Finite Group Databasebeta

← #10877436

Subgroups — #10877436

Order 768.

710 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#82774C211yesmaximal, characteristic
2#82248C211yesmaximal, characteristic
3C3×#1873C211yesmaximal, characteristic
4#71696C211yesmaximal, characteristic
5#71580C211yesmaximal, characteristic
6#70254C211yesmaximal, characteristic, complemented
7#70253C211yesmaximal, characteristic, complemented
8#70234C211yesmaximal, characteristic, complemented
9#70231C211yesmaximal, characteristic, complemented
10#69947C211yesmaximal, characteristic
11#69926C211yesmaximal, characteristic
12#68848C211yesmaximal, characteristic, complemented
13#68845C211yesmaximal, characteristic, complemented
14#68810C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27698—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25#5229C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5212C2211yescharacteristic
30C4×#217C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4984C2211yescharacteristic
34#4978—21no—
35#4978—21no—
36#4977C2211yescharacteristic
37#4977C2211yescharacteristic
38#4975C2211yescharacteristic
39#4972C2211yescharacteristic
40#4952—21no—
41#4952—21no—
42#4948—21no—
43#4947—21no—
44#4933—21no—
45#4929—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2499—31no—
61#2439—31no—
62#2101—31no—
63#2099—31no—
64#2074—31no—
65#2024—31no—
66#1990—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1873D311yescharacteristic
71#1835—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×Dic6—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×#117—22no—
83C3×#117—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217D411yescharacteristic
94C2×#217D411yescharacteristic
95#926—22no—
96#926—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—41no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#888—22no—
115#888—41no—
116C3×#103—22no—
117C3×#103—41no—
118C3×C22⋊C8—22no—
119#835—22no—
120#835—41no—
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⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C8×Dic3—22no—
135C8×Dic3—41no—
136C23×C8—31no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8C2×D311yescharacteristic
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141C2×#103—31no—
142C2×#103—31no—
143C2×#103C2×D311yescharacteristic
144C2×#103C2×D311yescharacteristic
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#397—61no—
149#397—61no—
150#396—61no—
151#393—61no—
152#391—31no—
153#391—31no—
154#391—61no—
155#389—31no—
156#389—31no—
157#387—61no—
158#387—61no—
159#387—61no—
160#384—31no—
161#382—31no—
162#382—31no—
163#380—31no—
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—
179C24×C3C2×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×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4112yes—
196C22×C3×C4#4112yes—
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
206C2×Dic6—42no—
207C2×Dic6—42no—
208C2×Dic6—81no—
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—42no—
221C3×C4⋊C4#4112yes—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C42—42no—
225#217—22no—
226#217—22no—
227#217—42no—
228#217—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—81no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
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—44no—
253C2×C3⋊C8#4112yes—
254C22×Q8—61no—
255C22×Q8C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—62no—
260C22×C8—62no—
261C22×C8—121no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4⋊Q8—62no—
265C4⋊Q8—121no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268#117—22no—
269#117—41no—
270#117—62no—
271#117—62no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—21no—
292C2×C42—31no—
293C2×C42—31no—
294C2×C42—61no—
295#103—22no—
296#103—41no—
297#103—62no—
298#103—121no—
299C22⋊C8—22no—
300C4×C8—62no—
301C4×C8—121no—
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—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3#12412yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
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×C4#12412yes—
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
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—
373C2×Dic3—41no—
374C2×Dic3#13512yes—
375C2×Dic3#13512yes—
376Dic6—84no—
377C3×C8—41no—
378C3×C8—41no—
379C3⋊C8—22no—
380C3⋊C8—41no—
381C24D3×D411yescharacteristic
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—122no—
389C2×Q8—122no—
390C2×Q8—241no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4#23512yes—
408C22×C4D3×D411yescharacteristic
409C22×C4#23712yes—
410C22×C4C23×D311yescharacteristic
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—32no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—124no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4—241no—
443C4⋊C4#23712yes—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—62no—
447C22⋊C4—62no—
448C22⋊C4—122no—
449C22⋊C4—241no—
450C42—42no—
451C42—61no—
452C42—61no—
453C42—61no—
454C42—61no—
455C42—122no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3C2×#12411yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3#54711yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3C2×#13511yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C3×C4—22no—
471C3×C4—42no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#52911yescharacteristic
476C3×C4#53911yescharacteristic
477C3×C4#55111yescharacteristic
478C3×C4#55211yescharacteristic
479Dic3—22no—
480Dic3—22no—
481Dic3—41no—
482Dic3—41no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#88512yes—
494C23C2×D3×D411yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—244no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—32no—
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—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—121no—
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#88512yes—
543C2×C4C2×#23511yescharacteristic
544C2×C4C2×#23711yescharacteristic
545C2×C4C2×#23711yescharacteristic
546C2×C4#100411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#101011yescharacteristic
549C8—41no—
550C8—41no—
551C8—62no—
552C8—121no—
553C2×C3—42no—
554C2×C3#248811yescharacteristic
555C2×C3#313111yescharacteristic
556C2×C3#313711yescharacteristic
557C2×C3#329611yescharacteristic
558C2×C3#330011yescharacteristic
559C2×C3#334011yescharacteristic
560C2×C3#334211yescharacteristic
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22C2×#88511yescharacteristic
569C22#559111yescharacteristic
570C22#565511yescharacteristic
571C22D3×#13511yescharacteristic
572C22D3×#13511yescharacteristic
573C22#577811yescharacteristic
574C22#577911yescharacteristic
575C4—22no—
576C4—42no—
577C4—62no—
578C4—62no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—121no—
583C4—121no—
584C4—241no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4#554111yescharacteristic
589C4#555611yescharacteristic
590C4#560011yescharacteristic
591C4#560211yescharacteristic
592C3#2769811yescharacteristic, complemented, Sylow
593C2—42no—
594C2#7474011yescharacteristic
595C2#7882511yescharacteristic
596C2#7895211yescharacteristic
597C2#8024611yescharacteristic
598C2#8024711yescharacteristic
599C2#8153211yescharacteristic
600C2#8153311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1873D3
hypocenterC3#27698
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
6. #70254C2semidirect
7. #70253C2semidirect
8. #70234C2semidirect
9. #70231C2semidirect
12. #68848C2semidirect
13. #68845C2semidirect
14. #68810C2semidirect
15. #68803C2semidirect
592. C3#27698semidirect

Maximal subgroups

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

subgroupindex
#827742
#822482
C3×#18732
#716962
#715802
#702542
#702532
#702342
#702312
#699472
#699262
#688482
#688452
#688102
#688032
#276983