Finite Group Databasebeta

← #10824313

Subgroups — #10824313

Order 768.

774 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79208C211yesmaximal, characteristic
2D3×#428C211yesmaximal, characteristic
3C2×#4881C211yesmaximal, characteristic
4#79080C211yesmaximal, characteristic
5C3×#2089C211yesmaximal, characteristic, complemented
6#70671C211yesmaximal, characteristic, complemented
7#70670C211yesmaximal, characteristic, complemented
8#70626C211yesmaximal, characteristic, complemented
9#70533C211yesmaximal, characteristic, complemented
10#68989C211yesmaximal, characteristic
11D3×#343C211yesmaximal, characteristic
12#68981C211yesmaximal, characteristic, complemented
13#68980C211yesmaximal, characteristic, complemented
14#68938C211yesmaximal, characteristic
15#68924C211yesmaximal, characteristic, complemented
16#23675—31nomaximal, Sylow
17C2×D3×M16C2211yescharacteristic
18C2×D3×M16C2211yescharacteristic
19C2×D3×M16C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21D3×#117C2211yescharacteristic
22C2×#892C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C3×#424C2211yescharacteristic
28#5121C2211yescharacteristic
29#5121C2211yescharacteristic
30#5119C2211yescharacteristic
31C2×#817C2211yescharacteristic
32#5102C2211yescharacteristic
33#4999C2211yescharacteristic
34#4995C2211yescharacteristic
35#4995C2211yescharacteristic
36#4964C2211yescharacteristic
37#4964C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4881—22no—
41#4881—22no—
42#4881—21no—
43#4881—21no—
44#4880—21no—
45D3×#107—21no—
46#4861—21no—
47#4861—21no—
48#4860—21no—
49D3×#106—21no—
50#4839—22no—
51#4837—22no—
52C3×#343C2211yescharacteristic
53C3×#343C2211yescharacteristic
54C3×#341C2211yescharacteristic
55#4558C2211yescharacteristic
56#4558C2211yescharacteristic
57#4554C2211yescharacteristic
58#4554C2211yescharacteristic
59#4554C2211yescharacteristic
60#4554C2211yescharacteristic
61#4552C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64#2911—31no—
65C2×#428—31no—
66C2×#428—31no—
67#2905—31no—
68#2090—31no—
69#2090—31no—
70#2090—31no—
71#2090—31no—
72#2089—31no—
73#2089—31no—
74#2089—31no—
75#2089D311yescharacteristic, complemented
76#1742—31no—
77C2×#343—31no—
78#1738—31no—
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C2×C3×M16C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#117C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89#926C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98D3×M16—22no—
99D3×M16—22no—
100D3×M16—22no—
101D3×M16—22no—
102D3×M16Q814yes—
103D3×M16Q814yes—
104C2×ΓL(1,𝔽25)—21no—
105C2×ΓL(1,𝔽25)—21no—
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—21no—
109C2×ΓL(1,𝔽25)D412yes—
110C2×ΓL(1,𝔽25)D412yes—
111C2×D3×C8—21no—
112C2×D3×C8—21no—
113C2×D3×C8—21no—
114#892—21no—
115#892—21no—
116#892—21no—
117#892—21no—
118#892D411yescharacteristic
119#892D411yescharacteristic
120D3×C4⋊C4—21no—
121D3×C4⋊C4—21no—
122D3×C4⋊C4—21no—
123D3×C4⋊C4—21no—
124D3×C4⋊C4D411yescharacteristic
125D3×C4⋊C4D411yescharacteristic
126D3×C22⋊C4—21no—
127#877—21no—
128#872—21no—
129C42×D3—21no—
130C3×#107—22no—
131C3×#107—21no—
132C3×#107—21no—
133C3×#106—21no—
134C3×#106—21no—
135C3×C4⋊C8—22no—
136#818—21no—
137#818—21no—
138#817—22no—
139#817—21no—
140#817—21no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#808—21no—
144#808—21no—
145#808—21no—
146#808—21no—
147#808—22no—
148#807—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807—22no—
153C4⋊(C3⋊C8)—22no—
154C22×M16—31no—
155C22×M16—31no—
156C22×M16—31no—
157C22×M16—31no—
158C2×#117—31no—
159C2×#117—31no—
160C22×C4⋊C4—31no—
161#428—31no—
162#428—31no—
163#428—31no—
164#428—31no—
165#428—31no—
166#428—31no—
167#428—61no—
168#428—61no—
169#428—61no—
170#428—61no—
171#428—62no—
172#428—62no—
173#428C2×D311yescharacteristic
174#428C2×D311yescharacteristic
175#427—61no—
176#427—61no—
177C2×#107—31no—
178C2×#107—61no—
179C2×#107C2×D311yescharacteristic
180C2×#106—31no—
181C2×#106—31no—
182C2×#106—61no—
183#424—31no—
184#424—31no—
185#424—31no—
186#424—62no—
187#424C2×D311yescharacteristic
188#423—62no—
189#343—31no—
190#343—31no—
191#343—31no—
192#343—31no—
193#343—31no—
194#343—31no—
195#343C2×D311yescharacteristic
196#343C2×D311yescharacteristic
197#341—31no—
198#341—31no—
199#341—31no—
200#341C2×D311yescharacteristic
201C23×D3#4111yescharacteristic
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3#4111yescharacteristic
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—41no—
211C2×C4×D3—41no—
212C2×C4×D3—82no—
213C2×C4×D3C2×D411yescharacteristic
214C2×C4×D3C2×D411yescharacteristic
215C2×C4×D3C2×D411yescharacteristic
216C2×C4×D3C2×D411yescharacteristic
217C2×C4×D3C2×Q812yes—
218C2×C4×D3C2×Q812yes—
219C2×C4×D3#4112yes—
220C2×C4×D3#4112yes—
221C3×M16—21no—
222C3×M16—21no—
223C3×M16—21no—
224C3×M16—21no—
225C3×M16C2×Q812yes—
226C3×M16C2×Q812yes—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C22⋊C4—21no—
240C3×C42—21no—
241#217—21no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×D411yescharacteristic
251C4⋊Dic3C2×D411yescharacteristic
252Dic3⋊C4—21no—
253Dic3⋊C4—21no—
254Dic3⋊C4—41no—
255Dic3⋊C4—41no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—41no—
260C4×Dic3—41no—
261#208—21no—
262#208—21no—
263#208—21no—
264#208—21no—
265#208C2×Q812yes—
266#208C2×Q812yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273ΓL(1,𝔽25)—24no—
274ΓL(1,𝔽25)—24no—
275ΓL(1,𝔽25)—24no—
276ΓL(1,𝔽25)—24no—
277ΓL(1,𝔽25)—42no—
278D3×C8—24no—
279D3×C8—24no—
280D3×C8—42no—
281C23×C4—31no—
282C23×C4—61no—
283C2×M16—32no—
284C2×M16—32no—
285C2×M16—31no—
286C2×M16—31no—
287C2×M16—34no—
288C2×M16—34no—
289C2×M16—31no—
290C2×M16—62no—
291C2×M16—62no—
292C2×M16—61no—
293C2×M16—61no—
294C2×M16—61no—
295C2×M16—62no—
296C2×M16—62no—
297C2×M16—62no—
298C2×M16—62no—
299C2×M16C22×D311yescharacteristic
300C2×M16C22×D311yescharacteristic
301C2×M16C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8—61no—
304C22×C8—61no—
305C22×C8—61no—
306C22×C8C22×D311yescharacteristic
307#117—31no—
308#117—31no—
309#117—31no—
310#117—61no—
311#117—61no—
312#117—61no—
313#117—61no—
314#117—61no—
315#117—61no—
316#117C22×D311yescharacteristic
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—61no—
331C2×C42—61no—
332#107—22no—
333#107—21no—
334#107—21no—
335#107—61no—
336#107—61no—
337#107—62no—
338#107—61no—
339#107—61no—
340#107—61no—
341#107—61no—
342#106—21no—
343#106—21no—
344#106—61no—
345#106—61no—
346#106—62no—
347#106—61no—
348#106—61no—
349#106—62no—
350#106—61no—
351#106—61no—
352C4⋊C8—22no—
353C4⋊C8—62no—
354C4⋊C8—62no—
355C4⋊C8—62no—
356C23×C3C2×#4111yescharacteristic
357C22×D3—41no—
358C22×D3—41no—
359C22×D3#12111yescharacteristic
360C22×D3#12111yescharacteristic
361C22×D3#12512yes—
362C22×D3#12512yes—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—81no—
368C2×C3×C4—81no—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×Q811yescharacteristic
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—81no—
380C2×Dic3—81no—
381C2×Dic3#12111yescharacteristic
382C2×Dic3#12111yescharacteristic
383C2×Dic3#12512yes—
384C2×Dic3#12512yes—
385C4×D3—81no—
386C4×D3—81no—
387C4×D3—82no—
388C4×D3—82no—
389C4×D3C22⋊Q812yes—
390C4×D3C22⋊Q812yes—
391C4×D3C22⋊Q812yes—
392C4×D3C22⋊Q812yes—
393C4×D3#12312yes—
394C4×D3#12312yes—
395C4×D3C4⋊Q812yes—
396C4×D3C4⋊Q812yes—
397C3×C8—22no—
398C3×C8—22no—
399C3×C8—22no—
400C3×C8—41no—
401C3⋊C8—22no—
402C3⋊C8—22no—
403C3⋊C8—22no—
404C3⋊C8—41no—
405C24—31no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4—242no—
424C22×C4C23×D311yescharacteristic
425M16—21no—
426M16—21no—
427M16—21no—
428M16—21no—
429M16—32no—
430M16—32no—
431M16—64no—
432M16—64no—
433M16—64no—
434M16—64no—
435M16—61no—
436M16—61no—
437M16—61no—
438M16—61no—
439M16—122no—
440M16D3×Q812yes—
441M16D3×Q812yes—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—32no—
448C2×C8—64no—
449C2×C8—64no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—122no—
456C2×C8D3×D412yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—31no—
462C4⋊C4—31no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—121no—
470C4⋊C4—121no—
471C4⋊C4D3×D411yescharacteristic
472C4⋊C4D3×D411yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—121no—
478C22⋊C4—121no—
479C42—21no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—121no—
484C42—121no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3C2×#12111yescharacteristic
488C22×C3C2×#12511yescharacteristic
489C22×C3C2×#12511yescharacteristic
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3#39812yes—
497C2×D3#39812yes—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4C2×C22⋊Q811yescharacteristic
503C3×C4C2×C22⋊Q811yescharacteristic
504C3×C4C2×#12311yescharacteristic
505C3×C4C2×C4⋊Q811yescharacteristic
506Dic3—41no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510Dic3#39812yes—
511Dic3#39812yes—
512C23—31no—
513C23—31no—
514C23—32no—
515C23—32no—
516C23—121no—
517C23—121no—
518C23D3×#4111yescharacteristic
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—32no—
522C2×C4—32no—
523C2×C4—31no—
524C2×C4—31no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—241no—
542C2×C4—241no—
543C2×C4—242no—
544C2×C4—242no—
545C2×C4—241no—
546C2×C4—241no—
547C2×C4C2×D3×D411yescharacteristic
548C2×C4C2×D3×D411yescharacteristic
549C2×C4C2×D3×Q811yescharacteristic
550C2×C4C2×D3×Q811yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C8—22no—
554C8—22no—
555C8—22no—
556C8—41no—
557C8—62no—
558C8—62no—
559C8—62no—
560C8—121no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3C2×#39811yescharacteristic
564C2×C3#306811yescharacteristic
565C2×C3#306911yescharacteristic
566D3—22no—
567D3—22no—
568C22—21no—
569C22—21no—
570C22—32no—
571C22—32no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—61no—
576C22—122no—
577C22—122no—
578C22D3×#12111yescharacteristic
579C22D3×#12511yescharacteristic
580C22D3×#12511yescharacteristic
581C4—32no—
582C4—32no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—121no—
588C4—121no—
589C4—241no—
590C4—241no—
591C4D3×C22⋊Q811yescharacteristic
592C4D3×C22⋊Q811yescharacteristic
593C4D3×#12311yescharacteristic
594C4D3×C4⋊Q811yescharacteristic
595C3#2367511yescharacteristic, complemented, Sylow
596C2—21no—
597C2—21no—
598C2—62no—
599C2—62no—
600C2D3×#39811yescharacteristic
601C2#8093811yescharacteristic
602C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2089C2
socleC22×C3C2×#121
prime radicalC23×D3#41
hypercenter#2089D3
hypocenterC3#23675
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2089

Complements

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

NH ≅ G/Ntype
5. C3×#2089C2semidirect
6. #70671C2semidirect
7. #70670C2semidirect
8. #70626C2semidirect
9. #70533C2semidirect
12. #68981C2semidirect
13. #68980C2semidirect
15. #68924C2semidirect
75. #2089D3semidirect
595. C3#23675semidirect

Maximal subgroups

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

subgroupindex
#792082
D3×#4282
C2×#48812
#790802
C3×#20892
#706712
#706702
#706262
#705332
#689892
D3×#3432
#689812
#689802
#689382
#689242
#236753