Finite Group Databasebeta

← #10824503

Subgroups — #10824503

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2328C211yesmaximal, characteristic
2C2×C4×#214C211yesmaximal, characteristic, complemented
3C2×#5018C211yesmaximal, characteristic, complemented
4#73386C211yesmaximal, characteristic, complemented
5#71768C211yesmaximal, characteristic
6#71762C211yesmaximal, characteristic
7#71609C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#69653C211yesmaximal, characteristic, complemented
10#69652C211yesmaximal, characteristic, complemented
11#69651C211yesmaximal, characteristic, complemented
12#69651C211yesmaximal, characteristic, complemented
13#69560C211yesmaximal, characteristic, complemented
14#69541C211yesmaximal, characteristic, complemented
15#69436C211yesmaximal, characteristic, complemented
16#23690—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#386—21no—
26C3×#386—21no—
27C3×#386—21no—
28C3×#386—21no—
29C3×#382C2211yescharacteristic
30C3×#376C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5018—24no—
38C4×#214—24no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4929C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×#803C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4652—21no—
53#4652—21no—
54#4651—21no—
55#4650—21no—
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2328D311yescharacteristic
65#2320—31no—
66#2085—31no—
67#2074—31no—
68#2067—31no—
69#2065—31no—
70#2055—31no—
71#2055—31no—
72#2041—31no—
73#2014—31no—
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C22×Dic6C2311yescharacteristic
84C3×#127—22no—
85C3×#127—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#214—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214—22no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C4×Dic6—22no—
120C4×Dic6D414yescomplemented
121C3×#95—21no—
122C3×#95—21no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127#832—22no—
128#832—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#810—22no—
141#810—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×#127C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391—31no—
172#390C2×D311yescharacteristic
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#387—61no—
177#387—61no—
178#386—21no—
179#386—21no—
180#386—21no—
181#386—21no—
182#383—61no—
183#382—61no—
184#382C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#376C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—81no—
208C2×C3×D4—81no—
209C2×C3×D4—81no—
210C2×C3×D4—81no—
211C2×C3×D4#4112yes—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4—81no—
218C22×C3×C4—81no—
219C22×C3×C4—81no—
220C22×C3×C4—81no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#214—28no—
255#214—28no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3C2×D412yes—
262Dic3⋊C4—24no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265C4×Dic3—24no—
266C4×Dic3—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⋊C8—21no—
277C2×C3⋊C8#4112yes—
278C2×C3⋊C8#4112yes—
279C22×Q8—31no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22×D311yescharacteristic
283C22×D4C22×D311yescharacteristic
284C23×C4—21no—
285C23×C4—21no—
286C2×SD16—62no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—31no—
296#127—22no—
297#127—22no—
298C4×Q8—34no—
299C4×Q8—62no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325C4⋊C8—62no—
326C4⋊C8—62no—
327#103—62no—
328#103—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—62no—
335#97—62no—
336C4×C8—62no—
337#95—21no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95#6711yescharacteristic
347#95#6711yescharacteristic
348#95#6711yescharacteristic
349#95#6711yescharacteristic
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—84no—
374C3×D4—84no—
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—21no—
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—84no—
398C2×C3×C4—84no—
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×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419Dic6—22no—
420Dic6—22no—
421Dic6—41no—
422Dic6—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—21no—
428C24—21no—
429C2×Q8—32no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—81no—
451C2×D4#23512yes—
452C2×D4#23512yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—68no—
473SD16—68no—
474C2×C8—32no—
475C2×C8—32no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—64no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—64no—
510C42—241no—
511C42D3×D412yes—
512C42#23712yes—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#52511yescharacteristic
529C22×C3#53211yescharacteristic
530C22×C3#57611yescharacteristic
531C22×C3#57711yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57711yescharacteristic
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4#52511yescharacteristic
543C3×C4#52511yescharacteristic
544C3×C4#52911yescharacteristic
545C3×C4#52911yescharacteristic
546Dic3—42no—
547Dic3—42no—
548Dic3—81no—
549Dic3—81no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C22×#6711yescharacteristic
565Q8—62no—
566Q8—62no—
567Q8—121no—
568Q8—121no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578D4—84no—
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—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4#93812yes—
616C2×C4#93812yes—
617C2×C4C2×#23511yescharacteristic
618C2×C4C2×#23511yescharacteristic
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—62no—
624C8—62no—
625C8—121no—
626C8—121no—
627C2×C3—82no—
628C2×C3—82no—
629C2×C3#239211yescharacteristic
630C2×C3#304311yescharacteristic
631C2×C3#304511yescharacteristic
632C2×C3#311811yescharacteristic
633C2×C3#312311yescharacteristic
634C2×C3#317411yescharacteristic
635C2×C3#317911yescharacteristic
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#579111yescharacteristic
651C22C2×#93811yescharacteristic
652C22#583211yescharacteristic
653C22#583511yescharacteristic
654C22#583611yescharacteristic
655C22#583611yescharacteristic
656C22#583611yescharacteristic
657C4—22no—
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—122no—
666C4—122no—
667C4—241no—
668C4—241no—
669C4#553811yescharacteristic
670C4#553811yescharacteristic
671C4#554711yescharacteristic
672C4#554711yescharacteristic
673C3#2369011yescharacteristic, complemented, Sylow
674C2—82no—
675C2—82no—
676C2#8240111yescharacteristic
677C2#8254111yescharacteristic
678C2#8254511yescharacteristic
679C2#8338511yescharacteristic
680C2#8339111yescharacteristic
681C2#8343111yescharacteristic
682C2#8343311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2328C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2328D3
hypocenterC3#23690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. C2×C4×#214C2semidirect
3. C2×#5018C2semidirect
4. #73386C2semidirect
7. #71609C2semidirect
8. C4×#832C2semidirect
9. #69653C2semidirect
10. #69652C2semidirect
11. #69651C2semidirect
12. #69651C2semidirect
13. #69560C2semidirect
14. #69541C2semidirect
15. #69436C2semidirect
120. C4×Dic6D4semidirect
673. C3#23690semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
C2×C4×#2142
C2×#50182
#733862
#717682
#717622
#716092
C4×#8322
#696532
#696522
#696512
#695602
#695412
#694362
#236903