Finite Group Databasebeta

← #10898413

Subgroups — #10898413

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76483C211yesmaximal, characteristic, complemented
2#76375C211yesmaximal, characteristic
3C3×#1968C211yesmaximal, characteristic
4#72051C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#70568C211yesmaximal, characteristic, complemented
7#70558C211yesmaximal, characteristic, complemented
8#70247C211yesmaximal, characteristic, complemented
9#70238C211yesmaximal, characteristic, complemented
10#70179C211yesmaximal, characteristic
11#70156C211yesmaximal, characteristic
12#69769C211yesmaximal, characteristic, complemented
13#69755C211yesmaximal, characteristic, complemented
14#69675C211yesmaximal, characteristic, complemented
15#69645C211yesmaximal, characteristic
16#29442—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#390C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4984C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40#4953—21no—
41#4952—21no—
42#4952—21no—
43#4948—21no—
44#4932C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4654—21no—
49#4653—21no—
50#4651—21no—
51#4650—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2644—31no—
61#2320—31no—
62#2088—31no—
63#2066—31no—
64#2065—31no—
65#2055—31no—
66#1968D311yescharacteristic
67#1941—31no—
68#1936—31no—
69#1935—31no—
70#1926—31no—
71#1920—31no—
72#1880—31no—
73#1865—31no—
74#1768—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C3×#106—22no—
110C3×#103—22no—
111C3×#102—22no—
112C3×C4×C8—41no—
113C3×#95—21no—
114C3×#95—21no—
115#835—22no—
116#835—41no—
117#832—22no—
118#832—41no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#817—22no—
132#817—41no—
133Dic3⋊C8—22no—
134Dic3⋊C8—22no—
135#803—41no—
136C22×SD16C2×D311yescharacteristic
137C2×C4×Q8—31no—
138C2×#106—31no—
139C2×#106—31no—
140C2×#106C2×D311yescharacteristic
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×C8C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#397—61no—
154#396—61no—
155#391—31no—
156#391C2×D311yescharacteristic
157#390—31no—
158#390—31no—
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389—31no—
165#387—61no—
166#387—61no—
167#383—61no—
168#382—61no—
169C4×C4⋊C4—31no—
170#376—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
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#4112yes—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C3×SD16—44no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8#4112yes—
212C2×C3×C8#4112yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C22⋊C4—81no—
218C3×C22⋊C4—81no—
219C3×C42—21no—
220C3×C42—21no—
221#217—81no—
222#217—81no—
223#217—81no—
224#217—81no—
225C4⋊Dic3—24no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—81no—
234C4⋊Dic3#4112yes—
235Dic3⋊C4—81no—
236Dic3⋊C4—81no—
237Dic3⋊C4—81no—
238Dic3⋊C4—81no—
239C4×Dic3—24no—
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—21no—
251C22×Q8C22×D311yescharacteristic
252C22×D4C22×D311yescharacteristic
253C2×SD16—22no—
254C2×SD16—22no—
255C2×SD16—41no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C4×Q8—62no—
261C4×Q8—62no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—61no—
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—21no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#106—22no—
286#106—62no—
287#106—121no—
288C4⋊C8—62no—
289C4⋊C8—62no—
290#103—22no—
291#103—62no—
292#103—121no—
293#102—22no—
294#102—62no—
295#102—121no—
296#97—121no—
297C4×C8—41no—
298#95—21no—
299#95—21no—
300#95—31no—
301#95—31no—
302#95—31no—
303#95—31no—
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—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
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×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C42−1+411yescharacteristic
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—22no—
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—
373C3×C8—22no—
374C3×C8—22no—
375C3⋊C8—41no—
376C3⋊C8—41no—
377C24—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8#23512yes—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C23×D311yescharacteristic
404SD16—44no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8#23712yes—
416C2×C8#23712yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—32no—
422C4⋊C4—64no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—241no—
431C4⋊C4—241no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4—241no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—241no—
438C22⋊C4—241no—
439C22⋊C4—241no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—64no—
444C42—61no—
445C42—61no—
446C42—61no—
447C42—61no—
448C42—241no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3#52511yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53711yescharacteristic
459C22×C3#57711yescharacteristic
460C22×C3#57711yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57711yescharacteristic
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4#55111yescharacteristic
469C3×C4#55211yescharacteristic
470C3×C4#55311yescharacteristic
471C3×C4#55611yescharacteristic
472Dic3—41no—
473Dic3—42no—
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×D3×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—61no—
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—62no—
523C2×C4—61no—
524C2×C4—121no—
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×C4C2×#23511yescharacteristic
536C2×C4C2×#23711yescharacteristic
537C2×C4C2×#23711yescharacteristic
538C2×C4#100411yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4#101011yescharacteristic
541C8—22no—
542C8—22no—
543C8—121no—
544C8—121no—
545C2×C3—161no—
546C2×C3#260111yescharacteristic
547C2×C3#312511yescharacteristic
548C2×C3#312611yescharacteristic
549C2×C3#317311yescharacteristic
550C2×C3#317411yescharacteristic
551C2×C3#326011yescharacteristic
552C2×C3#326311yescharacteristic
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#554711yescharacteristic
561C22#559711yescharacteristic
562C22#571611yescharacteristic
563C22#576211yescharacteristic
564C22#576211yescharacteristic
565C22#577011yescharacteristic
566C22#577011yescharacteristic
567C4—41no—
568C4—41no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—121no—
573C4—121no—
574C4—122no—
575C4—121no—
576C4—121no—
577C4—241no—
578C4—241no—
579C4—241no—
580C4#549511yescharacteristic
581C4#554111yescharacteristic
582C4#560111yescharacteristic
583C4#567111yescharacteristic
584C3#2944211yescharacteristic, complemented, Sylow
585C2—161no—
586C2#7636911yescharacteristic
587C2#7966111yescharacteristic
588C2#7969811yescharacteristic
589C2#8064611yescharacteristic
590C2#8064711yescharacteristic
591C2#8175311yescharacteristic
592C2#8181011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1968C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1968D3
hypocenterC3#29442
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
1. #76483C2semidirect
6. #70568C2semidirect
7. #70558C2semidirect
8. #70247C2semidirect
9. #70238C2semidirect
12. #69769C2semidirect
13. #69755C2semidirect
14. #69675C2semidirect
584. C3#29442semidirect

Maximal subgroups

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

subgroupindex
#764832
#763752
C3×#19682
#720512
#720482
#705682
#705582
#702472
#702382
#701792
#701562
#697692
#697552
#696752
#696452
#294423