Finite Group Databasebeta

← #10944303

Subgroups — #10944303

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×#948C211yesmaximal, characteristic, complemented
2C3×#2615C211yesmaximal, characteristic, complemented
3#84098C211yesmaximal, characteristic, complemented
4C2×#5039C212yesmaximal, complemented
5C2×#5038C211yesmaximal, characteristic, complemented
6#71682C211yesmaximal, characteristic, complemented
7#71681C212yesmaximal, complemented
8#69517C211yesmaximal, characteristic, complemented
9#69515C212yesmaximal, complemented
10#69513C212yesmaximal, complemented
11#69511C211yesmaximal, characteristic, complemented
12#32939—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#121C2211yescharacteristic
16C2×#948—28no—
17C2×#948—24no—
18C22×#213C2211yescharacteristic
19C2×#930C2212yes—
20C2×#930C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2212yes—
23C23×C3⋊C8C2212yes—
24C23×C3⋊C8C2211yescharacteristic
25C3×#393—21no—
26C3×#392—21no—
27C3×#392—22no—
28C3×#380C2211yescharacteristic
29C3×#380C2212yes—
30#5231—21no—
31#5223—22no—
32#5223—21no—
33C2×#832C2211yescharacteristic
34C2×#832C2212yes—
35#5212C2212yes—
36#5212C2211yescharacteristic
37#5039—28no—
38#5038—24no—
39C2×#809C2211yescharacteristic
40C2×#809C2212yes—
41C2×#808C2212yes—
42C2×#807C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#469—32no—
48C2×#466—31no—
49#2615—31no—
50#2615D311yescharacteristic, complemented
51#1895—31no—
52#1895—31no—
53#1893—32no—
54#1893—32no—
55#1809—32no—
56#1807—31no—
57C2×C3×#41—24no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4D412yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—22no—
70C22×#67—21no—
71C22×D12C2311yescharacteristic
72C3×#121—22no—
73C3×#121—24no—
74C3×#121—24no—
75C3×#121—22no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×#217—22no—
83C2×#217—21no—
84#948—232no—
85C2×#213—22no—
86C2×#213—24no—
87#930—22no—
88#930—24no—
89#930—24no—
90#930—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×#208—22no—
96C2×#208—24no—
97C2×#208—24no—
98C2×#208—24no—
99C2×#208D418yescomplemented
100C22×C3⋊C8—24no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8—24no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8D414yescomplemented
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C3×#95—21no—
109C3×#95—22no—
110#832—22no—
111#832—24no—
112C3⋊#95—22no—
113C3⋊#95—21no—
114#809—22no—
115#809—24no—
116#808—24no—
117#807—22no—
118C22×#41C2×D311yescharacteristic
119C22×D8—31no—
120C2×#131—64no—
121C2×#131—68no—
122C22×M16—32no—
123C22×M16—31no—
124C23×C8—31no—
125C23×C8—32no—
126C2×#121—32no—
127C2×#121—31no—
128C2×#121C2×D312yes—
129C2×#121C2×D311yescharacteristic
130#469—68no—
131#466—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×#102—31no—
135C2×#102—32no—
136C2×#102—31no—
137C2×#102—32no—
138#393—21no—
139#393—61no—
140#392—21no—
141#392—22no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—32no—
146#380C2×D311yescharacteristic
147#380C2×D312yes—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C24×C3C2×D411yescharacteristic
154C23×D3—21no—
155C3×#41—28no—
156C3×#41—216no—
157C3×#41—28no—
158C2×C3×Q8—22no—
159C2×C3×Q8—24no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—24no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—44no—
171C2×C3×D4—82no—
172C2×C3×D4—82no—
173C2×C3×D4—82no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4#4112yes—
187C22×C3×C4#4114yes—
188C22×C3×C4C2411yescharacteristic
189C2×#67—42no—
190C2×#67—44no—
191C2×#67—82no—
192C2×#67—82no—
193C2×#67—82no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—82no—
207C3×C22⋊C4—81no—
208#217—82no—
209#217—81no—
210#213—44no—
211(C2×D3)⋊C4—42no—
212(C2×D3)⋊C4—44no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217#208—28no—
218#208—216no—
219C2×C3⋊C8—216no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—28no—
224C2×C3⋊C8C2×D412yes—
225C2×C3⋊C8C2×D414yes—
226C2×C3⋊C8C2×D412yes—
227C2×C3⋊C8C2×D414yes—
228C2×#41—22no—
229C2×#41—24no—
230C2×#41—22no—
231C2×#41—24no—
232C22×Q8#6711yescharacteristic
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—31no—
236C22×D4—62no—
237C22×D4—61no—
238C22×D4#6711yescharacteristic
239C22×D4#6712yes—
240C22×D4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D312yes—
243C2×D8—62no—
244C2×D8—64no—
245#131—632no—
246C2×M16—38no—
247C2×M16—64no—
248C2×M16—64no—
249C2×M16—64no—
250C2×M16—62no—
251C22×C8—31no—
252C22×C8—34no—
253C22×C8—31no—
254C22×C8—32no—
255C22×C8—64no—
256C22×C8—64no—
257C22×C8—64no—
258C22×C8—64no—
259#121—22no—
260#121—24no—
261#121—24no—
262#121—22no—
263#121—62no—
264#121—64no—
265#121—64no—
266#121—62no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279#107—62no—
280#106—64no—
281#102—62no—
282#102—64no—
283#102—62no—
284#102—64no—
285#95—21no—
286#95—22no—
287#95—61no—
288#95—62no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3#12112yes—
305C23×C3#12114yes—
306C23×C3C22×D411yescharacteristic
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3—41no—
312C3×Q8—28no—
313C3×D4—216no—
314C3×D4—28no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—42no—
318C3×D4—84no—
319C3×D4—88no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4#12112yes—
336C2×C3×C4#12114yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D412yes—
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4112yes—
341#67—84no—
342#67—88no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348D12—41no—
349D12—41no—
350D12—42no—
351C3⋊C8—22no—
352C3⋊C8—24no—
353C3⋊C8#12014yes—
354C24—21no—
355C24—61no—
356C24C2×#6712yes—
357C24C2×#6711yescharacteristic
358#41—28no—
359#41—28no—
360#41—216no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—61no—
377C2×D4—62no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—122no—
382C2×D4—124no—
383C2×D4—242no—
384C2×D4—242no—
385C2×D4—242no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4D3×D412yes—
397C22×C4D3×D414yes—
398C22×C4#23712yes—
399C22×C4#23714yes—
400C22×C4C2×#6712yes—
401C22×C4C2×#6711yescharacteristic
402C22×C4C23×D311yescharacteristic
403D8—124no—
404M16—616no—
405M16—68no—
406C2×C8—34no—
407C2×C8—32no—
408C2×C8—32no—
409C2×C8—34no—
410C2×C8—68no—
411C2×C8—616no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—62no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—62no—
420C4⋊C4—61no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C22⋊C4—81no—
427C22⋊C4—122no—
428C22⋊C4—124no—
429C22⋊C4—242no—
430C22⋊C4—241no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3C2×#12111yescharacteristic
449C22×C3C2×#12112yes—
450C22×C3#57612yes—
451C22×C3#57711yescharacteristic
452C22×C3#57711yescharacteristic
453C2×D3—81no—
454C2×D3—81no—
455C2×D3—81no—
456C2×D3—82no—
457C2×D3—82no—
458C3×C4—24no—
459C3×C4—22no—
460C3×C4—81no—
461C3×C4—82no—
462C3×C4C2×#12011yescharacteristic
463C3×C4#52512yes—
464C3×C4#52511yescharacteristic
465Dic3—81no—
466Dic3—82no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—121no—
483C23—122no—
484C23—122no—
485C23—121no—
486C23—121no—
487C23#93912yes—
488C23#93914yes—
489C23C22×#6711yescharacteristic
490Q8—28no—
491D4—216no—
492D4—28no—
493D4—41no—
494D4—42no—
495D4—41no—
496D4—84no—
497D4—88no—
498D4—121no—
499D4—122no—
500D4—121no—
501D4—244no—
502D4—248no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4#93912yes—
524C2×C4#93914yes—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D412yes—
527C2×C4C2×#23711yescharacteristic
528C2×C4C2×#23712yes—
529C8—34no—
530C8—62no—
531C8—64no—
532C2×C3—42no—
533C2×C3—44no—
534C2×C3—161no—
535C2×C3#284711yescharacteristic
536C2×C3#305811yescharacteristic
537C2×C3#306011yescharacteristic
538C2×C3#306212yes—
539C2×C3#306212yes—
540D3—161no—
541C22—22no—
542C22—24no—
543C22—22no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—82no—
556C22—81no—
557C22—82no—
558C22—241no—
559C22—241no—
560C22—242no—
561C22—241no—
562C22—242no—
563C22C2×#93911yescharacteristic
564C22C2×#93912yes—
565C22#583512yes—
566C22#583611yescharacteristic
567C22#583711yescharacteristic
568C4—24no—
569C4—22no—
570C4—81no—
571C4—82no—
572C4—241no—
573C4—242no—
574C4#558611yescharacteristic
575C4#558612yes—
576C4D3×#12011yescharacteristic
577C3#3293911yescharacteristic, complemented, Sylow
578C2—42no—
579C2—44no—
580C2—161no—
581C2—481no—
582C2#8260111yescharacteristic
583C2#8337811yescharacteristic
584C2#8337912yes—
585C2#8342011yescharacteristic
586C2#8342212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2615C2
socleC23×C3C22×D4
hypercenter#2615D3
hypocenterC3#32939
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2615

Complements

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

NH ≅ G/Ntype
1. C22×#948C2semidirect
2. C3×#2615C2semidirect
3. #84098C2semidirect
4. C2×#5039C2semidirect
5. C2×#5038C2semidirect
6. #71682C2semidirect
7. #71681C2semidirect
8. #69517C2semidirect
9. #69515C2semidirect
10. #69513C2semidirect
11. #69511C2semidirect
50. #2615D3semidirect
99. C2×#208D4semidirect
104. C22×C3⋊C8D4semidirect
577. C3#32939semidirect

Maximal subgroups

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

subgroupindex
C22×#9482
C3×#26152
#840982
C2×#50392
C2×#50382
#716822
#716812
#695172
#695152
#695132
#695112
#329393