Finite Group Databasebeta

← #10877452

Subgroups — #10877452

Order 768.

730 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#76520C211yesmaximal, characteristic
2#76003C211yesmaximal, characteristic
3C3×#1933C211yesmaximal, characteristic, complemented
4#70270C211yesmaximal, characteristic
5#70268C211yesmaximal, characteristic
6#70253C211yesmaximal, characteristic, complemented
7#70232C211yesmaximal, characteristic, complemented
8C8×(C2×D3)⋊C4C211yesmaximal, characteristic
9#69763C211yesmaximal, characteristic, complemented
10#69671C211yesmaximal, characteristic, complemented
11#69613C211yesmaximal, characteristic, complemented
12#68871C211yesmaximal, characteristic
13D3×#340C211yesmaximal, characteristic
14#68845C211yesmaximal, characteristic, complemented
15#68804C211yesmaximal, characteristic, complemented
16#27699—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4987—21no—
33#4987C2211yescharacteristic
34#4984C2211yescharacteristic
35#4981—21no—
36#4977C2211yescharacteristic
37C4×(C2×D3)⋊C4C2211yescharacteristic
38#4936C2211yescharacteristic
39#4936C2211yescharacteristic
40#4668—21no—
41#4664—21no—
42#4664—21no—
43#4659—21no—
44#4658—21no—
45#4658—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2503—31no—
61#2442—31no—
62#2101—31no—
63#2101—31no—
64#2074—31no—
65#2026—31no—
66#1991—31no—
67#1933—31no—
68#1933D311yescharacteristic, complemented
69#1892—31no—
70#1872—31no—
71#1835—31no—
72#1823—31no—
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86#946—22no—
87#946—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C2×D3×C8—22no—
106C2×D3×C8—22no—
107C2×D3×C8—41no—
108D3×C4⋊C4—22no—
109D3×C4⋊C4—22no—
110D3×C4⋊C4—22no—
111D3×C4⋊C4—41no—
112#890—22no—
113#890—41no—
114C3×#103—22no—
115C3×#103—22no—
116C3×C4×C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120#835—41no—
121#835—41no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134(C2×D3)⋊C8—22no—
135C8×Dic3—41no—
136C23×C8—31no—
137C2×#125—31no—
138C2×C22⋊Q8—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103C2×D311yescharacteristic
144C2×#103C2×D311yescharacteristic
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8C2×D311yescharacteristic
148#400—61no—
149#400—61no—
150#400—61no—
151#398—61no—
152#391—31no—
153#391C2×D311yescharacteristic
154#389—31no—
155#389C2×D311yescharacteristic
156#385—61no—
157#382—31no—
158#382—31no—
159#382—31no—
160#381—61no—
161#381—61no—
162#380—31no—
163#380—61no—
164C4×C22⋊C4—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#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—81no—
210C2×C4×D3#4112yes—
211C2×C4×D3#4112yes—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8#4112yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C42—21no—
226C3×C42—21no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—42no—
230(C2×D3)⋊C4—42no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—42no—
240C4⋊Dic3—81no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253D3×C8—44no—
254C22×Q8C22×D311yescharacteristic
255C23×C4—31no—
256C23×C4—61no—
257C23×C4—61no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—62no—
261C22×C8—62no—
262C22×C8—121no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265#125—62no—
266#125—121no—
267C22⋊Q8—62no—
268C22⋊Q8—62no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—121no—
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×C42—31no—
292C2×C42C22×D311yescharacteristic
293#103—22no—
294#103—22no—
295#103—121no—
296#103—121no—
297C22⋊C8—62no—
298C4×C8—22no—
299C4×C8—121no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—31no—
304#95—31no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3C22×D411yescharacteristic
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3#12412yes—
321C22×D3#13512yes—
322C22×D3#13512yes—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—41no—
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—41no—
344C2×C3×C4—41no—
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C2×C3×C42−1+411yescharacteristic
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—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
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#12412yes—
372C4×D3—22no—
373C4×D3—22no—
374C4×D3—22no—
375C4×D3—22no—
376C4×D3—84no—
377C4×D3—84no—
378C3×C8—22no—
379C3×C8—41no—
380C3⋊C8—41no—
381C3⋊C8—41no—
382C24—31no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—31no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—241no—
412C22×C4D3×D411yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4C23×D311yescharacteristic
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—124no—
427C2×C8#23712yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—81no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4#23512yes—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—61no—
456C42—61no—
457C22×C3C2×#12411yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54711yescharacteristic
460C22×C3C2×#13511yescharacteristic
461C22×C3C2×#13511yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57811yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3#48814yes—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4#52911yescharacteristic
478C3×C4#53911yescharacteristic
479C3×C4#55111yescharacteristic
480C3×C4#55211yescharacteristic
481Dic3—22no—
482Dic3—41no—
483Dic3—41no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—32no—
489C23—32no—
490C23—32no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23C2×D3×D411yescharacteristic
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—32no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—244no—
544C2×C4—244no—
545C2×C4C2×#23511yescharacteristic
546C2×C4C2×#23711yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4#100911yescharacteristic
550C2×C4#101011yescharacteristic
551C8—22no—
552C8—41no—
553C8—121no—
554C8—121no—
555C2×C3#248811yescharacteristic
556C2×C3C2×#48811yescharacteristic
557C2×C3#313711yescharacteristic
558C2×C3#329611yescharacteristic
559C2×C3#330011yescharacteristic
560C2×C3#333611yescharacteristic
561C2×C3#333711yescharacteristic
562D3—24no—
563C22—34no—
564C22—62no—
565C22—62no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22#555011yescharacteristic
571C22#561111yescharacteristic
572C22D3×#12411yescharacteristic
573C22#576211yescharacteristic
574C22#576611yescharacteristic
575C22D3×#13511yescharacteristic
576C22D3×#13511yescharacteristic
577C4—41no—
578C4—42no—
579C4—41no—
580C4—62no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—121no—
585C4—121no—
586C4—241no—
587C4—241no—
588C4—241no—
589C4—241no—
590C4#552911yescharacteristic
591C4#555711yescharacteristic
592C4#565811yescharacteristic
593C4#570711yescharacteristic
594C3#2769911yescharacteristic, complemented, Sylow
595C2—64no—
596C2#7586811yescharacteristic
597C2#7967711yescharacteristic
598C2#7973611yescharacteristic
599C2#8067311yescharacteristic
600C2#8077011yescharacteristic
601C2#8159911yescharacteristic
602C2D3×#48811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1933D3
hypocenterC3#27699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
3. C3×#1933C2semidirect
6. #70253C2semidirect
7. #70232C2semidirect
9. #69763C2semidirect
10. #69671C2semidirect
11. #69613C2semidirect
14. #68845C2semidirect
15. #68804C2semidirect
68. #1933D3semidirect
594. C3#27699semidirect

Maximal subgroups

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

subgroupindex
#765202
#760032
C3×#19332
#702702
#702682
#702532
#702322
C8×(C2×D3)⋊C42
#697632
#696712
#696132
#688712
D3×#3402
#688452
#688042
#276993