Finite Group Databasebeta

← #10944255

Subgroups — #10944255

Order 768.

1494 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×#907C211yesmaximal, characteristic, complemented
2#82709C211yesmaximal, characteristic, complemented
3C2×#5038C211yesmaximal, characteristic, complemented
4C2×#4775C212yesmaximal, complemented
5#74030C211yesmaximal, characteristic, complemented
6C3×#1895C211yesmaximal, characteristic, complemented
7#71990C211yesmaximal, characteristic, complemented
8#70203C212yesmaximal, complemented
9#69512C211yesmaximal, characteristic, complemented
10#68885C212yesmaximal, complemented
11#68869C212yesmaximal, complemented
12#32933—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×#938C2212yes—
17C22×#213C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×#907—24no—
21C2×#907—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×D3×C8C2212yes—
24C2×#895C2212yes—
25C2×C3×#102C2212yes—
26C3×#380C2211yescharacteristic
27#5223—22no—
28C2×#832C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×#821C2212yes—
31#5038—24no—
32#4987C2212yes—
33#4987C2212yes—
34C2×#809C2211yescharacteristic
35C2×#807C2211yescharacteristic
36C2×#807C2212yes—
37#4960—22no—
38#4960—21no—
39#4947—21no—
40#4775—28no—
41#4673—21no—
42#4671—21no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#466—32no—
48C2×#466—31no—
49#2615—31no—
50#2615—31no—
51#1895—32no—
52#1895—32no—
53#1895—31no—
54#1895D311yescharacteristic, complemented
55#1808—31no—
56#1808—32no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×#67—22no—
62C22×#67D412yes—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—24no—
66C2×#235—24no—
67C22×D12—22no—
68C22×D12D411yescharacteristic
69C22×D12C2311yescharacteristic
70C22×C4×D3C2312yes—
71C22×Dic6D411yescharacteristic
72C2×C3×M16—22no—
73C2×C3×M16—24no—
74C22×C3×C8C2312yes—
75C3×#121—22no—
76C3×#121—22no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×#217—22no—
82C2×#217—21no—
83#938—24no—
84#938—24no—
85C2×#213—24no—
86C2×#213—22no—
87#930—22no—
88#930—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C22×C3⋊C8—24no—
96C22×C3⋊C8D414yescomplemented
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#907—232no—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)—24no—
102C2×ΓL(1,𝔽25)—24no—
103C2×D3×C8—24no—
104C2×D3×C8—24no—
105C2×D3×C8D418yescomplemented
106#895—24no—
107#895—24no—
108C3×#102—24no—
109C3×#95—21no—
110#832—22no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#821—24no—
115#809—22no—
116#807—24no—
117#807—22no—
118C22×#41—31no—
119C22×D8—31no—
120C2×#131—64no—
121C2×#131—68no—
122C22×M16—32no—
123C22×M16C2×D311yescharacteristic
124C23×C8—31no—
125C23×C8—32no—
126C2×#121—32no—
127C2×#121—31no—
128C2×#121—32no—
129C2×#121C2×D311yescharacteristic
130#466—68no—
131#466—64no—
132C2×#107—32no—
133C2×#107—31no—
134C2×#102—31no—
135C2×#102—32no—
136C2×#102—31no—
137C2×#102C2×D312yes—
138#393—61no—
139#393—61no—
140#392—61no—
141#392—62no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—32no—
146#380—32no—
147#380C2×D311yescharacteristic
148#340—32no—
149#340—32no—
150#340C2×D312yes—
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C23×D3C2×D412yes—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—42no—
160C2×C3×D4—82no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—24no—
172C2×#67—42no—
173C2×#67—44no—
174C2×#67—82no—
175C2×#67—82no—
176C2×#67—82no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3C2×D412yes—
180#235—28no—
181#235—216no—
182#235—28no—
183C2×D12—24no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—21no—
189C2×D12—44no—
190C2×D12—82no—
191C2×D12—82no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3C2×D414yes—
197C2×C4×D3#4114yes—
198C2×Dic6—22no—
199C2×Dic6—24no—
200C3×M16—28no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8C2×D414yes—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—81no—
210#217—44no—
211#217—81no—
212#213—44no—
213(C2×D3)⋊C4—42no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—82no—
216(C2×D3)⋊C4—82no—
217C4⋊Dic3—22no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—28no—
223C2×C3⋊C8C2×D412yes—
224C2×C3⋊C8C2×D412yes—
225C2×C3⋊C8C2×D414yes—
226ΓL(1,𝔽25)—216no—
227D3×C8—216no—
228C2×#41—64no—
229C2×#41—64no—
230C2×#41—62no—
231C2×#41—62no—
232C22×Q8—31no—
233C22×D4—21no—
234C22×D4—32no—
235C22×D4—31no—
236C22×D4—31no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—61no—
240C22×D4C22×D311yescharacteristic
241C23×C4—32no—
242C23×C4C22×D311yescharacteristic
243C2×D8—64no—
244C2×D8—62no—
245#131—632no—
246C2×M16—22no—
247C2×M16—24no—
248C2×M16—64no—
249C2×M16—64no—
250C2×M16—64no—
251C22×C8—31no—
252C22×C8—34no—
253C22×C8—31no—
254C22×C8—38no—
255C22×C8—64no—
256C22×C8—64no—
257C22×C8—64no—
258C22×C8C22×D312yes—
259#121—22no—
260#121—22no—
261#121—64no—
262#121—64no—
263#121—64no—
264#121—64no—
265#121—62no—
266#121—62no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—61no—
279#107—64no—
280#107—62no—
281#102—24no—
282#102—62no—
283#102—62no—
284#102—64no—
285#95—21no—
286#95—62no—
287#95—62no—
288#95—61no—
289C23×C3—22no—
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3#12112yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3#12114yes—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—84no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4#12112yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4112yes—
330#67—216no—
331#67—88no—
332#67—84no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3#12114yes—
341D12—28no—
342D12—41no—
343D12—41no—
344D12—42no—
345D12—88no—
346C4×D3—24no—
347C4×D3—24no—
348C4×D3—24no—
349C4×D3—24no—
350Dic6—28no—
351C3×C8—24no—
352C3⋊C8—22no—
353C3⋊C8#12014yes—
354C24—21no—
355C24—32no—
356C24—61no—
357C24D3×D411yescharacteristic
358#41—68no—
359#41—68no—
360#41—616no—
361C2×Q8—64no—
362C2×Q8—62no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—64no—
371C2×D4—64no—
372C2×D4—62no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—82no—
378C2×D4—124no—
379C2×D4—124no—
380C2×D4—122no—
381C2×D4—242no—
382C2×D4—242no—
383C2×D4—242no—
384C2×D4—242no—
385C2×D4—242no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—32no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4#23512yes—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D412yes—
402C22×C4C23×D311yescharacteristic
403D8—124no—
404M16—28no—
405M16—616no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—32no—
409C2×C8—34no—
410C2×C8—32no—
411C2×C8—616no—
412C2×C8—68no—
413C2×C8—62no—
414C2×C8D3×D414yes—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C22⋊C4—124no—
426C22⋊C4—124no—
427C22⋊C4—122no—
428C22⋊C4—242no—
429C22⋊C4—242no—
430C22⋊C4—241no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—81no—
437C22×C3—82no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—82no—
441C22×C3C2×#12111yescharacteristic
442C22×C3C2×#12112yes—
443C22×C3#57611yescharacteristic
444C22×C3#57612yes—
445C22×C3#57811yescharacteristic
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—24no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—81no—
454C2×D3—82no—
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—82no—
458C3×C4—22no—
459C3×C4—82no—
460C3×C4—81no—
461C3×C4C2×#12011yescharacteristic
462C3×C4#52511yescharacteristic
463C3×C4#52512yes—
464Dic3—24no—
465Dic3—81no—
466Dic3—82no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—34no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—62no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—62no—
482C23—62no—
483C23—121no—
484C23—121no—
485C23—122no—
486C23—121no—
487C23—122no—
488C23#88412yes—
489C23C2×D3×D411yescharacteristic
490Q8—68no—
491D4—41no—
492D4—42no—
493D4—41no—
494D4—68no—
495D4—616no—
496D4—84no—
497D4—121no—
498D4—122no—
499D4—121no—
500D4—248no—
501D4—244no—
502D4—248no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—34no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4#88412yes—
525C2×C4C2×#23511yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×D3×D412yes—
528C2×C4D3×#4112yes—
529C8—24no—
530C8—34no—
531C8—62no—
532C2×C3—42no—
533C2×C3—161no—
534C2×C3#284711yescharacteristic
535C2×C3#305811yescharacteristic
536C2×C3#305812yes—
537C2×C3#306111yescharacteristic
538C2×C3#306112yes—
539D3—44no—
540D3—161no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—64no—
547C22—64no—
548C22—64no—
549C22—81no—
550C22—81no—
551C22—82no—
552C22—81no—
553C22—82no—
554C22—122no—
555C22—122no—
556C22—122no—
557C22—122no—
558C22—241no—
559C22—241no—
560C22—242no—
561C22—241no—
562C22—242no—
563C22C2×#88411yescharacteristic
564C22D3×#12112yes—
565C22#577411yescharacteristic
566C22#577412yes—
567C22#577911yescharacteristic
568C4—22no—
569C4—64no—
570C4—81no—
571C4—82no—
572C4—241no—
573C4—242no—
574C4#554712yes—
575C4D3×#12011yescharacteristic
576C4#565611yescharacteristic
577C3#3293311yescharacteristic, complemented, Sylow
578C2—42no—
579C2—124no—
580C2—161no—
581C2—481no—
582C2#7493511yescharacteristic
583C2#7889111yescharacteristic
584C2#7893211yescharacteristic
585C2#8080812yes—
586C2#8086512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1895C2
socleC23×C3C22×D4
hypercenter#1895D3
hypocenterC3#32933
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
2. #82709C2semidirect
3. C2×#5038C2semidirect
4. C2×#4775C2semidirect
5. #74030C2semidirect
6. C3×#1895C2semidirect
7. #71990C2semidirect
8. #70203C2semidirect
9. #69512C2semidirect
10. #68885C2semidirect
11. #68869C2semidirect
54. #1895D3semidirect
96. C22×C3⋊C8D4semidirect
105. C2×D3×C8D4semidirect
577. C3#32933semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
#827092
C2×#50382
C2×#47752
#740302
C3×#18952
#719902
#702032
#695122
#688852
#688692
#329333