Finite Group Databasebeta

← #10813489

Subgroups — #10813489

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#440C211yesmaximal, characteristic
2C2×#5164C211yesmaximal, characteristic
3C2×Dic3×SD16C211yesmaximal, characteristic
4#82461C211yesmaximal, characteristic, complemented
5#82334C211yesmaximal, characteristic
6#71629C211yesmaximal, characteristic, complemented
7#71607C211yesmaximal, characteristic
8#70614C211yesmaximal, characteristic, complemented
9#70607C211yesmaximal, characteristic, complemented
10#70237C211yesmaximal, characteristic, complemented
11#70222C211yesmaximal, characteristic, complemented
12#70169C211yesmaximal, characteristic
13#70154C211yesmaximal, characteristic
14#69833C211yesmaximal, characteristic, complemented
15#69820C211yesmaximal, characteristic, complemented
16#22673—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×D4×Dic3C2211yescharacteristic
24C3×#440—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5210—21no—
34C2×C3⋊#95—21no—
35#5164—24no—
36Dic3×SD16C418yescomplemented
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42#4977—21no—
43#4977C2211yescharacteristic
44#4977C2211yescharacteristic
45#4972—21no—
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4946—21no—
51#4942—21no—
52C4×Dic3⋊C4—21no—
53C4×Dic3⋊C4C2211yescharacteristic
54#4928—21no—
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63C2×#440—31no—
64C2×#440D311yescharacteristic
65C2×C4×SD16—31no—
66#2351—31no—
67#2349—31no—
68#1944—31no—
69#1944—31no—
70#1865—31no—
71#1865—31no—
72#1864—31no—
73#1864—31no—
74#1846—31no—
75#1805—31no—
76#1804—31no—
77#1770—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C2×C3×SD16C2×C414yes—
84C2×C3×SD16D414yes—
85C2×C3×SD16Q814yes—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—41no—
90C3×C4×D4—22no—
91C3×C4×D4—41no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102Q8×Dic3—41no—
103Q8×Dic3C2×C414yescomplemented
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—41no—
107D4×Dic3C2×C414yes—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C3×#107—22no—
127C3×#103—22no—
128C3×#102—22no—
129C3×#97—22no—
130#835—22no—
131#835C2×C414yescomplemented
132#832—22no—
133#832C2×C414yes—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#818—22no—
145#817C2×C414yescomplemented
146#815—22no—
147Dic3⋊C8—22no—
148C8×Dic3C2×C414yescomplemented
149#808—41no—
150#808—41no—
151C22×SD16C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157C2×C4×D4C2×D311yescharacteristic
158#440—24no—
159#440—64no—
160C4×SD16—38no—
161C2×#107—31no—
162C2×#107C2×D311yescharacteristic
163C2×#106—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103C2×D311yescharacteristic
170C2×#102—31no—
171C2×#102—31no—
172C2×#102C2×D311yescharacteristic
173C2×#97—31no—
174C2×#97C2×D311yescharacteristic
175C2×C4×C8—31no—
176#389—31no—
177#389—31no—
178#389—31no—
179#389—31no—
180#389—61no—
181#389—61no—
182#385—61no—
183#385—61no—
184#382—31no—
185#382—31no—
186#382—61no—
187#380—61no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C2×#95—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#336—31no—
196#336—31no—
197C24×C3—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8C22×C412yes—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8C2×Q812yes—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4C22×C412yes—
208C2×C3×D4C2×D412yes—
209C2×C3×D4C2×Q812yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—81no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—24no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3—81no—
227C22×Dic3—81no—
228C22×Dic3—81no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C3×SD16C4⋊C4116yescomplemented
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C22×C412yes—
238C2×C3×C8C2×D412yes—
239C2×C3×C8C2×Q812yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—42no—
249#217—24no—
250#217—42no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—24no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3C22×C412yes—
260C4⋊Dic3C22×C412yes—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—24no—
270C4×Dic3—42no—
271C4×Dic3C22×C412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C22×C412yes—
279C22×Q8C22×D311yescharacteristic
280C22×D4C22×D311yescharacteristic
281C23×C4—21no—
282C23×C4—61no—
283C23×C4—61no—
284C2×SD16Dic614yes—
285C2×SD16C4×D314yes—
286C2×SD16#6714yes—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291C4×Q8—22no—
292C4×Q8—34no—
293C4×Q8—41no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—121no—
297C4×D4—22no—
298C4×D4—34no—
299C4×D4—41no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—121no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—22no—
328#107—62no—
329#106—34no—
330#106—121no—
331#106—121no—
332C4⋊C8—62no—
333#103—22no—
334#103—34no—
335#103—62no—
336#102—22no—
337#102—34no—
338#102—62no—
339#97—22no—
340#97—62no—
341C4×C8—34no—
342#95—31no—
343#95—31no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—22no—
363C3×Q8C2×C4⋊C414yes—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4C2×C4⋊C414yes—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—84no—
384C2×C3×C4C23×C411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×Q811yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—84no—
413C2×Dic3—84no—
414C2×Dic3C4×D412yes—
415C2×Dic3C4×D412yes—
416C3×C8—41no—
417C3×C8C2×C4⋊C414yes—
418C3⋊C8—22no—
419C3⋊C8—41no—
420C24—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×Dic612yes—
425C2×Q8C2×C4×D312yes—
426C2×Q8C2×#6712yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×Dic612yes—
431C2×D4C2×C4×D312yes—
432C2×D4C2×#6712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—81no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4—241no—
452C22×C4—241no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16Dic3⋊C4116yescomplemented
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8C2×Dic612yes—
467C2×C8C2×C4×D312yes—
468C2×C8C2×#6712yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—32no—
474C4⋊C4—32no—
475C4⋊C4—42no—
476C4⋊C4—64no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—121no—
484C4⋊C4—121no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—241no—
489C4⋊C4—241no—
490C22⋊C4—42no—
491C22⋊C4—64no—
492C22⋊C4—122no—
493C42—21no—
494C42—21no—
495C42—32no—
496C42—32no—
497C42—42no—
498C42—64no—
499C42—61no—
500C42—61no—
501C42—122no—
502C22×C3—24no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3C2×C4×D411yescharacteristic
510C22×C3D4211yescharacteristic
511C22×C3D4×Q811yescharacteristic
512C22×C3#57511yescharacteristic
513C22×C3#57511yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57711yescharacteristic
516C3×C4—24no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4C22×C4⋊C411yescharacteristic
522C3×C4#52011yescharacteristic
523C3×C4#54711yescharacteristic
524C3×C4#55411yescharacteristic
525Dic3—22no—
526Dic3—22no—
527Dic3—42no—
528Dic3—42no—
529Dic3—41no—
530Dic3—41no—
531Dic3—81no—
532Dic3—81no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×D3×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8C2×Dic3⋊C414yes—
545D4—22no—
546D4—22no—
547D4—22no—
548D4C2×Dic3⋊C414yes—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—84no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—244no—
590C2×C4—244no—
591C2×C4C22×Dic611yescharacteristic
592C2×C4C22×C4×D311yescharacteristic
593C2×C4#100411yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4#101011yescharacteristic
596C2×C4C22×#6711yescharacteristic
597C8—41no—
598C8—62no—
599C8—121no—
600C8C2×Dic3⋊C414yes—
601C2×C3—44no—
602C2×C3D4×C4⋊C411yescharacteristic
603C2×C3#294311yescharacteristic
604C2×C3#294911yescharacteristic
605C2×C3#327511yescharacteristic
606C2×C3#327511yescharacteristic
607C2×C3#338811yescharacteristic
608C2×C3#338811yescharacteristic
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22D4×Dic611yescharacteristic
617C22C4×D3×D411yescharacteristic
618C22#577311yescharacteristic
619C22#577511yescharacteristic
620C22#577711yescharacteristic
621C22#577811yescharacteristic
622C22D4×#6711yescharacteristic
623C4—24no—
624C4—41no—
625C4—41no—
626C4—62no—
627C4—62no—
628C4—81no—
629C4—81no—
630C4—121no—
631C4—122no—
632C4—122no—
633C4—121no—
634C4—241no—
635C4—241no—
636C4#553911yescharacteristic
637C4#554611yescharacteristic
638C4C22×Dic3⋊C411yescharacteristic
639C4#582411yescharacteristic
640C3#2267311yescharacteristic, complemented, Sylow
641C2—44no—
642C2#7976111yescharacteristic
643C2#7976211yescharacteristic
644C2#7977211yescharacteristic
645C2#7977711yescharacteristic
646C2D4×Dic3⋊C411yescharacteristic
647C2#8396211yescharacteristic
648C2#8399111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC2×#440D3
hypocenterC3#22673
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
4. #82461C2semidirect
6. #71629C2semidirect
8. #70614C2semidirect
9. #70607C2semidirect
10. #70237C2semidirect
11. #70222C2semidirect
14. #69833C2semidirect
15. #69820C2semidirect
36. Dic3×SD16C4semidirect
103. Q8×Dic3C2×C4semidirect
131. #835C2×C4semidirect
145. #817C2×C4semidirect
148. C8×Dic3C2×C4semidirect
233. C3×SD16C4⋊C4semidirect
456. SD16Dic3⋊C4semidirect
640. C3#22673semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4402
C2×#51642
C2×Dic3×SD162
#824612
#823342
#716292
#716072
#706142
#706072
#702372
#702222
#701692
#701542
#698332
#698202
#226733