Finite Group Databasebeta

← #10894955

Subgroups — #10894955

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76297C211yesmaximal, characteristic
2#76104C211yesmaximal, characteristic, complemented
3C3×#1934C211yesmaximal, characteristic
4#70216C211yesmaximal, characteristic
5#70178C211yesmaximal, characteristic
6#70154C211yesmaximal, characteristic
7#70153C211yesmaximal, characteristic
8#69767C211yesmaximal, characteristic, complemented
9#69755C211yesmaximal, characteristic, complemented
10#69645C211yesmaximal, characteristic
11#69616C211yesmaximal, characteristic, complemented
12#68905C211yesmaximal, characteristic, complemented
13#68904C211yesmaximal, characteristic, complemented
14#68842C211yesmaximal, characteristic, complemented
15#68811C211yesmaximal, characteristic, complemented
16#29134—31nomaximal, Sylow
17C2×#936C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C2×#832C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4985—21no—
32#4984C2211yescharacteristic
33#4981—21no—
34#4977C2211yescharacteristic
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40#4946—21no—
41#4946—21no—
42#4945—21no—
43#4945—21no—
44#4932C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4658—21no—
49#4657—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2647—31no—
61#2326—31no—
62#2100—31no—
63#2099—31no—
64#2042—31no—
65#2042—31no—
66#1936—31no—
67#1935—31no—
68#1934D311yescharacteristic
69#1933—31no—
70#1805—31no—
71#1804—31no—
72#1803—31no—
73#1802—31no—
74#1768—31no—
75C22×C3×D4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
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—
90#936—22no—
91#936—22no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94C2×C4⋊Dic3—21no—
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×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#887—22no—
111#887—22no—
112C3×#102—22no—
113C3×#102—41no—
114C3×C4×C8—41no—
115C3×#95—21no—
116#832—22no—
117#832—41no—
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#816—22no—
130#816—41no—
131Dic3⋊C8—22no—
132Dic3⋊C8—22no—
133#810—22no—
134#810—41no—
135#803—41no—
136C2×#124—31no—
137C2×C4×Q8—31no—
138C2×C4×D4—31no—
139C2×C4⋊C8—31no—
140C2×C4⋊C8—31no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102C2×D311yescharacteristic
148C2×#102C2×D311yescharacteristic
149C2×#97—31no—
150C2×C4×C8C2×D311yescharacteristic
151#400—61no—
152#400—61no—
153#398—61no—
154#398—61no—
155#391—31no—
156#390C2×D311yescharacteristic
157#389—31no—
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#388—61no—
163#385—61no—
164#385—61no—
165#384—31no—
166#384—31no—
167#382—31no—
168#376—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—
179C24×C3—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—42no—
197C22×Dic3—81no—
198C22×Dic3—81no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6#4112yes—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—81no—
222C3×C22⋊C4—81no—
223C3×C22⋊C4—81no—
224C3×C42—21no—
225C3×C42—21no—
226#217—24no—
227#217—24no—
228#217—42no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—81no—
236Dic3⋊C4—42no—
237Dic3⋊C4—81no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240Dic3⋊C4—81no—
241Dic3⋊C4—81no—
242Dic3⋊C4—81no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—42no—
246C4×Dic3#4112yes—
247C4×Dic3#4112yes—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C22×Q8—31no—
255C22×D4C22×D311yescharacteristic
256C23×C4—61no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#124—62no—
262#124—62no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×D4—62no—
266C4×D4—62no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—31no—
288C2×C42—31no—
289C2×C42—61no—
290C2×C42C22×D311yescharacteristic
291C4⋊C8—62no—
292C4⋊C8—62no—
293#103—62no—
294#103—62no—
295#103—121no—
296#103—121no—
297#102—22no—
298#102—41no—
299#102—62no—
300#102—121no—
301#97—121no—
302C4×C8—41no—
303#95—21no—
304#95—31no—
305#95—31no—
306#95—31no—
307#95—31no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3C22×D411yescharacteristic
323C3×D4—22no—
324C3×D4—22no—
325C3×D4—41no—
326C3×D4—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
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×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—22no—
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—84no—
371C2×Dic3#12412yes—
372C2×Dic3#12412yes—
373Dic6—22no—
374Dic6—22no—
375Dic6—41no—
376Dic6—41no—
377C3×C8—41no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—21no—
382C2×Q8—32no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4#23512yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—122no—
406C22×C4—241no—
407C22×C4—241no—
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4C23×D311yescharacteristic
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—81no—
436C4⋊C4—122no—
437C4⋊C4—241no—
438C4⋊C4—241no—
439C4⋊C4—241no—
440C4⋊C4—241no—
441C4⋊C4—241no—
442C4⋊C4—241no—
443C4⋊C4—241no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—122no—
449C42—21no—
450C42—21no—
451C42—32no—
452C42—32no—
453C42—61no—
454C42—61no—
455C42—122no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3#52511yescharacteristic
464C22×C3C2×#12411yescharacteristic
465C22×C3#53711yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#55211yescharacteristic
476C3×C4#55211yescharacteristic
477C3×C4#55311yescharacteristic
478C3×C4#55311yescharacteristic
479Dic3—22no—
480Dic3—22no—
481Dic3—42no—
482Dic3—41no—
483Dic3—41no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D3×D411yescharacteristic
495Q8—62no—
496Q8—62no—
497Q8—121no—
498Q8—121no—
499D4—22no—
500D4—22no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—32no—
510C2×C4—32no—
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—41no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—62no—
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—244no—
543C2×C4C2×#23511yescharacteristic
544C2×C4#100411yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#101011yescharacteristic
549C8—41no—
550C8—41no—
551C8—121no—
552C8—121no—
553C2×C3—82no—
554C2×C3#256211yescharacteristic
555C2×C3#311911yescharacteristic
556C2×C3#312011yescharacteristic
557C2×C3#313611yescharacteristic
558C2×C3#313711yescharacteristic
559C2×C3#326211yescharacteristic
560C2×C3#326311yescharacteristic
561C22—42no—
562C22—42no—
563C22—42no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22#554711yescharacteristic
569C22#559711yescharacteristic
570C22D3×#12411yescharacteristic
571C22#575711yescharacteristic
572C22#576211yescharacteristic
573C22#577011yescharacteristic
574C22#577111yescharacteristic
575C4—41no—
576C4—41no—
577C4—62no—
578C4—62no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—121no—
583C4—122no—
584C4—121no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4#553111yescharacteristic
589C4#554011yescharacteristic
590C4#566111yescharacteristic
591C4#570811yescharacteristic
592C3#2913411yescharacteristic, complemented, Sylow
593C2—82no—
594C2#7560411yescharacteristic
595C2#7959511yescharacteristic
596C2#7965711yescharacteristic
597C2#8064711yescharacteristic
598C2#8070711yescharacteristic
599C2#8156311yescharacteristic
600C2#8163911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1934C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1934D3
hypocenterC3#29134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #76104C2semidirect
8. #69767C2semidirect
9. #69755C2semidirect
11. #69616C2semidirect
12. #68905C2semidirect
13. #68904C2semidirect
14. #68842C2semidirect
15. #68811C2semidirect
592. C3#29134semidirect

Maximal subgroups

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

subgroupindex
#762972
#761042
C3×#19342
#702162
#701782
#701542
#701532
#697672
#697552
#696452
#696162
#689052
#689042
#688422
#688112
#291343