Finite Group Databasebeta

← #10859033

Subgroups — #10859033

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76615C211yesmaximal, characteristic
2#76223C211yesmaximal, characteristic
3C3×#2068C211yesmaximal, characteristic, complemented
4#70632C211yesmaximal, characteristic
5#70621C211yesmaximal, characteristic, complemented
6#70478C211yesmaximal, characteristic
7#70460C211yesmaximal, characteristic, complemented
8#70271C211yesmaximal, characteristic
9#70269C211yesmaximal, characteristic
10#70235C211yesmaximal, characteristic, complemented
11#70222C211yesmaximal, characteristic, complemented
12#69668C211yesmaximal, characteristic, complemented
13#69603C211yesmaximal, characteristic, complemented
14#68867C211yesmaximal, characteristic
15#68812C211yesmaximal, characteristic, complemented
16#26428—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4994—21no—
34#4987C2211yescharacteristic
35#4980—21no—
36#4977C2211yescharacteristic
37#4972C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C4×(C2×D3)⋊C4C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4673—21no—
43#4668—21no—
44#4666—21no—
45#4652—21no—
46#4647—21no—
47#4646—21no—
48C3×#340C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2439—31no—
61#2360—31no—
62#2101—31no—
63#2086—31no—
64#2068D311yescharacteristic, complemented
65#2024—31no—
66#1988—31no—
67#1921—31no—
68#1894—31no—
69#1873—31no—
70#1863—31no—
71#1852—31no—
72#1851—31no—
73#1805—31no—
74#1736—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
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#946Q814yescomplemented
88Q8×Dic3—22no—
89Q8×Dic3—41no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—41no—
113C2×ΓL(1,𝔽25)—41no—
114#892—22no—
115#892—41no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—41no—
118#869—22no—
119#869—41no—
120C3×#107—41no—
121C3×C4⋊C8—41no—
122C3×#103—41no—
123C3×#103—41no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127#835—22no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137(C2×D3)⋊C8—22no—
138Dic3⋊C8—41no—
139#807—22no—
140C22×M16—31no—
141C2×C4⋊Q8—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C22×C4⋊C4—31no—
146C2×#107—31no—
147C2×#107C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×C22⋊C8—31no—
155#396—61no—
156#393—61no—
157#391—61no—
158#391C2×D311yescharacteristic
159#389—31no—
160#389—31no—
161#389—61no—
162#389C2×D311yescharacteristic
163#387—61no—
164#387—61no—
165#385—61no—
166#382—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—31no—
170C4×C22⋊C4—61no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—81no—
211C2×C4×D3C2×Q812yes—
212C2×C4×D3#4112yes—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C2×Dic6—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C42—21no—
230C3×C42—21no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—41no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—42no—
242C4⋊Dic3—42no—
243C4⋊Dic3C2×Q812yes—
244C4⋊Dic3#4112yes—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261ΓL(1,𝔽25)—44no—
262C22×Q8—61no—
263C22×Q8C22×D311yescharacteristic
264C23×C4—31no—
265C23×C4—61no—
266C2×M16—62no—
267C2×M16—121no—
268C2×M16—121no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4⋊Q8—62no—
274C4⋊Q8—121no—
275C22⋊Q8—34no—
276C22⋊Q8—62no—
277C4×Q8—62no—
278C4×Q8—121no—
279#117—62no—
280#117—121no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—121no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—31no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—41no—
308#107—62no—
309C4⋊C8—41no—
310C4⋊C8—121no—
311#103—41no—
312#103—41no—
313#103—62no—
314#103—121no—
315C22⋊C8—62no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3C22⋊Q812yes—
335C3×Q8—22no—
336C3×Q8—22no—
337C3×Q8—41no—
338C3×Q8—41no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4#13712yes—
357C2×C3×C4C22×Q811yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3C22⋊Q812yes—
384C4×D3—22no—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—84no—
389Dic6—84no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—31no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—122no—
401C2×Q8—122no—
402C2×Q8—241no—
403C2×Q8D3×Q812yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—241no—
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4C23×D311yescharacteristic
427M16—124no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—241no—
460C4⋊C4—241no—
461C22⋊C4—64no—
462C22⋊C4—62no—
463C22⋊C4—121no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C42—21no—
467C42—21no—
468C42—61no—
469C42—61no—
470C42—122no—
471C42—122no—
472C42—241no—
473C22×C3C2×C22⋊Q811yescharacteristic
474C22×C3#54611yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C22×C3#57511yescharacteristic
478C22×C3#57611yescharacteristic
479C22×C3#57811yescharacteristic
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—22no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#52911yescharacteristic
494C3×C4#53811yescharacteristic
495C3×C4#55411yescharacteristic
496C3×C4#55411yescharacteristic
497Dic3—22no—
498Dic3—41no—
499Dic3—42no—
500Dic3—42no—
501Dic3—42no—
502Dic3—41no—
503Dic3—81no—
504C23—32no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23—61no—
511C23C2×D3×D411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—244no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—244no—
560C2×C4#91812yes—
561C2×C4#100411yescharacteristic
562C2×C4#100411yescharacteristic
563C2×C4C2×D3×Q811yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C8—41no—
568C8—41no—
569C8—121no—
570C8—121no—
571C2×C3#245011yescharacteristic
572C2×C3#307811yescharacteristic
573C2×C3#308311yescharacteristic
574C2×C3#331411yescharacteristic
575C2×C3#331911yescharacteristic
576C2×C3#332711yescharacteristic
577C2×C3#332811yescharacteristic
578D3—42no—
579C22—62no—
580C22—62no—
581C22—62no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22#561011yescharacteristic
587C22#561211yescharacteristic
588C22D3×C22⋊Q811yescharacteristic
589C22C2×#91811yescharacteristic
590C22#576911yescharacteristic
591C22#577311yescharacteristic
592C22#577911yescharacteristic
593C4—22no—
594C4—42no—
595C4—41no—
596C4—62no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—122no—
601C4—121no—
602C4—122no—
603C4—121no—
604C4—122no—
605C4—241no—
606C4#561311yescharacteristic
607C4#563611yescharacteristic
608C4#563611yescharacteristic
609C4#570611yescharacteristic
610C3#2642811yescharacteristic, complemented, Sylow
611C2—122no—
612C2#7742011yescharacteristic
613C2#8049511yescharacteristic
614C2#8056311yescharacteristic
615C2#8077711yescharacteristic
616C2#8084511yescharacteristic
617C2#8110811yescharacteristic
618C2#8123411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2068D3
hypocenterC3#26428
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
3. C3×#2068C2semidirect
5. #70621C2semidirect
7. #70460C2semidirect
10. #70235C2semidirect
11. #70222C2semidirect
12. #69668C2semidirect
13. #69603C2semidirect
15. #68812C2semidirect
64. #2068D3semidirect
87. #946Q8semidirect
610. C3#26428semidirect

Maximal subgroups

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

subgroupindex
#766152
#762232
C3×#20682
#706322
#706212
#704782
#704602
#702712
#702692
#702352
#702222
#696682
#696032
#688672
#688122
#264283