Finite Group Databasebeta

← #10799569

Subgroups — #10799569

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82454C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3#73761C211yesmaximal, characteristic
4C3×C4×#103C211yesmaximal, characteristic, complemented
5#70260C211yesmaximal, characteristic
6#70257C211yesmaximal, characteristic
7#70256C211yesmaximal, characteristic, complemented
8#70254C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic
10#69749C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68860C211yesmaximal, characteristic, complemented
13#68859C211yesmaximal, characteristic, complemented
14#68857C211yesmaximal, characteristic
15#68853C211yesmaximal, characteristic
16#21291—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24#5229C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#815C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#5026C418yescomplemented
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4980—21no—
39#4980—21no—
40#4972C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4940—21no—
47#4940—21no—
48#4937—21no—
49#4936—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51#4932C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4667—21no—
54#4664—21no—
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#440—31no—
62#2320—31no—
63#2319—31no—
64#1973—31no—
65#1968—31no—
66#1943—31no—
67#1942—31no—
68#1936—31no—
69#1935—31no—
70#1934—31no—
71#1933—31no—
72#1776—31no—
73C4×#103D311yescharacteristic, complemented
74C4×#102—31no—
75#1726—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×Q8—22no—
82C3×C4×Q8C2×C414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#215—22no—
91C2×#215—22no—
92C2×#215C2×C414yes—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C4×D12—22no—
111C4×D12C2×C414yes—
112C3×#103—22no—
113C3×#103—22no—
114C3×C4×C8—22no—
115C3×#95—21no—
116#835C2×C414yes—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#821—22no—
131#821—22no—
132#815—22no—
133#815—22no—
134C8×Dic3—22no—
135C8×Dic3—22no—
136#809C2×C414yes—
137#808—22no—
138#807—22no—
139#807C2×C414yes—
140#803C2×C414yes—
141C22×SD16—31no—
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—31no—
144#440—38no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#103—31no—
149C2×#103C2×D311yescharacteristic
150C2×#103C2×D311yescharacteristic
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#97—31no—
155C2×#97—31no—
156C2×#97—31no—
157C2×C4×C8—31no—
158C2×C4×C8—31no—
159C2×C4×C8C2×D311yescharacteristic
160#391—31no—
161#391—31no—
162#390—31no—
163#390—31no—
164#389—31no—
165#389—31no—
166#389—31no—
167#389—31no—
168#388—61no—
169#388—61no—
170#387—61no—
171#387—61no—
172#383—61no—
173#382—61no—
174#381—61no—
175#380—61no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4C2×D311yescharacteristic
178#376—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340C2×D311yescharacteristic
184#340C2×D311yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8C22×C412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C22×C412yes—
212C2×C4×D3—24no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8#4112yes—
220C2×C3×C8#4112yes—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4C22×C412yes—
228C3×C42—24no—
229C3×C42—81no—
230C3×C42C22×C412yes—
231C3×C42#4112yes—
232#215—28no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—81no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—81no—
244C4⋊Dic3—81no—
245C4⋊Dic3C22×C412yes—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8C22×C412yes—
256C2×C3⋊C8C22×C412yes—
257C2×C3⋊C8#4112yes—
258C2×C3⋊C8#4112yes—
259C22×Q8C22×D311yescharacteristic
260C22×D4—31no—
261C23×C4—61no—
262C2×SD16—34no—
263C2×SD16—62no—
264C2×SD16—62no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C4×Q8—22no—
270C4×Q8C4×D314yes—
271C4×D4—34no—
272C4×D4—62no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—34no—
296#107—62no—
297#106—62no—
298#103—22no—
299#103—22no—
300#103—34no—
301#102—34no—
302#102—62no—
303#102—62no—
304#97—34no—
305#97—62no—
306#97—62no—
307C4×C8—22no—
308C4×C8—62no—
309C4×C8—62no—
310#95—21no—
311#95—31no—
312#95—31no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324C23×C3C22×D411yescharacteristic
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—41no—
335C3×Q8—41no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4C4×D412yes—
353C2×C3×C4C4×D412yes—
354C2×C3×C4C23×C411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381D12—22no—
382D12—22no—
383D12—41no—
384D12—41no—
385C4×D3—84no—
386C3×C8—22no—
387C3×C8—22no—
388C3⋊C8—22no—
389C3⋊C8#11714yes—
390C24—61no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×C4×D312yes—
397C2×D4—32no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—61no—
412C22×C4—64no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—241no—
416C22×C4—241no—
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4C23×D311yescharacteristic
420SD16—68no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—32no—
426C2×C8—32no—
427C2×C8—32no—
428C2×C8—32no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8#23512yes—
432C2×C8#23512yes—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4C2×C4×D312yes—
450C22⋊C4—64no—
451C22⋊C4—241no—
452C22⋊C4—241no—
453C22⋊C4—241no—
454C22⋊C4—241no—
455C42—24no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—61no—
460C42—81no—
461C42—241no—
462C42C2×C4×D312yes—
463C42#23712yes—
464C22×C3C2×C4×D411yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53211yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57711yescharacteristic
470C22×C3#57811yescharacteristic
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4C2×#11711yescharacteristic
485C3×C4C4×#4111yescharacteristic
486C3×C4#52911yescharacteristic
487C3×C4#52911yescharacteristic
488Dic3—41no—
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—42no—
493Dic3—81no—
494Dic3—81no—
495C23—62no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23C2×D3×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—62no—
508D4—62no—
509D4—121no—
510D4—121no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—244no—
549C2×C4#67⋊C412yes—
550C2×C4#67⋊C412yes—
551C2×C4C22×C4×D311yescharacteristic
552C2×C4C2×#23511yescharacteristic
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4D3×#4111yescharacteristic
556C2×C4D3×#4111yescharacteristic
557C8—22no—
558C8—22no—
559C8—34no—
560C8—62no—
561C2×C3#229311yescharacteristic
562C2×C3#294611yescharacteristic
563C2×C3#294711yescharacteristic
564C2×C3#317411yescharacteristic
565C2×C3#317511yescharacteristic
566C2×C3#317811yescharacteristic
567C2×C3#317911yescharacteristic
568D3—82no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22—241no—
573C22—241no—
574C22—241no—
575C22—241no—
576C22C2×#67⋊C411yescharacteristic
577C22#554811yescharacteristic
578C22#554811yescharacteristic
579C22#576211yescharacteristic
580C22#576611yescharacteristic
581C22#576911yescharacteristic
582C22#577011yescharacteristic
583C4—22no—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—121no—
590C4—121no—
591C4—121no—
592C4—121no—
593C4—122no—
594C4—241no—
595C4—241no—
596C4C4×#23511yescharacteristic
597C4D3×#11711yescharacteristic
598C4#553811yescharacteristic
599C4#553811yescharacteristic
600C3#2129111yescharacteristic, complemented, Sylow
601C2—242no—
602C2#7343911yescharacteristic
603C2#7877411yescharacteristic
604C2#7882011yescharacteristic
605C2#7966711yescharacteristic
606C2#7966911yescharacteristic
607C2#7972711yescharacteristic
608C2#7972911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenterC4×#103D3
hypocenterC3#21291
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. #82454C2semidirect
4. C3×C4×#103C2semidirect
7. #70256C2semidirect
8. #70254C2semidirect
10. #69749C2semidirect
11. #69481C2semidirect
12. #68860C2semidirect
13. #68859C2semidirect
33. #5026C4semidirect
73. C4×#103D3semidirect
600. C3#21291semidirect

Maximal subgroups

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

subgroupindex
#824542
C2×#50262
#737612
C3×C4×#1032
#702602
#702572
#702562
#702542
C4×#8212
#697492
#694812
#688602
#688592
#688572
#688532
#212913