Finite Group Databasebeta

← #10877348

Subgroups — #10877348

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74225C211yesmaximal, characteristic
2#73762C211yesmaximal, characteristic
3C3×#1949C211yesmaximal, characteristic, complemented
4#70644C211yesmaximal, characteristic, complemented
5#70632C211yesmaximal, characteristic
6#70591C211yesmaximal, characteristic, complemented
7#70580C211yesmaximal, characteristic
8#70485C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic
10#69559C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#68879C211yesmaximal, characteristic
13#68878C211yesmaximal, characteristic
14#68845C211yesmaximal, characteristic, complemented
15#68842C211yesmaximal, characteristic, complemented
16#27690—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#872C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977C2211yescharacteristic
34#4975C2211yescharacteristic
35#4973C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4940—21no—
41#4935C2211yescharacteristic
42#4928—21no—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4670—21no—
45#4668—21no—
46#4668—21no—
47#4653—21no—
48#4647—21no—
49#4647—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2323—31no—
61#2317—31no—
62#2101—31no—
63#2099—31no—
64#2097—31no—
65#2095—31no—
66#1949D311yescharacteristic, complemented
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1854—31no—
71#1852—31no—
72#1805—31no—
73#1803—31no—
74#1798—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
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⋊C4D411yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#897—22no—
110#897Q814yes—
111#892—41no—
112#892—41no—
113D3×C4⋊C4—41no—
114D3×C4⋊C4—41no—
115#872—22no—
116#872—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#107—41no—
120C3×#106—41no—
121C3×C4⋊C8—22no—
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—
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—22no—
133#816—41no—
134#816—41no—
135#810—41no—
136#810—41no—
137#807—41no—
138#807—41no—
139C4⋊(C3⋊C8)—22no—
140C2×C22⋊Q8—31no—
141C2×C4×Q8—31no—
142C2×#117—31no—
143C2×#117—31no—
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#107C2×D311yescharacteristic
148C2×#106C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#391—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#389—61no—
162#388—61no—
163#388—61no—
164#387—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382C2×D311yescharacteristic
171#380—31no—
172C4×C4⋊C4C2×D311yescharacteristic
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—42no—
203C2×C4×D3—81no—
204C2×C4×D3—81no—
205C2×C4×D3C2×Q812yes—
206C2×C4×D3#4112yes—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4C2×Q812yes—
227C3×C42—81no—
228C3×C42#4112yes—
229C3×C42#4112yes—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—81no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244C4⋊Dic3—81no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—42no—
254C4×Dic3—42no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×Q8—31no—
262C23×C4—31no—
263C23×C4—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C22⋊Q8—34no—
269C22⋊Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272#117—62no—
273#117—62no—
274#117—121no—
275#117—121no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—121no—
290C2×C4⋊C4—121no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—21no—
301C2×C42—31no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—41no—
306#107—121no—
307#107—121no—
308#106—41no—
309C4⋊C8—22no—
310C4⋊C8—62no—
311#103—121no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3C22×D411yescharacteristic
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3C22⋊Q812yes—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
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—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4#12312yes—
351C2×C3×C4#12312yes—
352C2×C3×C4C22×Q811yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3C22⋊Q812yes—
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—22no—
382C4×D3—22no—
383C4×D3—84no—
384Dic6—22no—
385Dic6—22no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—31no—
393C2×Q8—32no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—64no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C4⋊C4D3×Q812yes—
458C22⋊C4—64no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—122no—
462C22⋊C4—241no—
463C42—61no—
464C42—61no—
465C42—81no—
466C42—122no—
467C42—122no—
468C42#23512yes—
469C42#23912yes—
470C22×C3C2×C22⋊Q811yescharacteristic
471C22×C3C2×#12311yescharacteristic
472C22×C3#53511yescharacteristic
473C22×C3#57611yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C22×C3#57811yescharacteristic
477C2×D3—22no—
478C2×D3—22no—
479C2×D3—22no—
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4#55311yescharacteristic
491C3×C4#55311yescharacteristic
492C3×C4#55411yescharacteristic
493C3×C4#55411yescharacteristic
494Dic3—22no—
495Dic3—41no—
496Dic3—41no—
497Dic3—42no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—32no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23C2×D3×D411yescharacteristic
509Q8—62no—
510Q8—62no—
511Q8—121no—
512Q8—121no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—244no—
555C2×C4#89412yes—
556C2×C4#89412yes—
557C2×C4C2×#23511yescharacteristic
558C2×C4#100411yescharacteristic
559C2×C4C2×D3×Q811yescharacteristic
560C2×C4C2×#23911yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4#100911yescharacteristic
563C8—41no—
564C8—41no—
565C8—121no—
566C8—121no—
567C2×C3#248711yescharacteristic
568C2×C3#308211yescharacteristic
569C2×C3#308311yescharacteristic
570C2×C3#309711yescharacteristic
571C2×C3#309811yescharacteristic
572C2×C3#322311yescharacteristic
573C2×C3#322411yescharacteristic
574D3—42no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22—121no—
582C22C2×#89411yescharacteristic
583C22#560711yescharacteristic
584C22D3×C22⋊Q811yescharacteristic
585C22#577011yescharacteristic
586C22#577111yescharacteristic
587C22#577411yescharacteristic
588C22#577911yescharacteristic
589C4—22no—
590C4—22no—
591C4—42no—
592C4—62no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—121no—
597C4—121no—
598C4—122no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#555811yescharacteristic
603C4#557911yescharacteristic
604C4#569511yescharacteristic
605C4#573011yescharacteristic
606C3#2769011yescharacteristic, complemented, Sylow
607C2—122no—
608C2#7623211yescharacteristic
609C2#7917511yescharacteristic
610C2#7922111yescharacteristic
611C2#8054611yescharacteristic
612C2#8060911yescharacteristic
613C2#8117011yescharacteristic
614C2#8123311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1949D3
hypocenterC3#27690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
3. C3×#1949C2semidirect
4. #70644C2semidirect
6. #70591C2semidirect
8. #70485C2semidirect
10. #69559C2semidirect
11. #69466C2semidirect
14. #68845C2semidirect
15. #68842C2semidirect
66. #1949D3semidirect
606. C3#27690semidirect

Maximal subgroups

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

subgroupindex
#742252
#737622
C3×#19492
#706442
#706322
#705912
#705802
#704852
#704762
#695592
#694662
#688792
#688782
#688452
#688422
#276903