Finite Group Databasebeta

← #10844768

Subgroups — #10844768

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2592C211yesmaximal, characteristic, complemented
2C2×#5057C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic, complemented
4C2×#5024C211yesmaximal, characteristic, complemented
5#71725C211yesmaximal, characteristic, complemented
6#69682C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic, complemented
8#69679C211yesmaximal, characteristic, complemented
9#69675C211yesmaximal, characteristic, complemented
10#69653C211yesmaximal, characteristic, complemented
11#69650C211yesmaximal, characteristic
12#69648C211yesmaximal, characteristic
13#69645C211yesmaximal, characteristic, complemented
14C4×#809C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#25298—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#397—21no—
23C3×#393—21no—
24C3×#391C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31#5231—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5057—24no—
39#5024—24no—
40#4993—21no—
41#4989—21no—
42#4989—21no—
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4936C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592D311yescharacteristic, complemented
65#2580—31no—
66#2067—31no—
67#2066—31no—
68#2055—31no—
69#2054—31no—
70#1973—31no—
71#1972—31no—
72#1958—31no—
73#1949—31no—
74#1936—31no—
75#1935—31no—
76C4×#102—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×#124—22no—
83C3×#124—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#215—22no—
102C2×#215—22no—
103C2×#215—41no—
104C2×#213—22no—
105C2×#213—22no—
106C2×#213—41no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×D12—22no—
122C4×D12—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—41no—
131#832—22no—
132#832—22no—
133#832—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#809—22no—
139#809—22no—
140#808—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)D414yes—
145#803—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124C2×D311yescharacteristic
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4—31no—
153#463—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#399—61no—
172#397—21no—
173#393—21no—
174#393—61no—
175#391C2×D311yescharacteristic
176#391C2×D311yescharacteristic
177#390—31no—
178#390—31no—
179#390C2×D311yescharacteristic
180#390C2×D311yescharacteristic
181#389—31no—
182#389—31no—
183#388—21no—
184#388—21no—
185#385—61no—
186#385—61no—
187#382—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C23×D3—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—41no—
203C2×C3×Q8—41no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12#4112yes—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C22⋊C4—81no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—81no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#215—44no—
253#213—44no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8#4112yes—
279C22×Q8#6711yescharacteristic
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C22×D4C22×D311yescharacteristic
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—121no—
287C2×D8—62no—
288C2×D8—62no—
289C2×D8—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8—31no—
294#124—22no—
295#124—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×D4—62no—
299C4×D4—62no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4#6711yescharacteristic
306C2×C4⋊C4#6711yescharacteristic
307C2×C4⋊C4#6711yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#107—121no—
326#106—121no—
327C4⋊C8—34no—
328C4⋊C8—62no—
329C4⋊C8—62no—
330#103—62no—
331#103—62no—
332#103—121no—
333#102—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—62no—
339#97—62no—
340C4×C8—62no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—44no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
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—21no—
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×C4—41no—
399C2×C3×C4#12112yes—
400C2×C3×C4#12112yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416D12—22no—
417D12—22no—
418D12—41no—
419D12—41no—
420C4×D3—84no—
421C3⋊C8—22no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C24—21no—
426C24—61no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—32no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4—241no—
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—124no—
465D8—124no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—81no—
509C42D3×D412yes—
510C42#23712yes—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3#55011yescharacteristic
521C22×C3#57511yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57611yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4#52911yescharacteristic
542C3×C4#54211yescharacteristic
543C3×C4#54611yescharacteristic
544C3×C4#54811yescharacteristic
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×#6711yescharacteristic
563Q8—44no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—62no—
573D4—62no—
574D4—121no—
575D4—121no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—244no—
611C2×C4#93912yes—
612C2×C4#93912yes—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4#100911yescharacteristic
618C2×C4#101011yescharacteristic
619C8—62no—
620C8—62no—
621C8—62no—
622C8—121no—
623C2×C3—161no—
624C2×C3#242611yescharacteristic
625C2×C3#305411yescharacteristic
626C2×C3#306211yescharacteristic
627C2×C3#337011yescharacteristic
628C2×C3#337611yescharacteristic
629C2×C3#340111yescharacteristic
630C2×C3#340311yescharacteristic
631D3—82no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22C2×#93911yescharacteristic
647C22#582211yescharacteristic
648C22#582211yescharacteristic
649C22#583111yescharacteristic
650C22#583411yescharacteristic
651C22#583511yescharacteristic
652C22#583711yescharacteristic
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#567611yescharacteristic
666C4#567711yescharacteristic
667C4#568011yescharacteristic
668C4#568211yescharacteristic
669C3#2529811yescharacteristic, complemented, Sylow
670C2—161no—
671C2—242no—
672C2#8292311yescharacteristic
673C2#8299511yescharacteristic
674C2#8342211yescharacteristic
675C2#8366911yescharacteristic
676C2#8367211yescharacteristic
677C2#8371411yescharacteristic
678C2#8371611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2592C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2592D3
hypocenterC3#25298
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. C3×#2592C2semidirect
3. #82501C2semidirect
4. C2×#5024C2semidirect
5. #71725C2semidirect
6. #69682C2semidirect
7. #69680C2semidirect
8. #69679C2semidirect
9. #69675C2semidirect
10. #69653C2semidirect
13. #69645C2semidirect
14. C4×#809C2semidirect
15. #69468C2semidirect
64. #2592D3semidirect
669. C3#25298semidirect

Maximal subgroups

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

subgroupindex
C3×#25922
C2×#50572
#825012
C2×#50242
#717252
#696822
#696802
#696792
#696752
#696532
#696502
#696482
#696452
C4×#8092
#694682
#252983