Finite Group Databasebeta

← #10824371

Subgroups — #10824371

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
1C2×C4×#204C211yesmaximal, characteristic, complemented
2C2×C4×D24C211yesmaximal, characteristic
3C4×#875C211yesmaximal, characteristic, complemented
4#73299C211yesmaximal, characteristic, complemented
5C3×C4×C4⋊C8C211yesmaximal, characteristic, complemented
6#70489C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic, complemented
9#70481C211yesmaximal, characteristic
10C4×#816C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#68675C211yesmaximal, characteristic
13#68641C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic, complemented
15#68630C211yesmaximal, characteristic, complemented
16#23680—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#872—21no—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×C43C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39#4940—21no—
40#4940—21no—
41#4940—21no—
42#4940—21no—
43#4939C2211yescharacteristic
44#4939C2211yescharacteristic
45#4939C2211yescharacteristic
46C4×(C2×D3)⋊C4—21no—
47C4×(C2×D3)⋊C4—21no—
48C4×(C2×D3)⋊C4—21no—
49#4936C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54C4×D24—24no—
55C4×#204—24no—
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
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⋊C8D311yescharacteristic, complemented
75C4×#103—31no—
76C4×#102—31no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×D24—22no—
114C2×D24—22no—
115C2×D24—22no—
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—22no—
119#875—22no—
120#875—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12D414yes—
125#872—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×C4⋊C8—22no—
129C3×C4⋊C8—22no—
130C3×C4×C8—22no—
131C3×C4×C8—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#818—22no—
145#817—22no—
146#816—22no—
147#816—22no—
148#816—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×#117—61no—
156C4×SD16—64no—
157C4×D8—64no—
158C2×#107—31no—
159C2×#106—31no—
160C2×C4⋊C8C2×D311yescharacteristic
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8C2×D311yescharacteristic
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⋊C4—31no—
184C4×C22⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187C43C2×D311yescharacteristic
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336C2×D311yescharacteristic
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×D412yes—
220C2×D12#4112yes—
221C2×C4×D3—24no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
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—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8#4112yes—
244C2×C3×C8#4112yes—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—41no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280D24—28no—
281#204—28no—
282C22×Q8—31no—
283C22×D4—31no—
284C22×D4—31no—
285C23×C4—61no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—62no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×D4—34no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305#117—124no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C22⋊C4—61no—
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—61no—
325C2×C42—61no—
326C2×C42D1211yescharacteristic
327C2×C42D1211yescharacteristic
328C2×C42D1211yescharacteristic
329C2×C42D1211yescharacteristic
330C2×C42C22×D311yescharacteristic
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337#103—62no—
338#103—62no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345C4×C8—22no—
346C4×C8—22no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
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—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4#13512yes—
385C2×C3×C4#13512yes—
386C2×C3×C4AGL(1,ℤ/8)12yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
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—
411D12—22no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419C4×D3—48no—
420C4×D3—84no—
421Dic6—22no—
422Dic6—22no—
423Dic6—41no—
424Dic6—41no—
425C3×C8—22no—
426C3×C8—22no—
427C3×C8—41no—
428C3×C8—41no—
429C24—61no—
430C24—61no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×D4—32no—
438C2×D4—32no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—122no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C2×D1211yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—68no—
469D8—68no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8#23512yes—
481C2×C8#23512yes—
482C4⋊C4—32no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—122no—
514C42—122no—
515C42D3×D412yes—
516C42#23912yes—
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#53211yescharacteristic
520C22×C3C2×#13511yescharacteristic
521C22×C3C2×#13511yescharacteristic
522C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
523C22×C3#57511yescharacteristic
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4#46314yes—
544C3×C4#52511yescharacteristic
545C3×C4#52511yescharacteristic
546C3×C4#52911yescharacteristic
547C3×C4#52911yescharacteristic
548Dic3—42no—
549Dic3—42no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—62no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C22×D1211yescharacteristic
569Q8—62no—
570Q8—62no—
571Q8—121no—
572Q8—121no—
573D4—62no—
574D4—62no—
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
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—62no—
595C2×C4—62no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—128no—
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—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—244no—
614C2×C4#89512yes—
615C2×C4#89512yes—
616C2×C4#90412yes—
617C2×C4#90412yes—
618C2×C4#90812yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×#23511yescharacteristic
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23911yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—22no—
626C8—22no—
627C8—41no—
628C8—41no—
629C2×C3#239211yescharacteristic
630C2×C3C2×#46311yescharacteristic
631C2×C3#303511yescharacteristic
632C2×C3#310211yescharacteristic
633C2×C3#310611yescharacteristic
634C2×C3#316811yescharacteristic
635C2×C3#317111yescharacteristic
636D3—82no—
637D3—82no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#547711yescharacteristic
653C22C2×#89511yescharacteristic
654C22#552411yescharacteristic
655C22C2×#90411yescharacteristic
656C22C2×#90411yescharacteristic
657C22C2×#90811yescharacteristic
658C22#574811yescharacteristic
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—122no—
665C4—122no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4—241no—
670C4#484414yes—
671C4#554711yescharacteristic
672C4#554711yescharacteristic
673C4#558011yescharacteristic
674C4#558011yescharacteristic
675C3#2368011yescharacteristic, complemented, Sylow
676C2—242no—
677C2—242no—
678C2#7377611yescharacteristic
679C2#7846911yescharacteristic
680C2#7853211yescharacteristic
681C2C2×#484411yescharacteristic
682C2#7902111yescharacteristic
683C2#7929911yescharacteristic
684C2#7930111yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊C8C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenterC4×C4⋊C8D3
hypocenterC3#23680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C4×#204C2semidirect
3. C4×#875C2semidirect
4. #73299C2semidirect
5. C3×C4×C4⋊C8C2semidirect
6. #70489C2semidirect
7. #70483C2semidirect
8. #70482C2semidirect
10. C4×#816C2semidirect
11. C4×#821C2semidirect
13. #68641C2semidirect
14. #68637C2semidirect
15. #68630C2semidirect
74. C4×C4⋊C8D3semidirect
675. C3#23680semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2042
C2×C4×D242
C4×#8752
#732992
C3×C4×C4⋊C82
#704892
#704832
#704822
#704812
C4×#8162
C4×#8212
#686752
#686412
#686372
#686302
#236803