Finite Group Databasebeta

← #10944309

Subgroups — #10944309

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic
2C2×C3×#470C211yesmaximal, characteristic, complemented
3C2×#5189C211yesmaximal, characteristic, complemented
4C2×#5188C211yesmaximal, characteristic, complemented
5#82820C212yesmaximal
6#71988C211yesmaximal, characteristic, complemented
7#71696C212yesmaximal, complemented
8#70578C211yesmaximal, characteristic, complemented
9#70577C211yesmaximal, characteristic, complemented
10#70271C212yesmaximal, complemented
11#70270C212yesmaximal, complemented
12#32940—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×Q16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×#946C2212yes—
18C2×#946C2212yes—
19C23×C3⋊C8C2211yescharacteristic
20C2×#907—24no—
21C2×#907—24no—
22C2×#907—24no—
23C22×ΓL(1,𝔽25)C2211yescharacteristic
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C22×D3×C8C2211yescharacteristic
26C22×D3×C8C2211yescharacteristic
27C3×#470—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5212C2212yes—
33#5189—24no—
34#5188—24no—
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37#4991—22no—
38#4987C2212yes—
39#4987C2212yes—
40#4979—22no—
41#4959—22no—
42#4959—22no—
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#470—31no—
48C2×#470D311yescharacteristic, complemented
49C2×#467—31no—
50#2616—32no—
51#1894—32no—
52#1892—32no—
53#1892—32no—
54#1809—31no—
55#1809—31no—
56#1807—31no—
57C22×C3×Q8C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67D411yescharacteristic
60C22×#67D411yescharacteristic
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C22×D12D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C22×Dic6D411yescharacteristic
71C2×C3×Q16—22no—
72C2×C3×Q16—22no—
73C2×C3×Q16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16D414yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—24no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×#217—22no—
85#946—24no—
86#946—24no—
87#946—24no—
88#946—24no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8—22no—
99C22×C3⋊C8—22no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#907—232no—
103C2×ΓL(1,𝔽25)—22no—
104C2×ΓL(1,𝔽25)—22no—
105C2×ΓL(1,𝔽25)—22no—
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)D414yes—
109C2×D3×C8—22no—
110C2×D3×C8—22no—
111C2×D3×C8—22no—
112C2×D3×C8—22no—
113C2×D3×C8—22no—
114C2×D3×C8D414yes—
115C3×#106—22no—
116C3×#103—24no—
117#835—24no—
118#835—24no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122#818—22no—
123#817—22no—
124C22×#41—31no—
125C22×Q16C2×D311yescharacteristic
126C2×#131—64no—
127C2×#131—64no—
128C2×#131—64no—
129C22×M16—31no—
130C22×M16—31no—
131C22×M16C2×D311yescharacteristic
132C23×C8—31no—
133C23×C8—31no—
134C23×C8—31no—
135C2×C22⋊Q8—32no—
136C2×C22⋊Q8—32no—
137C2×C22⋊Q8C2×D312yes—
138#470—24no—
139#470—64no—
140#467—64no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×#103—32no—
145C2×#103—32no—
146C2×#103C2×D312yes—
147#395—62no—
148#394—62no—
149#394—62no—
150#394—62no—
151#380—32no—
152#380—32no—
153#380—32no—
154#340—32no—
155#340—32no—
156#340—32no—
157C24×C3C2×D411yescharacteristic
158C23×D3C2×D411yescharacteristic
159C23×D3C2×D411yescharacteristic
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—22no—
178C2×#67—22no—
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186#235—28no—
187#235—28no—
188#235—28no—
189#235—28no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3C2×D412yes—
202C2×C4×D3C2×D412yes—
203C2×C4×D3#4112yes—
204C2×C4×D3#4112yes—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—22no—
208C3×Q16—44no—
209C3×M16—28no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8C2×D412yes—
213C2×C3×C8C2×D412yes—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—44no—
219C3×C22⋊C4—44no—
220#217—82no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—44no—
223(C2×D3)⋊C4—82no—
224(C2×D3)⋊C4—82no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229Dic3⋊C4—44no—
230Dic3⋊C4—44no—
231C2×C3⋊C8—28no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8C2×D412yes—
237C2×C3⋊C8C2×D412yes—
238ΓL(1,𝔽25)—28no—
239ΓL(1,𝔽25)—28no—
240D3×C8—28no—
241D3×C8—28no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C22×Q8—31no—
249C22×Q8C22×D312yes—
250C22×D4—31no—
251C22×D4—31no—
252C22×D4—31no—
253C23×C4—31no—
254C23×C4—31no—
255C23×C4C22×D311yescharacteristic
256C2×Q16—22no—
257C2×Q16—22no—
258C2×Q16—22no—
259#131—632no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—34no—
263C2×M16—62no—
264C2×M16—62no—
265C2×M16—62no—
266C2×M16—62no—
267C2×M16—62no—
268C2×M16#6714yes—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—34no—
272C22×C8—62no—
273C22×C8—62no—
274C22×C8—62no—
275C22×C8—62no—
276C22×C8—62no—
277C22×C8—62no—
278C22×C8—62no—
279C22×C8—62no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—64no—
285C22⋊Q8—64no—
286C22⋊Q8—64no—
287C22⋊Q8—64no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4C22×D312yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300#107—62no—
301#106—22no—
302#106—62no—
303#103—24no—
304#103—64no—
305#103—64no—
306#95—62no—
307#95—62no—
308#95—62no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3#12112yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3#12112yes—
324C22×D3#12112yes—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12112yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348#67—28no—
349#67—28no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3#12112yes—
361C2×Dic3#12112yes—
362D12—28no—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371Dic6—28no—
372C3×C8—22no—
373C3×C8#12014yes—
374C3⋊C8—22no—
375C3⋊C8—22no—
376C24—31no—
377C24—31no—
378C24D3×D411yescharacteristic
379#41—68no—
380#41—68no—
381#41—68no—
382#41—68no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×Q8—62no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—62no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4D3×D411yescharacteristic
424C22×C4#23712yes—
425C22×C4C2×#6712yes—
426C22×C4C23×D311yescharacteristic
427Q16—44no—
428M16—28no—
429M16—68no—
430M16—68no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—32no—
434C2×C8—32no—
435C2×C8—68no—
436C2×C8—68no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—68no—
442C2×C8D3×D412yes—
443C2×C8D3×D412yes—
444C2×C8C2×#6712yes—
445C2×C8C2×#6712yes—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—62no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C22⋊C4—44no—
456C22⋊C4—124no—
457C22⋊C4—124no—
458C22⋊C4—242no—
459C22⋊C4—242no—
460C22⋊C4—242no—
461C22×C3—22no—
462C22×C3—24no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×#12111yescharacteristic
468C22×C3C2×#12111yescharacteristic
469C22×C3#57712yes—
470C22×C3#57812yes—
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—24no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—42no—
481C3×C4—22no—
482C3×C4—82no—
483C3×C4—82no—
484C3×C4C2×#12011yescharacteristic
485C3×C4#52511yescharacteristic
486C3×C4#52511yescharacteristic
487C3×C4#52511yescharacteristic
488Dic3—22no—
489Dic3—22no—
490Dic3—82no—
491Dic3—82no—
492C23—22no—
493C23—22no—
494C23—21no—
495C23—21no—
496C23—32no—
497C23—32no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—62no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23#93812yes—
507C23C2×D3×D411yescharacteristic
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—68no—
513D4—68no—
514D4—68no—
515D4—68no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4#93812yes—
549C2×C4C2×D3×D411yescharacteristic
550C2×C4C2×D3×D411yescharacteristic
551C2×C4C2×#23711yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4C22×#6711yescharacteristic
555C8—22no—
556C8—62no—
557C8—62no—
558C8#93714yes—
559C2×C3—42no—
560C2×C3#284711yescharacteristic
561C2×C3#305911yescharacteristic
562C2×C3#306011yescharacteristic
563C2×C3#306312yes—
564C2×C3#306312yes—
565D3—42no—
566D3—42no—
567C22—24no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—62no—
573C22—62no—
574C22—64no—
575C22—62no—
576C22—62no—
577C22—121no—
578C22—121no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22D3×#12111yescharacteristic
583C22D3×#12111yescharacteristic
584C22#577812yes—
585C22#577912yes—
586C22C2×#93811yescharacteristic
587C4—22no—
588C4—62no—
589C4—62no—
590C4—82no—
591C4—82no—
592C4—242no—
593C4—242no—
594C4#560611yescharacteristic
595C4#560611yescharacteristic
596C4C2×#93711yescharacteristic
597C4#582811yescharacteristic
598C3#3294011yescharacteristic, complemented, Sylow
599C2—42no—
600C2—122no—
601C2—122no—
602C2#8090511yescharacteristic
603C2#8090611yescharacteristic
604C2#8090712yes—
605C2#8272311yescharacteristic
606C2#8398612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#470C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#470D3
hypocenterC3#32940
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
2. C2×C3×#470C2semidirect
3. C2×#5189C2semidirect
4. C2×#5188C2semidirect
6. #71988C2semidirect
7. #71696C2semidirect
8. #70578C2semidirect
9. #70577C2semidirect
10. #70271C2semidirect
11. #70270C2semidirect
48. C2×#470D3semidirect
598. C3#32940semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
C2×C3×#4702
C2×#51892
C2×#51882
#828202
#719882
#716962
#705782
#705772
#702712
#702702
#329403