Finite Group Databasebeta

← #10804118

Subgroups — #10804118

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C2×#4727C212yesmaximal, complemented
3C3×#1780C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic, complemented
5#69958C212yesmaximal, complemented
6#69958C212yesmaximal, complemented
7#69957C212yesmaximal, complemented
8#69957C212yesmaximal, complemented
9#69910C212yesmaximal, complemented
10#21745—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13#5832—28no—
14C2×#947C411yescharacteristic
15C2×#944C411yescharacteristic
16C2×#940C411yescharacteristic
17C2×#939C412yes—
18C2×#939C412yes—
19C2×#936C411yescharacteristic
20C2×C4×#67C2212yes—
21C2×C4×#67C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×ΓL(1,𝔽25)C2212yes—
24C22×D3×C8C2212yes—
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#830C2212yes—
30C2×#830C2211yescharacteristic
31C2×#830C2212yes—
32C2×#830C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#4727—28no—
36#4550C2212yes—
37#4550C2212yes—
38#4550C2212yes—
39#4550C2212yes—
40C2×#532—31no—
41C2×#408—32no—
42#1870—32no—
43#1870—32no—
44#1869—32no—
45#1869—32no—
46#1817—32no—
47#1780—31no—
48#1780D311yescharacteristic, complemented
49C2×C3×#41—24no—
50C2×C3×#41—41no—
51C2×C3×#41—41no—
52C2×C3×#41—42no—
53C22×C3×Q8C2×C411yescharacteristic
54C22×C3×D4C2×C411yescharacteristic
55C22×C3×D4C2×C411yescharacteristic
56C22×C3×D4C2×C411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C412yes—
61C22×#67C2×C412yes—
62C22×#67C2×C412yes—
63C23×Dic3C2×C412yes—
64C2×#235—41no—
65C2×#235—42no—
66C2×#235—41no—
67C2×#235D418yes—
68C22×D12C2×C411yescharacteristic
69C22×C4×D3C2312yes—
70C22×Dic6C2×C411yescharacteristic
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C2×#217C2×C412yes—
74C2×#217C2×C412yes—
75#947—24no—
76#944—24no—
77#940—24no—
78#939—28no—
79#939—28no—
80#936—24no—
81C4×#67—24no—
82C4×#67—24no—
83C4×#67—24no—
84C4×#67—24no—
85C2×(C2×D3)⋊C4C2×C412yes—
86C2×(C2×D3)⋊C4C2×C412yes—
87C2×Dic3⋊C4C2×C412yes—
88C2×Dic3⋊C4C2×C412yes—
89C2×C4×Dic3—24no—
90C2×C4×Dic3C2×C411yescharacteristic
91C2×C4×Dic3C2×C411yescharacteristic
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2312yes—
95C2×ΓL(1,𝔽25)—24no—
96C2×ΓL(1,𝔽25)—42no—
97C2×ΓL(1,𝔽25)—42no—
98C2×D3×C8—24no—
99C2×D3×C8—42no—
100C2×D3×C8—42no—
101C3×C22⋊C8—24no—
102C3×C22⋊C8—41no—
103C3×C22⋊C8—42no—
104C3×C22⋊C8—41no—
105#830—24no—
106#830—41no—
107#830—42no—
108#830—41no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—24no—
111C22×#41—31no—
112C22×#41C2×D311yescharacteristic
113C22×M16—32no—
114C23×C8—32no—
115#532—68no—
116C2×C4⋊Q8—31no—
117C2×#127—31no—
118C2×#124—31no—
119C2×#124—31no—
120C2×#121—32no—
121C2×#121—32no—
122C2×C4×D4—32no—
123C2×C4×D4—32no—
124C22×C42—31no—
125#408—68no—
126C2×C22⋊C8—32no—
127C2×C22⋊C8—32no—
128C2×C22⋊C8—32no—
129C2×C22⋊C8—31no—
130C2×C22⋊C8—32no—
131C2×C22⋊C8—31no—
132C2×C22⋊C8C2×D312yes—
133C2×C22⋊C8C2×D311yescharacteristic
134C2×C22⋊C8C2×D312yes—
135C2×C22⋊C8C2×D311yescharacteristic
136#336—32no—
137#336—32no—
138#336—32no—
139#336—32no—
140C24×C3C22×C411yescharacteristic
141C24×C3C22×C411yescharacteristic
142C24×C3C22×C411yescharacteristic
143C23×D3C22×C412yes—
144C3×#41—48no—
145C3×#41—48no—
146C2×C3×Q8—22no—
147C2×C3×Q8—41no—
148C2×C3×Q8—41no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—41no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—24no—
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—42no—
179C2×#67—42no—
180C2×#67—42no—
181C2×#67—42no—
182C2×#67C22⋊C418yes—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—42no—
186C22×Dic3C22×C412yes—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189#235—48no—
190#235—48no—
191C2×D12—41no—
192C2×D12—41no—
193C2×D12C22⋊C414yes—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3C2×D414yes—
199C2×C4×D3C2×D414yes—
200C2×Dic6—41no—
201C2×Dic6—41no—
202C2×Dic6C22⋊C414yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209#217—24no—
210#217—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225ΓL(1,𝔽25)—48no—
226D3×C8—48no—
227C2×#41—24no—
228C2×#41—38no—
229C2×#41—41no—
230C2×#41—41no—
231C2×#41—42no—
232C2×#41—121no—
233C2×#41—122no—
234C2×#41—121no—
235C22×Q8—31no—
236C22×Q8C4×D311yescharacteristic
237C22×D4—32no—
238C22×D4—32no—
239C22×D4—32no—
240C22×D4—31no—
241C22×D4C4×D311yescharacteristic
242C22×D4C4×D311yescharacteristic
243C22×D4C4×D311yescharacteristic
244C23×C4—32no—
245C23×C4—32no—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D311yescharacteristic
248C23×C4C22×D311yescharacteristic
249C2×M16—64no—
250C2×M16—122no—
251C2×M16—122no—
252C22×C8—32no—
253C22×C8—32no—
254C22×C8—64no—
255C22×C8—122no—
256C22×C8—122no—
257C22×C8C22×D312yes—
258C22×C8C22×D312yes—
259C4⋊Q8—64no—
260#127—64no—
261#124—64no—
262#124—64no—
263#121—68no—
264#121—68no—
265C4×D4—64no—
266C4×D4—64no—
267C4×D4—64no—
268C4×D4—64no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—32no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—32no—
274C2×C22⋊C4—32no—
275C2×C42—31no—
276C2×C42—31no—
277C2×C42—32no—
278C2×C42—64no—
279C22⋊C8—24no—
280C22⋊C8—41no—
281C22⋊C8—42no—
282C22⋊C8—41no—
283C22⋊C8—64no—
284C22⋊C8—64no—
285C22⋊C8—64no—
286C22⋊C8—121no—
287C22⋊C8—122no—
288C22⋊C8—121no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3C2×C22⋊C412yes—
304C23×C3C23×C411yescharacteristic
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3C2×C22⋊C414yes—
312C3×Q8—44no—
313C3×D4—44no—
314C3×D4—44no—
315C3×D4—44no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4C2×C22⋊C412yes—
331C2×C3×C4C22×D412yes—
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C4H(5,𝔽2)12yes—
335#67—48no—
336#67—48no—
337#67—48no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—48no—
346C2×Dic3C2×C22⋊C414yes—
347C2×Dic3#13114yes—
348D12—44no—
349C4×D3—24no—
350C4×D3—24no—
351C4×D3—24no—
352C4×D3—24no—
353Dic6—44no—
354C3×C8—42no—
355C3×C8—42no—
356C3⋊C8—42no—
357C3⋊C8—42no—
358C24—32no—
359C24C2×C4×D311yescharacteristic
360C24C2×C4×D311yescharacteristic
361C24C2×C4×D311yescharacteristic
362#41—48no—
363#41—48no—
364#41—128no—
365#41—128no—
366C2×Q8—22no—
367C2×Q8—34no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—121no—
371C2×Q8—121no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—38no—
376C2×D4—34no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—121no—
388C2×D4—121no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—122no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—34no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—64no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—64no—
415C22×C4—122no—
416C22×C4C2×C4×D311yescharacteristic
417C22×C4C2×C4×D311yescharacteristic
418C22×C4C2×C4×D311yescharacteristic
419C22×C4D3×D414yes—
420C22×C4C23×D311yescharacteristic
421M16—128no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—62no—
429C2×C8—62no—
430C2×C8—62no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—128no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C22⋊C4—64no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C42—62no—
442C42—62no—
443C42—62no—
444C42—62no—
445C22×C3—22no—
446C22×C3—24no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3C22×C22⋊C411yescharacteristic
461C22×C3#51811yescharacteristic
462C22×C3#51811yescharacteristic
463C22×C3C2×#13112yes—
464C22×C3#56812yes—
465C2×D3—24no—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C3×C4—22no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4#53412yes—
476C3×C4#53412yes—
477Dic3—24no—
478Dic3—24no—
479Dic3—42no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—34no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23D3×C22⋊C412yes—
502C23C22×C4×D311yescharacteristic
503Q8—44no—
504Q8—124no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—124no—
509D4—128no—
510D4—128no—
511D4—128no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—34no—
523C2×C4—34no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—128no—
540C2×C4D3×C22⋊C412yes—
541C2×C4C2×D3×D412yes—
542C2×C4#100411yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4#100412yes—
545C8—42no—
546C8—42no—
547C8—122no—
548C8—122no—
549C2×C3—42no—
550C2×C3—81no—
551C2×C3—81no—
552C2×C3#230811yescharacteristic
553C2×C3#288012yes—
554C2×C3#297911yescharacteristic
555C2×C3#298011yescharacteristic
556C2×C3#298112yes—
557D3—44no—
558C22—24no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—64no—
574C22—64no—
575C22—64no—
576C22—122no—
577C22—122no—
578C22—122no—
579C22—122no—
580C22C2×D3×C22⋊C411yescharacteristic
581C22#548611yescharacteristic
582C22#548611yescharacteristic
583C22D3×#13112yes—
584C22#575012yes—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—64no—
589C4—64no—
590C4—122no—
591C4#558912yes—
592C4#558912yes—
593C3#2174511yescharacteristic, complemented, Sylow
594C2—42no—
595C2—81no—
596C2—81no—
597C2—124no—
598C2#7384911yescharacteristic
599C2#7866312yes—
600C2#7866711yescharacteristic
601C2#7866811yescharacteristic
602C2#7866912yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1780C2
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenter#1780D3
hypocenterC3#21745
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. C2×#4727C2semidirect
3. C3×#1780C2semidirect
4. #72136C2semidirect
5. #69958C2semidirect
6. #69958C2semidirect
7. #69957C2semidirect
8. #69957C2semidirect
9. #69910C2semidirect
48. #1780D3semidirect
593. C3#21745semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C2×#47272
C3×#17802
#721362
#699582
#699572
#699102
#217453