C22×C412yes—
178C22×Dic3C22×C411yescharacteristic
179C22×Dic3C22×C411yescharacteristic
180C22×Dic3C22×C412yes—
181C2×C4×D3—24no—
182C2×C4×D3—24no—
183C2×C4×D3—48no—
184C2×C4×D3—48no—
185C2×C4×D3—42no—
186C2×C4×D3—42no—
187C2×C4×D3C22⋊C414yes—
188C2×C4×D3C22⋊C414yes—
189C2×C4×D3C2×D412yes—
190C2×C4×D3C2×D412yes—
191C2×C4×D3C2×D412yes—
192C2×C4×D3C2×D412yes—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—24no—
196C2×C3×C8—24no—
197#217—24no—
198#217—24no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—24no—
201Dic3⋊C4—24no—
202Dic3⋊C4—24no—
203C4×Dic3—24no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—24no—
210ΓL(1,𝔽25)—48no—
211D3×C8—48no—
212C25—31no—
213C25C4×D311yescharacteristic
214C22×D4—32no—
215C22×D4—32no—
216C22×D4—68no—
217C22×D4—64no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—34no—
226C23×C4—34no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—44no—
230C23×C4—124no—
231C23×C4C4×D311yescharacteristic
232C23×C4C4×D312yes—
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235C2×M16—64no—
236C2×M16—124no—
237C22×C8—32no—
238C22×C8—32no—
239C22×C8—64no—
240C22×C8—124no—
241C22×C8C22×D312yes—
242C22×C8C22×D312yes—
243C4×D4—68no—
244C4×D4—68no—
245C4×D4—64no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—64no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—64no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—32no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C42—32no—
259C2×C42—32no—
260C2×C42—62no—
261C2×C42—62no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—44no—
265C22⋊C8—64no—
266C22⋊C8—64no—
267C22⋊C8—62no—
268C22⋊C8—62no—
269C22⋊C8—124no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—28no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—48no—
280C23×C3—48no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3C2×C22⋊C412yes—
284C23×C3C2×M1614yes—
285C23×C3#13112yes—
286C23×C3#13112yes—
287C23×C3C23×C411yescharacteristic
288C22×D3—22no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—28no—
296C22×D3—28no—
297C22×D3—44no—
298C22×D3—48no—
299C22×D3—48no—
300C22×D3C2×C22⋊C412yes—
301C22×D3C2×C22⋊C412yes—
302C22×D3C2×M1614yes—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—48no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4C2×C22⋊C412yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4H(5,𝔽2)14yes—
317#67—216no—
318#67—216no—
319C2×Dic3—28no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—44no—
328C2×Dic3—44no—
329C2×Dic3C2×C22⋊C412yes—
330C2×Dic3C2×C22⋊C412yes—
331C2×Dic3C2×M1614yes—
332C4×D3—24no—
333C4×D3—24no—
334C4×D3—24no—
335C4×D3—24no—
336C4×D3—44no—
337C4×D3—44no—
338C3×C8—42no—
339C3×C8—42no—
340C3⋊C8—42no—
341C3⋊C8—42no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—31no—
347C24—31no—
348C24—34no—
349C24—42no—
350C24—64no—
351C24—62no—
352C24—62no—
353C24—122no—
354C24C2×C4×D312yes—
355C24C2×C4×D311yescharacteristic
356C2×D4—38no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—632no—
362C2×D4—64no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—32no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—34no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—32no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—64no—
387C22×C4—62no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—128no—
394C22×C4—128no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4—122no—
398C22×C4C2×C4×D312yes—
399C22×C4C2×C4×D311yescharacteristic
400C22×C4D3×D412yes—
401C22×C4D3×D412yes—
402C22×C4C23×D311yescharacteristic
403M16—128no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—64no—
411C2×C8—64no—
412C2×C8—128no—
413C4⋊C4—64no—
414C4⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—64no—
420C42—62no—
421C42—62no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—416no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3#40814yes—
435C22×C3C22×C22⋊C411yescharacteristic
436C22×C3#51812yes—
437C22×C3C22×M1612yes—
438C22×C3C2×#13111yescharacteristic
439C22×C3C2×#13111yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—24no—
443C2×D3—24no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—416no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3#40714yes—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4#53412yes—
454C3×C4#53412yes—
455Dic3—22no—
456Dic3—22no—
457Dic3—24no—
458Dic3—42no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—32no—
469C23—34no—
470C23—32no—
471C23—48no—
472C23—48no—
473C23—44no—
474C23—44no—
475C23—68no—
476C23—61no—
477C23—62no—
478C23—61no—
479C23—61no—
480C23—62no—
481C23—62no—
482C23—61no—
483C23—68no—
484C23—124no—
485C23—128no—
486C23—128no—
487C23D3×C22⋊C412yes—
488C23#90112yes—
489C23D3×M1614yes—
490C23#90712yes—
491C23C22×C4×D311yescharacteristic
492D4—616no—
493D4—616no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—34no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—68no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—124no—
520C2×C4—124no—
521C2×C4—124no—
522C2×C4—124no—
523C2×C4D3×C22⋊C412yes—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4#100414yes—
527C8—42no—
528C8—42no—
529C8—122no—
530C8—122no—
531C2×C3—24no—
532C2×C3—44no—
533C2×C3#230811yescharacteristic
534C2×C3C2×#40711yescharacteristic
535C2×C3C2×#40811yescharacteristic
536C2×C3#297412yes—
537C2×C3#297412yes—
538D3—24no—
539D3—42no—
540C22—24no—
541C22—22no—
542C22—24no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—34no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—62no—
554C22—64no—
555C22—64no—
556C22—62no—
557C22—62no—
558C22—62no—
559C22—62no—
560C22—1216no—
561C22—122no—
562C22—122no—
563C22#472714yes—
564C22C2×D3×C22⋊C411yescharacteristic
565C22#548612yes—
566C22C2×#90111yescharacteristic
567C22C2×D3×M1612yes—
568C22C2×#90711yescharacteristic
569C4—22no—
570C4—42no—
571C4—64no—
572C4—62no—
573C4—62no—
574C4—122no—
575C4#558912yes—
576C4#558912yes—
577C3#2174011yescharacteristic, complemented, Sylow
578C2—24no—
579C2—44no—
580C2—64no—
581C2—122no—
582C2#7384911yescharacteristic
583C2C2×#472711yescharacteristic
584C2#7860312yes—
585C2D3×#40711yescharacteristic
586C2#7863312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1779C2
prime radicalC23×#67C4
cosocleC22×C3×C4C24
hypercenter#1779D3
hypocenterC3#21740
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#4726C2semidirect
3. C2×D3×C22⋊C8C2semidirect
4. C3×#1779C2semidirect
5. #72126C2semidirect
6. #69958C2semidirect
7. #69957C2semidirect
8. #69910C2semidirect
9. #69910C2semidirect
24. C22×ΓL(1,𝔽25)C22semidirect
25. C22×D3×C8C22semidirect
28. C2×C3×C22⋊C8C22semidirect
31. C2×#830C22semidirect
32. C2×(C2×D3)⋊C8C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
36. #4550C22semidirect
37. #4550C22semidirect
46. #1779D3semidirect
129. C2×C22⋊C8C2×D3semidirect
169. C2×#67M16semidirect
577. C3#21740semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#672
C2×#47262
C2×D3×C22⋊C82
C3×#17792
#721262
#699582
#699572
#699102
#217403