Finite Group Databasebeta

← #10824639

Subgroups — #10824639

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#204C211yesmaximal, characteristic, complemented
2C2×#4685C211yesmaximal, characteristic, complemented
3C4×#897C211yesmaximal, characteristic, complemented
4C4×#895C211yesmaximal, characteristic
5C3×C4×#106C211yesmaximal, characteristic, complemented
6#70589C211yesmaximal, characteristic, complemented
7#70585C211yesmaximal, characteristic, complemented
8#70582C211yesmaximal, characteristic, complemented
9#70581C211yesmaximal, characteristic
10C4×#810C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic, complemented
12#68887C211yesmaximal, characteristic
13#68863C211yesmaximal, characteristic, complemented
14#68840C211yesmaximal, characteristic, complemented
15#68832C211yesmaximal, characteristic, complemented
16#23700—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#895C2211yescharacteristic
20C2×C4×D12—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4988C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C4×(C2×D3)⋊C4—21no—
47C4×(C2×D3)⋊C4—21no—
48C4×C4⋊Dic3—21no—
49C2×C4×C3⋊C8C2211yescharacteristic
50C4×#204—24no—
51#4685—24no—
52#4670—21no—
53#4668—21no—
54#4665—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2066—31no—
69#2065—31no—
70#2055—31no—
71#2035—31no—
72#2034—31no—
73#2000—31no—
74C4×#106D311yescharacteristic, complemented
75C4×#103—31no—
76C4×#102—31no—
77C22×D12—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3D411yescharacteristic
81C22×C4×D3D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42—21no—
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×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×#204—22no—
113C2×#204—22no—
114C2×#204—22no—
115#897—22no—
116#897—22no—
117#895—22no—
118#895—22no—
119C4×D12—22no—
120C4×D12—22no—
121C4×D12—44no—
122C42×D3—22no—
123C42×D3—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×#106—22no—
127C3×#106—22no—
128C3×C4×C8D414yescomplemented
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#817—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#810—22no—
146#809—22no—
147#809—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C2×C22⋊Q8—31no—
151C2×#121—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42—31no—
156C4×SD16—64no—
157#431—64no—
158C2×#106—31no—
159C2×#106C2×D311yescharacteristic
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—31no—
171C2×C4×C8—31no—
172C2×C4×C8C2×D311yescharacteristic
173#391—31no—
174#390—31no—
175#389—31no—
176#389—31no—
177#388—61no—
178#387—61no—
179#386—61no—
180#385—61no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—61no—
187C4×C22⋊C4—61no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—24no—
214C2×D12—21no—
215C2×D12—42no—
216C2×D12—42no—
217C2×D12#4112yes—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—41no—
228C2×C4×D3—41no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3SD1614yescomplemented
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4#4112yes—
252C3×C4⋊C4#4112yes—
253C3×C42—42no—
254C3×C42—42no—
255C3×C42C2×D412yes—
256C3×C42#4112yes—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—24no—
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—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—42no—
270C4⋊Dic3—42no—
271Dic3⋊C4—42no—
272C4×Dic3—22no—
273C4×Dic3—22no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—42no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283#204—28no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4—61no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300#121—62no—
301#121—62no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
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—21no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—62no—
330C2×C42—62no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#106—22no—
334#106—22no—
335#106—62no—
336C4⋊C8—62no—
337C4⋊C8—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344C22⋊C8—62no—
345C22⋊C8—62no—
346C4×C8—62no—
347C4×C8D1214yescomplemented
348#95—21no—
349#95—21no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—22no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3C2×SD1612yes—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
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—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C2×SD1612yes—
388C2×C3×C4#13512yes—
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—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
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×Dic3C2×SD1612yes—
413D12—22no—
414D12—22no—
415D12—48no—
416D12—41no—
417D12—41no—
418C4×D3—28no—
419C4×D3—44no—
420C4×D3—41no—
421C4×D3—41no—
422C4×D3—41no—
423C4×D3—41no—
424C4×D3—84no—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—31no—
434C24—61no—
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—122no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—34no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—121no—
471C22×C4—121no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8C2×D1212yes—
490C2×C8C2×D1212yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—64no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4#23512yes—
506C4⋊C4#23512yes—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—42no—
515C42—42no—
516C42—62no—
517C42—62no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42C2×D1212yes—
522C42#23912yes—
523C22×C3C2×#12111yescharacteristic
524C22×C3#52511yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3C22×SD1611yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57511yescharacteristic
530C2×D3—22no—
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—41no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4#46514yes—
550C3×C4#52511yescharacteristic
551C3×C4#52911yescharacteristic
552C3×C4#52911yescharacteristic
553C3×C4#53211yescharacteristic
554Dic3—22no—
555Dic3—42no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560C23—32no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—62no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—62no—
580D4—62no—
581D4—128no—
582D4—121no—
583D4—121no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—32no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—68no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—124no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—244no—
623C2×C4#89512yes—
624C2×C4#89512yes—
625C2×C4D3×SD1612yes—
626C2×C4#91612yes—
627C2×C4C22×D1211yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×#23911yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—22no—
635C8—121no—
636C8—121no—
637C2×C3#239211yescharacteristic
638C2×C3C2×#46511yescharacteristic
639C2×C3#305411yescharacteristic
640C2×C3#310111yescharacteristic
641C2×C3#310211yescharacteristic
642C2×C3#310211yescharacteristic
643C2×C3#310311yescharacteristic
644D3—42no—
645D3—82no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—121no—
650C22—121no—
651C22—121no—
652C22—121no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#89511yescharacteristic
661C22#554711yescharacteristic
662C22#554711yescharacteristic
663C22C2×D3×SD1611yescharacteristic
664C22C2×#91611yescharacteristic
665C22#576811yescharacteristic
666C22#576811yescharacteristic
667C4—22no—
668C4—42no—
669C4—42no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—121no—
674C4—122no—
675C4—121no—
676C4—241no—
677C4—241no—
678C4#486414yes—
679C4#547711yescharacteristic
680C4#552411yescharacteristic
681C4#558011yescharacteristic
682C4#558011yescharacteristic
683C3#2370011yescharacteristic, complemented, Sylow
684C2—122no—
685C2—242no—
686C2#7377611yescharacteristic
687C2C2×#486411yescharacteristic
688C2#7918111yescharacteristic
689C2#7965111yescharacteristic
690C2#7965211yescharacteristic
691C2#7965311yescharacteristic
692C2#7969411yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenterC4×#106D3
hypocenterC3#23700
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
1. C2×C4×#204C2semidirect
2. C2×#4685C2semidirect
3. C4×#897C2semidirect
5. C3×C4×#106C2semidirect
6. #70589C2semidirect
7. #70585C2semidirect
8. #70582C2semidirect
10. C4×#810C2semidirect
11. C4×#809C2semidirect
13. #68863C2semidirect
14. #68840C2semidirect
15. #68832C2semidirect
74. C4×#106D3semidirect
128. C3×C4×C8D4semidirect
232. C2×C4×D3SD16semidirect
347. C4×C8D12semidirect
683. C3#23700semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2042
C2×#46852
C4×#8972
C4×#8952
C3×C4×#1062
#705892
#705852
#705822
#705812
C4×#8102
C4×#8092
#688872
#688632
#688402
#688322
#237003