Finite Group Databasebeta

← #10852290

Subgroups — #10852290

Order 768.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4829C211yesmaximal, characteristic, complemented
2C2×#4772C211yesmaximal, characteristic
3#76301C211yesmaximal, characteristic
4#76224C211yesmaximal, characteristic, complemented
5C3×#2065C211yesmaximal, characteristic, complemented
6#70588C211yesmaximal, characteristic, complemented
7#70577C211yesmaximal, characteristic, complemented
8#70477C211yesmaximal, characteristic, complemented
9#70199C211yesmaximal, characteristic
10#70184C211yesmaximal, characteristic, complemented
11#70180C211yesmaximal, characteristic, complemented
12#69639C211yesmaximal, characteristic, complemented
13#69632C211yesmaximal, characteristic, complemented
14#68904C211yesmaximal, characteristic, complemented
15#68877C211yesmaximal, characteristic, complemented
16#25953—31nomaximal, Sylow
17C22×#237C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4993—21no—
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4976—21no—
42#4975C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4957—21no—
48#4957—21no—
49#4942—21no—
50Dic3×C22⋊C4—21no—
51C4×(C2×D3)⋊C4C2211yescharacteristic
52#4829—24no—
53#4772—24no—
54#4676—21no—
55#4652—21no—
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#416—31no—
64#2441—31no—
65#2341—31no—
66#2094—31no—
67#2067—31no—
68#2065D311yescharacteristic, complemented
69#2025—31no—
70#1865—31no—
71#1865—31no—
72#1854—31no—
73#1850—31no—
74#1807—31no—
75#1803—31no—
76#1788—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C2×#237—22no—
81C2×#237—22no—
82C2×#237—22no—
83C2×#237—41no—
84C2×#237—41no—
85C2×#237Q814yes—
86C22×C4×D3C2311yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
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—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D3×C8—22no—
121C2×D3×C8—22no—
122C2×D3×C8—41no—
123#892—22no—
124#892—41no—
125#887—22no—
126#887—41no—
127#870—22no—
128#870—41no—
129C4×Dic6—22no—
130C4×Dic6—41no—
131C3×#106—41no—
132C3×C4⋊C8—22no—
133C3×#102—22no—
134C3×#102—41no—
135C3×#95—21no—
136#832—22no—
137#832—41no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143(C2×D3)⋊C8—22no—
144#818—22no—
145#817—22no—
146#816—22no—
147#816D414yes—
148Dic3⋊C8—41no—
149#810—22no—
150#810—22no—
151#808—41no—
152C22×#41—31no—
153C23×C8—31no—
154C2×#125—31no—
155C2×C4×Q8—31no—
156C2×C4×Q8—31no—
157C2×#117—31no—
158#479—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×C4⋊C8C2×D311yescharacteristic
165#416—64no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102C2×D311yescharacteristic
173C2×#102C2×D311yescharacteristic
174C2×C22⋊C8—31no—
175#399—61no—
176#396—61no—
177#390—31no—
178#390—61no—
179#390—61no—
180#390C2×D311yescharacteristic
181#389—31no—
182#389—61no—
183#389C2×D311yescharacteristic
184#387—61no—
185#384—31no—
186#380—31no—
187#380—31no—
188#380—61no—
189C4×C22⋊C4—31no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336—31no—
197C24×C3—21no—
198C23×D3C2×D411yescharacteristic
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4C2×Q812yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C2×#67—24no—
212C2×#67—42no—
213C2×#67—42no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—24no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224#237—28no—
225#237—44no—
226#237—44no—
227#237—44no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3C2×Q812yes—
233C2×C4×D3#4112yes—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×Q812yes—
239C2×Dic6#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×D412yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—81no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—81no—
255C3×C42—21no—
256C3×C42—21no—
257#217—42no—
258#217—81no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—41no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D412yes—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—81no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—42no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288D3×C8—44no—
289C2×#41—34no—
290C2×#41—62no—
291C2×#41—62no—
292C2×#41—62no—
293C2×#41—121no—
294C2×#41—121no—
295C22×Q8—31no—
296C22×D4—61no—
297C22×D4C22×D311yescharacteristic
298C23×C4—31no—
299C23×C4—61no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—62no—
303C22×C8—62no—
304C22×C8—121no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#125—62no—
308#125—121no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—121no—
312C4×Q8—121no—
313#117—62no—
314#117—121no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—31no—
339C2×C42—61no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343#107—62no—
344#106—41no—
345#106—62no—
346#106—121no—
347C4⋊C8—22no—
348C4⋊C8—121no—
349#103—34no—
350#103—62no—
351#103—62no—
352#103—62no—
353#102—22no—
354#102—41no—
355#102—62no—
356#102—121no—
357C22⋊C8—62no—
358#95—21no—
359#95—31no—
360#95—31no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—22no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3C22×D411yescharacteristic
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3C22⋊Q812yes—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×Q811yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404#67—48no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—48no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3C22⋊Q812yes—
426C2×Dic3#13512yes—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6C22⋊Q814yes—
435C3×C8—22no—
436C3×C8—41no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C24—21no—
440C24—31no—
441#41—68no—
442#41—124no—
443#41—124no—
444#41—124no—
445C2×Q8—32no—
446C2×Q8—32no—
447C2×Q8—32no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—64no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4D3×Q812yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—31no—
468C22×C4—64no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—124no—
494C2×C8D3×D412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—32no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—81no—
508C4⋊C4—122no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C22⋊C4—42no—
516C22⋊C4—62no—
517C22⋊C4—81no—
518C22⋊C4—121no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C42—21no—
525C42—21no—
526C42—61no—
527C42—61no—
528C42—122no—
529C42—122no—
530C42—122no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3C2×C22⋊Q811yescharacteristic
539C22×C3D4211yescharacteristic
540C22×C3#54611yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57711yescharacteristic
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—81no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4C2×C22⋊Q811yescharacteristic
558C3×C4#54311yescharacteristic
559C3×C4#54611yescharacteristic
560C3×C4#55111yescharacteristic
561Dic3—24no—
562Dic3—22no—
563Dic3—22no—
564Dic3—42no—
565Dic3—41no—
566Dic3—81no—
567Dic3—81no—
568C23—22no—
569C23—32no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—61no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23C2×D3×D411yescharacteristic
583Q8—34no—
584Q8—62no—
585Q8—62no—
586Q8—62no—
587D4—22no—
588D4—22no—
589D4—41no—
590D4—41no—
591D4—128no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—62no—
623C2×C4—61no—
624C2×C4—128no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4C2×D3×D411yescharacteristic
634C2×C4#100411yescharacteristic
635C2×C4C2×D3×Q811yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4#100911yescharacteristic
639C8—22no—
640C8—41no—
641C8—121no—
642C8—121no—
643C2×C3—82no—
644C2×C3#243711yescharacteristic
645C2×C3#307611yescharacteristic
646C2×C3#308411yescharacteristic
647C2×C3#327111yescharacteristic
648C2×C3#328411yescharacteristic
649C2×C3#330111yescharacteristic
650C2×C3#331711yescharacteristic
651D3—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—62no—
656C22—62no—
657C22—62no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—121no—
663C22—121no—
664C22—121no—
665C22—121no—
666C22#560911yescharacteristic
667C22#561211yescharacteristic
668C22D3×C22⋊Q811yescharacteristic
669C22#576211yescharacteristic
670C22D3×#13511yescharacteristic
671C22#577311yescharacteristic
672C22#577511yescharacteristic
673C4—41no—
674C4—41no—
675C4—62no—
676C4—64no—
677C4—62no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—121no—
682C4—122no—
683C4—241no—
684C4—241no—
685C4#561011yescharacteristic
686C4D3×C22⋊Q811yescharacteristic
687C4#563411yescharacteristic
688C4#570511yescharacteristic
689C3#2595311yescharacteristic, complemented, Sylow
690C2—82no—
691C2—122no—
692C2#7741811yescharacteristic
693C2#8039811yescharacteristic
694C2#8056011yescharacteristic
695C2#8067911yescharacteristic
696C2#8088511yescharacteristic
697C2#8106311yescharacteristic
698C2#8116511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2065C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#2065D3
hypocenterC3#25953
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. C2×#4829C2semidirect
4. #76224C2semidirect
5. C3×#2065C2semidirect
6. #70588C2semidirect
7. #70577C2semidirect
8. #70477C2semidirect
10. #70184C2semidirect
11. #70180C2semidirect
12. #69639C2semidirect
13. #69632C2semidirect
14. #68904C2semidirect
15. #68877C2semidirect
68. #2065D3semidirect
689. C3#25953semidirect

Maximal subgroups

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

subgroupindex
C2×#48292
C2×#47722
#763012
#762242
C3×#20652
#705882
#705772
#704772
#701992
#701842
#701802
#696392
#696322
#689042
#688772
#259533