Finite Group Databasebeta

← #10698490

Subgroups — #10698490

Order 768.

694 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
1#82117C211yesmaximal, characteristic
2C3×#1439C211yesmaximal, characteristic
3#71559C211yesmaximal, characteristic, complemented
4#71555C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#10399—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C2×C3×#95—21no—
14C2×C3×#95C2211yescharacteristic
15#5247—21no—
16#5212C2211yescharacteristic
17#5210—21no—
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#810—21no—
21C2×#807—21no—
22#4949—21no—
23#4948—21no—
24#4948—21no—
25#4947—21no—
26#4529—21no—
27#4529—21no—
28#4525—22no—
29#4452—22no—
30#2626—31no—
31#1872—31no—
32#1852—31no—
33C2×#328—31no—
34C2×#323—31no—
35#1439D311yescharacteristic
36#1294—31no—
37C25×C3D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4—21no—
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—42no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—42no—
53C2×C3×C22⋊C4—42no—
54C2×C3×C22⋊C4—42no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—42no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830D412yes—
83#810—42no—
84#807—42no—
85C2×C22⋊Q8—31no—
86C22×C4⋊C4C2×D311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4C2×D311yescharacteristic
89C2×#107—61no—
90C2×#103—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93#399—61no—
94#396—61no—
95#396—61no—
96#393—61no—
97#385—61no—
98#380—31no—
99#379—61no—
100C2×#95—21no—
101C2×#95C2×D311yescharacteristic
102#340—61no—
103#340—61no—
104#328—62no—
105#323—62no—
106C24×C3—21no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3—21no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3Q1612yes—
117C24×C3C2×D411yescharacteristic
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C2×Dic6—21no—
150C2×Dic6—21no—
151C2×Dic6—41no—
152C2×Dic6—41no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—48no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162#217—41no—
163#217—41no—
164#217—41no—
165#217—41no—
166#217—81no—
167#217—81no—
168#217—81no—
169#217—81no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174Dic3⋊C4—41no—
175Dic3⋊C4—41no—
176Dic3⋊C4—81no—
177Dic3⋊C4—81no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C2×C3⋊C8—41no—
184C25#6711yescharacteristic
185C22×Q8—31no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—61no—
191C22×C8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—61no—
220#107—122no—
221#103—122no—
222C22⋊C8—32no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226#95—21no—
227#95—21no—
228#95—42no—
229#95—61no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—81no—
254C23×C3—81no—
255C23×C3—81no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3—81no—
259C23×C3#12011yescharacteristic
260C23×C3#12111yescharacteristic
261C23×C3#12311yescharacteristic
262C23×C3C2×Q1611yescharacteristic
263C23×C3#13511yescharacteristic
264C23×C3AGL(1,ℤ/8)11yescharacteristic
265C23×C3#13711yescharacteristic
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—81no—
285C2×C3×C4—81no—
286C2×C3×C4—81no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4—81no—
292C2×C3×C4—81no—
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12111yescharacteristic
295C2×C3×C4#12311yescharacteristic
296C2×C3×C4#12411yescharacteristic
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309Dic6—42no—
310Dic6—81no—
311C3⋊C8—42no—
312C3⋊C8—81no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24#21612yes—
324C24C2×#6711yescharacteristic
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—121no—
328C2×Q8—121no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4#23511yescharacteristic
355C22×C4#23511yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23711yescharacteristic
360C2×C8—62no—
361C2×C8—121no—
362C2×C8—121no—
363C2×C8—121no—
364C2×C8—121no—
365C2×C8—121no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—121no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C4⋊C4—121no—
376C4⋊C4—241no—
377C4⋊C4—241no—
378C22⋊C4—48no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—121no—
384C22⋊C4—121no—
385C22⋊C4—121no—
386C22⋊C4—121no—
387C22⋊C4—241no—
388C22⋊C4—241no—
389C22⋊C4—241no—
390C22⋊C4—241no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3#39411yescharacteristic
415C22×C3#44911yescharacteristic
416C22×C3#45111yescharacteristic
417C22×C3#45311yescharacteristic
418C22×C3#45511yescharacteristic
419C22×C3B(4,𝔽2)11yescharacteristic
420C22×C3#45811yescharacteristic
421C22×C3#46711yescharacteristic
422C22×C3#46811yescharacteristic
423C22×C3#48411yescharacteristic
424C22×C3#48511yescharacteristic
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—82no—
430C3×C4—82no—
431C3×C4—81no—
432Dic3—81no—
433Dic3—81no—
434Dic3—81no—
435Dic3—81no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23#92911yescharacteristic
464C23#93211yescharacteristic
465C23#93711yescharacteristic
466C23#93911yescharacteristic
467C23C2×#21611yescharacteristic
468C23#95011yescharacteristic
469C23#95111yescharacteristic
470Q8—122no—
471Q8—241no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4#88311yescharacteristic
512C2×C4#88411yescharacteristic
513C2×C4#88411yescharacteristic
514C2×C4#88511yescharacteristic
515C8—122no—
516C8—241no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—82no—
522C2×C3#160011yescharacteristic
523C2×C3#160211yescharacteristic
524C2×C3#161211yescharacteristic
525C2×C3#161411yescharacteristic
526C2×C3#203611yescharacteristic
527C2×C3#203711yescharacteristic
528C2×C3#204411yescharacteristic
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22#495911yescharacteristic
553C22#497011yescharacteristic
554C22#499611yescharacteristic
555C22#503211yescharacteristic
556C22#503711yescharacteristic
557C22#504211yescharacteristic
558C22#505011yescharacteristic
559C22#505111yescharacteristic
560C22#505311yescharacteristic
561C22#506111yescharacteristic
562C22#507711yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—82no—
568C4—81no—
569C4—82no—
570C4—241no—
571C4—241no—
572C4—241no—
573C4—241no—
574C3#1039911yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2#6952711yescharacteristic
581C2#6959911yescharacteristic
582C2#6966511yescharacteristic
583C2#6969711yescharacteristic
584C2#6970111yescharacteristic
585C2#6970911yescharacteristic
586C2#6971611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1439C2
socleC23×C3#120
prime radicalC25×C3D4
hypercenter#1439D3
hypocenterC3#10399
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #71559C2semidirect
4. #71555C2semidirect
6. C2×#4452C2semidirect
7. #67612C2semidirect
574. C3#10399semidirect

Maximal subgroups

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

subgroupindex
#821172
C3×#14392
#715592
#715552
C2×#45252
C2×#44522
#676122
#103993