Finite Group Databasebeta

← #10853142

Subgroups — #10853142

Order 768.

740 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#74326C211yesmaximal, characteristic
2#73633C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70632C211yesmaximal, characteristic
5#70631C211yesmaximal, characteristic
6#70622C211yesmaximal, characteristic, complemented
7#70617C211yesmaximal, characteristic, complemented
8#70478C211yesmaximal, characteristic
9#70464C211yesmaximal, characteristic, complemented
10#69571C211yesmaximal, characteristic, complemented
11#69458C211yesmaximal, characteristic, complemented
12#68867C211yesmaximal, characteristic
13#68818C211yesmaximal, characteristic, complemented
14#68813C211yesmaximal, characteristic
15#68799C211yesmaximal, characteristic, complemented
16#26011—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#896—21no—
19C2×#892C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30#4989—21no—
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33#4983—21no—
34#4979—21no—
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4940—21no—
40C4×(C2×D3)⋊C4C2211yescharacteristic
41#4932—21no—
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4669—21no—
45#4650—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440—31no—
61#2346—31no—
62#2099—31no—
63#2088—31no—
64#2087—31no—
65#2057—31no—
66#1987—31no—
67#1947D311yescharacteristic, complemented
68#1920—31no—
69#1918—31no—
70#1852—31no—
71#1851—31no—
72#1825—31no—
73#1809—31no—
74#1736—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
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
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C2×ΓL(1,𝔽25)—22no—
109C2×ΓL(1,𝔽25)—22no—
110C2×ΓL(1,𝔽25)—41no—
111#896—44no—
112#892—22no—
113#892Q814yescomplemented
114D3×C4⋊C4—22no—
115D3×C4⋊C4—41no—
116#888—22no—
117#888—41no—
118C3×#107—41no—
119C3×#107—41no—
120C3×C4⋊C8—22no—
121C3×#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131(C2×D3)⋊C8—22no—
132#817—22no—
133Dic3⋊C8—41no—
134#807—22no—
135C22×M16—31no—
136C2×C4⋊Q8—31no—
137C2×C22⋊Q8—61no—
138C2×#117—31no—
139C22×C4⋊C4—31no—
140C2×#107—31no—
141C2×#107C2×D311yescharacteristic
142C2×#107C2×D311yescharacteristic
143C2×#106—31no—
144C2×C4⋊C8—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×C22⋊C8—31no—
147#398—61no—
148#395—61no—
149#389—31no—
150#389—31no—
151#389—31no—
152#389—31no—
153#388—61no—
154#388—61no—
155#385—61no—
156#384C2×D311yescharacteristic
157#382—61no—
158#380—31no—
159#380—31no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4C2×D311yescharacteristic
162C4×C22⋊C4—31no—
163#376—61no—
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178C23×D3C2×D411yescharacteristic
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—42no—
198C2×C4×D3—42no—
199C2×C4×D3—81no—
200C2×C4×D3C2×Q812yes—
201C2×C4×D3#4112yes—
202C2×Dic6—42no—
203C2×Dic6—42no—
204C2×Dic6—81no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8#4112yes—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4—81no—
220C3×C4⋊C4C2×Q812yes—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3C2×Q812yes—
236C4⋊Dic3#4112yes—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—81no—
242Dic3⋊C4—81no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—24no—
246C4×Dic3—42no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253ΓL(1,𝔽25)—44no—
254C22×Q8—61no—
255C23×C4—31no—
256C23×C4—61no—
257C2×M16—62no—
258C2×M16—62no—
259C2×M16—121no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4⋊Q8—62no—
265C4⋊Q8—121no—
266C22⋊Q8—124no—
267#117—34no—
268#117—62no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—121no—
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C42—21no—
293C2×C42—31no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#107—41no—
298#107—41no—
299#107—62no—
300#106—62no—
301C4⋊C8—22no—
302C4⋊C8—121no—
303C22⋊C8—62no—
304#95—21no—
305#95—31no—
306#95—31no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3C22⋊Q812yes—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4AGL(1,ℤ/8)12yes—
340C2×C3×C4C22×Q811yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C42−1+411yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—22no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3C22⋊Q812yes—
366C2×Dic3#13712yes—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371C4×D3—84no—
372Dic6—84no—
373C3×C8—22no—
374C3×C8—41no—
375C3⋊C8—41no—
376C3⋊C8—41no—
377C24—31no—
378C2×Q8—122no—
379C2×Q8—122no—
380C2×Q8—241no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—31no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—122no—
399C22×C4—122no—
400C22×C4—241no—
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4C23×D311yescharacteristic
404M16—124no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8#23512yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—32no—
425C4⋊C4—32no—
426C4⋊C4—42no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—81no—
434C4⋊C4—122no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—241no—
439C4⋊C4—241no—
440C4⋊C4D3×Q812yes—
441C22⋊C4—62no—
442C22⋊C4—64no—
443C22⋊C4—62no—
444C22⋊C4—122no—
445C22⋊C4—241no—
446C42—21no—
447C42—21no—
448C42—42no—
449C42—64no—
450C42—61no—
451C42—61no—
452C42—122no—
453C22×C3C2×C22⋊Q811yescharacteristic
454C22×C3#54611yescharacteristic
455C22×C3#54711yescharacteristic
456C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
457C22×C3C2×#13711yescharacteristic
458C22×C3#57611yescharacteristic
459C22×C3#57811yescharacteristic
460C2×D3—22no—
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4#53311yescharacteristic
474C3×C4#54411yescharacteristic
475C3×C4#55111yescharacteristic
476C3×C4#55211yescharacteristic
477Dic3—22no—
478Dic3—22no—
479Dic3—42no—
480Dic3—41no—
481Dic3—42no—
482Dic3—81no—
483Dic3—81no—
484C23—32no—
485C23—61no—
486C23—61no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23C2×D3×D411yescharacteristic
492Q8—244no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—41no—
502C2×C4—41no—
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—41no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—62no—
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—244no—
536C2×C4#91112yes—
537C2×C4C2×#23511yescharacteristic
538C2×C4#100411yescharacteristic
539C2×C4C2×D3×Q811yescharacteristic
540C2×C4#100711yescharacteristic
541C2×C4D3×#4111yescharacteristic
542C2×C4D3×#4111yescharacteristic
543C8—22no—
544C8—41no—
545C8—121no—
546C8—121no—
547C2×C3#243811yescharacteristic
548C2×C3#307911yescharacteristic
549C2×C3#308311yescharacteristic
550C2×C3#331211yescharacteristic
551C2×C3#332511yescharacteristic
552C2×C3#334311yescharacteristic
553C2×C3#335711yescharacteristic
554D3—42no—
555C22—62no—
556C22—62no—
557C22—62no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22#550711yescharacteristic
563C22#561211yescharacteristic
564C22D3×C22⋊Q811yescharacteristic
565C22C2×#91111yescharacteristic
566C22#576611yescharacteristic
567C22#577411yescharacteristic
568C22D3×#13711yescharacteristic
569C4—22no—
570C4—42no—
571C4—41no—
572C4—62no—
573C4—62no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—122no—
578C4—121no—
579C4—122no—
580C4—241no—
581C4—241no—
582C4#555611yescharacteristic
583C4#557811yescharacteristic
584C4#569311yescharacteristic
585C4#572911yescharacteristic
586C3#2601111yescharacteristic, complemented, Sylow
587C2—122no—
588C2#7622211yescharacteristic
589C2#7915811yescharacteristic
590C2#7923811yescharacteristic
591C2#8035611yescharacteristic
592C2#8049311yescharacteristic
593C2#8096611yescharacteristic
594C2#8111111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1947D3
hypocenterC3#26011
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. C3×#1947C2semidirect
6. #70622C2semidirect
7. #70617C2semidirect
9. #70464C2semidirect
10. #69571C2semidirect
11. #69458C2semidirect
13. #68818C2semidirect
15. #68799C2semidirect
67. #1947D3semidirect
113. #892Q8semidirect
586. C3#26011semidirect

Maximal subgroups

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

subgroupindex
#743262
#736332
C3×#19472
#706322
#706312
#706222
#706172
#704782
#704642
#695712
#694582
#688672
#688182
#688132
#687992
#260113