Finite Group Databasebeta

← #10824377

Subgroups — #10824377

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#936C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic
3C2×C4×#204C211yesmaximal, characteristic, complemented
4#73617C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70216C211yesmaximal, characteristic, complemented
7#70214C211yesmaximal, characteristic
8#70212C211yesmaximal, characteristic
9#70179C211yesmaximal, characteristic, complemented
10C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
11C4×#810C211yesmaximal, characteristic, complemented
12#68913C211yesmaximal, characteristic, complemented
13#68903C211yesmaximal, characteristic, complemented
14#68863C211yesmaximal, characteristic, complemented
15#68836C211yesmaximal, characteristic, complemented
16#23680—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#872—21no—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C4×#217—21no—
31C4×#217—21no—
32C2×#821C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C4×#213—24no—
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4977C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4945—21no—
49#4945—21no—
50C4×(C2×D3)⋊C4—21no—
51C42×Dic3C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53C4×#204—24no—
54#4669—21no—
55#4669—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×D8—31no—
64C4×#124—31no—
65#2293—31no—
66#2085—31no—
67#2073—31no—
68#2068—31no—
69#2065—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#102D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96#936—22no—
97#936—22no—
98C2×#213—22no—
99C2×#213—22no—
100C2×#213—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×#204—22no—
121C2×#204—22no—
122C2×#204—22no—
123C4×D12—22no—
124C4×D12D414yes—
125#872—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×C4×C8—22no—
131C3×#95—21no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#817—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#810—22no—
146#809—22no—
147#807—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155C2×#117—61no—
156C4×SD16—64no—
157C4×D8—64no—
158C2×#107—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171C2×C4×C8C2×D311yescharacteristic
172#391—31no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#388—61no—
179#388—61no—
180#388—61no—
181#388—61no—
182#382—31no—
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187C43—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C2×D412yes—
224C2×C4×D3—24no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6#4112yes—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C22⋊C4—42no—
246C3×C42—42no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249#217—24no—
250#217—24no—
251#217—42no—
252#217—42no—
253#213—28no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2×D412yes—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266C4×Dic3—22no—
267C4×Dic3—22no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—22no—
273C4×Dic3—41no—
274C4×Dic3—42no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8#4112yes—
281#204—28no—
282C22×Q8—31no—
283C22×D4—31no—
284C22×D4C22×D311yescharacteristic
285C23×C4—21no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—62no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—34no—
304C4×D4—62no—
305#117—124no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—21no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340#102—22no—
341#102—22no—
342#102—62no—
343#102—62no—
344#102—62no—
345C4×C8—22no—
346C4×C8—62no—
347#95—21no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—41no—
372C3×D4—41no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
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—41no—
384C2×C3×C4—84no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4#13512yes—
388C2×C3×C4AGL(1,ℤ/8)12yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3#13512yes—
416D12—22no—
417D12—22no—
418D12—41no—
419D12—41no—
420C4×D3—48no—
421Dic6—22no—
422Dic6—22no—
423Dic6—41no—
424Dic6—41no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—21no—
430C24—61no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
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—
448C2×D4#23512yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—64no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—68no—
469D8—68no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—32no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8#23512yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—32no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C22⋊C4—42no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—42no—
506C42—62no—
507C42—62no—
508C42—62no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42—121no—
514C42—122no—
515C42#23512yes—
516C42D3×D412yes—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53211yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
530C22×C3#57511yescharacteristic
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4#46314yes—
543C3×C4#52511yescharacteristic
544C3×C4#52511yescharacteristic
545C3×C4#52911yescharacteristic
546C3×C4#52911yescharacteristic
547Dic3—22no—
548Dic3—41no—
549Dic3—41no—
550Dic3—42no—
551Dic3—41no—
552Dic3—42no—
553Dic3—81no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—62no—
570Q8—62no—
571Q8—121no—
572Q8—121no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—62no—
578D4—62no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—32no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—62no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—84no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—128no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4#88412yes—
616C2×C4#88412yes—
617C2×C4#91112yes—
618C2×C4#91612yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×#23511yescharacteristic
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—22no—
626C8—41no—
627C8—62no—
628C8—121no—
629C2×C3—82no—
630C2×C3#239211yescharacteristic
631C2×C3C2×#46311yescharacteristic
632C2×C3#303511yescharacteristic
633C2×C3#310211yescharacteristic
634C2×C3#310611yescharacteristic
635C2×C3#316811yescharacteristic
636C2×C3#317111yescharacteristic
637D3—82no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22C2×#88411yescharacteristic
653C22#554711yescharacteristic
654C22#554811yescharacteristic
655C22C2×#91111yescharacteristic
656C22C2×#91611yescharacteristic
657C22D3×#13511yescharacteristic
658C22#576811yescharacteristic
659C4—22no—
660C4—62no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—121no—
665C4—122no—
666C4—122no—
667C4—121no—
668C4—121no—
669C4—241no—
670C4#478714yes—
671C4#547911yescharacteristic
672C4#552811yescharacteristic
673C4#554711yescharacteristic
674C4#554711yescharacteristic
675C3#2368011yescharacteristic, complemented, Sylow
676C2—82no—
677C2—242no—
678C2#7358111yescharacteristic
679C2C2×#478711yescharacteristic
680C2#7886111yescharacteristic
681C2#7959311yescharacteristic
682C2#7959911yescharacteristic
683C2#7965211yescharacteristic
684C2#7966311yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC4×#102D3
hypocenterC3#23680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. C4×#936C2semidirect
3. C2×C4×#204C2semidirect
4. #73617C2semidirect
5. C3×C4×#102C2semidirect
6. #70216C2semidirect
9. #70179C2semidirect
10. C4×Dic3⋊C8C2semidirect
11. C4×#810C2semidirect
12. #68913C2semidirect
13. #68903C2semidirect
14. #68863C2semidirect
15. #68836C2semidirect
76. C4×#102D3semidirect
675. C3#23680semidirect

Maximal subgroups

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

subgroupindex
C4×#9362
C2×C4×#2132
C2×C4×#2042
#736172
C3×C4×#1022
#702162
#702142
#702122
#701792
C4×Dic3⋊C82
C4×#8102
#689132
#689032
#688632
#688362
#236803