Finite Group Databasebeta

← #10898397

Subgroups — #10898397

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82325C211yesmaximal, characteristic
2#76560C211yesmaximal, characteristic, complemented
3C3×#1877C211yesmaximal, characteristic
4#72051C211yesmaximal, characteristic
5#72049C211yesmaximal, characteristic
6#71606C211yesmaximal, characteristic
7#70262C211yesmaximal, characteristic, complemented
8#70255C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic
10#70188C211yesmaximal, characteristic
11#69667C211yesmaximal, characteristic, complemented
12#68859C211yesmaximal, characteristic, complemented
13#68840C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#29441—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#103C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C3×#391C2211yescharacteristic
22#5229C2211yescharacteristic
23#5229C2211yescharacteristic
24C2×#835C2211yescharacteristic
25C2×#835C2211yescharacteristic
26#5221C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5212—21no—
31#5211—21no—
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4951—21no—
44#4951—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46#4932C2211yescharacteristic
47#4928C2211yescharacteristic
48C3⋊#401—21no—
49#4660—21no—
50#4659—21no—
51#4658—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2645—31no—
61#2319—31no—
62#2088—31no—
63#2068—31no—
64#2067—31no—
65#2055—31no—
66#1968—31no—
67#1942—31no—
68#1936—31no—
69#1935—31no—
70#1928—31no—
71#1920—31no—
72#1877D311yescharacteristic
73#1862—31no—
74#1768—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C3×#103—22no—
108C3×#102—41no—
109C3×#95—21no—
110C3×#95—21no—
111#835—22no—
112#835—41no—
113#832—22no—
114#832—22no—
115C3⋊#95—21no—
116C3⋊#95—21no—
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⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129#817—22no—
130#815—41no—
131Dic3⋊C8—22no—
132Dic3⋊C8—22no—
133C8×Dic3—41no—
134#807—22no—
135#807—41no—
136C22×SD16C2×D311yescharacteristic
137C2×C4×D4C2×D311yescharacteristic
138C2×#107—31no—
139C2×#107—31no—
140C2×#106—31no—
141C2×C4⋊C8—31no—
142C2×C4⋊C8—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#103C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#97—31no—
150C2×C4×C8—31no—
151#401—61no—
152#400—61no—
153#400—61no—
154#400—61no—
155#391—31no—
156#391—31no—
157#391C2×D311yescharacteristic
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389—31no—
165#388—61no—
166#388—61no—
167#381—61no—
168#380—61no—
169C4×C4⋊C4—31no—
170#376—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—24no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C3×SD16—44no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8#4112yes—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4#4112yes—
220C3×C22⋊C4—24no—
221C3×C42—21no—
222C3×C42—21no—
223#217—81no—
224#217—81no—
225#217—81no—
226#217—81no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—81no—
236C4⋊Dic3—81no—
237C4⋊Dic3—81no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8#4112yes—
251C22×Q8C22×D311yescharacteristic
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C2×SD16—22no—
255C2×SD16—22no—
256C2×SD16—41no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261C4×D4—22no—
262C4×D4—22no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283#107—62no—
284#107—121no—
285#106—62no—
286C4⋊C8—62no—
287C4⋊C8—62no—
288#103—22no—
289#103—62no—
290#103—121no—
291#102—41no—
292#102—62no—
293#102—62no—
294#97—121no—
295C4×C8—121no—
296#95—21no—
297#95—21no—
298#95—31no—
299#95—31no—
300#95—31no—
301#95—31no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3—22no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3C22×D411yescharacteristic
320C3×Q8—41no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—84no—
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C3×C8—22no—
375C3×C8—41no—
376C3⋊C8—22no—
377C3⋊C8—41no—
378C24—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4#23712yes—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—44no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—32no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8#23512yes—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4—241no—
435C4⋊C4—241no—
436C4⋊C4—241no—
437C4⋊C4#23712yes—
438C22⋊C4—24no—
439C22⋊C4—241no—
440C22⋊C4—241no—
441C22⋊C4—241no—
442C22⋊C4—241no—
443C42—21no—
444C42—21no—
445C42—61no—
446C42—61no—
447C42—61no—
448C42—61no—
449C42—241no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3#52511yescharacteristic
458C22×C3#53211yescharacteristic
459C22×C3#53711yescharacteristic
460C22×C3#57611yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57811yescharacteristic
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#55111yescharacteristic
470C3×C4#55211yescharacteristic
471C3×C4#55311yescharacteristic
472C3×C4#55611yescharacteristic
473Dic3—41no—
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480Dic3—81no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23C2×D3×D411yescharacteristic
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—84no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4C2×#23511yescharacteristic
538C2×C4C2×#23711yescharacteristic
539C2×C4C2×#23711yescharacteristic
540C2×C4#100411yescharacteristic
541C2×C4D3×#4111yescharacteristic
542C2×C4#101011yescharacteristic
543C8—22no—
544C8—41no—
545C8—62no—
546C8—121no—
547C2×C3—82no—
548C2×C3#260111yescharacteristic
549C2×C3#311811yescharacteristic
550C2×C3#312011yescharacteristic
551C2×C3#317411yescharacteristic
552C2×C3#317511yescharacteristic
553C2×C3#325711yescharacteristic
554C2×C3#325811yescharacteristic
555C22—42no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22#554811yescharacteristic
563C22#558611yescharacteristic
564C22#560811yescharacteristic
565C22#576211yescharacteristic
566C22#577011yescharacteristic
567C22#577411yescharacteristic
568C22#577911yescharacteristic
569C4—41no—
570C4—42no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—121no—
575C4—121no—
576C4—121no—
577C4—121no—
578C4—241no—
579C4—241no—
580C4—241no—
581C4—241no—
582C4#549511yescharacteristic
583C4#554211yescharacteristic
584C4#561411yescharacteristic
585C4#565011yescharacteristic
586C3#2944111yescharacteristic, complemented, Sylow
587C2—82no—
588C2#7477711yescharacteristic
589C2#7966711yescharacteristic
590C2#7976811yescharacteristic
591C2#8013611yescharacteristic
592C2#8021611yescharacteristic
593C2#8044211yescharacteristic
594C2#8051511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1877D3
hypocenterC3#29441
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #76560C2semidirect
7. #70262C2semidirect
8. #70255C2semidirect
11. #69667C2semidirect
12. #68859C2semidirect
13. #68840C2semidirect
14. #68623C2semidirect
15. #68604C2semidirect
586. C3#29441semidirect

Maximal subgroups

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

subgroupindex
#823252
#765602
C3×#18772
#720512
#720492
#716062
#702622
#702552
#701942
#701882
#696672
#688592
#688402
#686232
#686042
#294413