Finite Group Databasebeta

← #10858856

Subgroups — #10858856

Order 768.

774 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#73495C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3C3×#1850C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic
6#70488C211yesmaximal, characteristic, complemented
7#70486C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70461C211yesmaximal, characteristic, complemented
10#69930C211yesmaximal, characteristic
11#69928C211yesmaximal, characteristic
12#68690C211yesmaximal, characteristic, complemented
13#68684C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#26416—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4946—21no—
35#4946—21no—
36#4942—21no—
37Dic3×C22⋊C4—21no—
38#4936C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4660—21no—
45#4657—21no—
46#4655—21no—
47#4647—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2442—31no—
61#2360—31no—
62#2101—31no—
63#2088—31no—
64#2066—31no—
65#2026—31no—
66#1989—31no—
67#1920—31no—
68#1892—31no—
69#1872—31no—
70#1865—31no—
71#1853—31no—
72#1850D311yescharacteristic
73#1805—31no—
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—41no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—41no—
97C2×C4⋊Dic3—41no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924Q814yes—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116#870—22no—
117#870—41no—
118C4×Dic6—22no—
119C4×Dic6—41no—
120C3×C4⋊C8—41no—
121C3×C4⋊C8—41no—
122C3×C22⋊C8—22no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#817—22no—
135#817—41no—
136#816—22no—
137#816—41no—
138#816—41no—
139#816—41no—
140C23×C8C2×D311yescharacteristic
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#400—61no—
156#398—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389—61no—
162#387—61no—
163#385—61no—
164#385—61no—
165#382—31no—
166#380—31no—
167#380—61no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—61no—
170C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×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×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×Q812yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3—81no—
209C22×Dic3—81no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C2×Q812yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—44no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235#217—24no—
236#217—42no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3C2×Q812yes—
251C4⋊Dic3#4112yes—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—42no—
261C4×Dic3—81no—
262C22×Q8—31no—
263C23×C4—61no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266C22×C8—22no—
267C22×C8—41no—
268C22×C8—41no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273#125—62no—
274#125—121no—
275C22⋊Q8—34no—
276C22⋊Q8—62no—
277C4×Q8—62no—
278C4×Q8—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—121no—
298C2×C4⋊C4—121no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4D1211yescharacteristic
302C2×C22⋊C4D1211yescharacteristic
303C2×C42—61no—
304C2×C42—61no—
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307#106—62no—
308#106—121no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311#103—62no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—22no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95D1211yescharacteristic
326#95D1211yescharacteristic
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3#13512yes—
333C23×C3#13712yes—
334C23×C3C22×D411yescharacteristic
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4C22⋊Q812yes—
354C2×C3×C4C22×Q811yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
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—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×Dic3—41no—
384C2×Dic3—84no—
385C2×Dic3—84no—
386Dic6—22no—
387Dic6—22no—
388Dic6—41no—
389Dic6—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C24C2×D1211yescharacteristic
395C2×Q8—32no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4#23712yes—
425C22×C4D3×Q812yes—
426C22×C4C23×D311yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—44no—
440C4⋊C4—32no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—64no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—241no—
461C4⋊C4—241no—
462C4⋊C4—241no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—64no—
466C22⋊C4—122no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—122no—
472C42—241no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3#47914yes—
480C22×C3C2×C22⋊Q811yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×#13711yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C3×C4—22no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4#52911yescharacteristic
493C3×C4#53811yescharacteristic
494C3×C4#55411yescharacteristic
495C3×C4#55411yescharacteristic
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—42no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
503Dic3—81no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23#89712yes—
509C23#90412yes—
510C23#90912yes—
511C23C22×D1211yescharacteristic
512Q8—62no—
513Q8—62no—
514Q8—121no—
515Q8—121no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
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—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—244no—
559C2×C4—244no—
560C2×C4#89712yes—
561C2×C4C2×#23711yescharacteristic
562C2×C4#100411yescharacteristic
563C2×C4#100411yescharacteristic
564C2×C4C2×D3×Q811yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C2×C3—24no—
572C2×C3#245011yescharacteristic
573C2×C3C2×#47911yescharacteristic
574C2×C3#308411yescharacteristic
575C2×C3#329611yescharacteristic
576C2×C3#329611yescharacteristic
577C2×C3#331611yescharacteristic
578C2×C3#332011yescharacteristic
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22#482914yes—
586C22C2×#89711yescharacteristic
587C22#555411yescharacteristic
588C22#555411yescharacteristic
589C22C2×#90411yescharacteristic
590C22C2×#90911yescharacteristic
591C22#575111yescharacteristic
592C22#575111yescharacteristic
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—122no—
599C4—122no—
600C4—122no—
601C4—122no—
602C4—241no—
603C4—241no—
604C4—241no—
605C4—241no—
606C4#559311yescharacteristic
607C4#563611yescharacteristic
608C4#563611yescharacteristic
609C4#564911yescharacteristic
610C3#2641611yescharacteristic, complemented, Sylow
611C2—24no—
612C2#7442411yescharacteristic
613C2C2×#482911yescharacteristic
614C2#7909211yescharacteristic
615C2#7944011yescharacteristic
616C2#7944011yescharacteristic
617C2#7951311yescharacteristic
618C2#7951511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1850C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1850D3
hypocenterC3#26416
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
6. #70488C2semidirect
7. #70486C2semidirect
8. #70464C2semidirect
9. #70461C2semidirect
12. #68690C2semidirect
13. #68684C2semidirect
14. #68644C2semidirect
15. #68623C2semidirect
610. C3#26416semidirect

Maximal subgroups

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

subgroupindex
#734952
#734652
C3×#18502
#719722
C2×#45512
#704882
#704862
#704642
#704612
#699302
#699282
#686902
#686842
#686442
#686232
#264163