Finite Group Databasebeta

← C2×#30829 #7501958

Subgroups — C2×#30829 #7501958

Order 512.

2248 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
1C22×#489C211yesmaximal, characteristic
2C22×#476C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2447C211yesmaximal, characteristic
5#30829C2116yesmaximal, direct product factor
6C2×#2096C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#2042C211yesmaximal, characteristic
11C2×#2024C211yesmaximal, characteristic, complemented
12C2×#1943C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1925C211yesmaximal, characteristic
15C2×#1917C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#489—28no—
23C2×#489C2218yes—
24C2×#476—28no—
25C2×#476C2218yes—
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2551—24no—
41#2551C2218yescomplemented
42#2447—24no—
43#2447C2218yes—
44C2×#397—21no—
45C2×#397—21no—
46C2×#395—21no—
47C2×#395—21no—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#385—21no—
56C2×#384C2211yescharacteristic
57C2×#381—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2096C2218yescomplemented
61#2071C2218yes—
62#2068C2218yescomplemented
63#2058C2218yescomplemented
64#2042C2218yes—
65#2024C2218yescomplemented
66#1943C2218yescomplemented
67#1927C2218yescomplemented
68#1925C2218yes—
69#1917C2218yescomplemented
70#1864C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8—48no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2314yes—
98C2×#124—24no—
99C2×#124—22no—
100C2×#124—42no—
101C2×#124C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—42no—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#489—232no—
131#476—232no—
132C2×#107—24no—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8—24no—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—42no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102D418yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#97—24no—
159C2×#97C2314yes—
160#397—28no—
161#397—28no—
162#395—28no—
163#395—28no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#390—22no—
169#390C2314yes—
170#389—28no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#385—28no—
178#384—22no—
179#384C2314yes—
180#381—28no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D414yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2314yes—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8—82no—
218C22×Q8—82no—
219C22×Q8C2412yes—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4C2×D414yes—
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—82no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8C2×D414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—432no—
269#124—28no—
270#124—44no—
271#124—48no—
272C4×D4—28no—
273C4×D4—44no—
274C4×D4—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341#107—28no—
342#106—28no—
343#106—28no—
344#106—44no—
345C4⋊C8—28no—
346#103—28no—
347#103—28no—
348#103—28no—
349#103—44no—
350#102—28no—
351#102—28no—
352#102—28no—
353#102C2×D4116yes—
354#97—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—24no—
366C24—22no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24C22×D411yescharacteristic
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×Q8—816no—
394C2×Q8—816no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4C22×D414yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—84no—
455C22×C4—816no—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4#13714yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C22×C42−1+412yes—
471C22×C4C2511yescharacteristic
472C2×C8—28no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8C22×D414yes—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4C22×D414yes—
514C4⋊C4C2×#4114yes—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C23—24no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23#53612yes—
545C23D4212yes—
546C23#54812yes—
547C23C2×AGL(1,ℤ/8)12yes—
548C23C2×#13712yes—
549C23#57712yes—
550C23#57712yes—
551C23C23×D411yescharacteristic
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—816no—
557Q8—816no—
558Q8—84no—
559D4—24no—
560D4—24no—
561D4—48no—
562D4—42no—
563D4—42no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#53612yes—
616C2×C4#53614yes—
617C2×C4#54214yes—
618C2×C4D4212yes—
619C2×C4#54714yes—
620C2×C4#54812yes—
621C2×C4#56314yes—
622C2×C4C2×AGL(1,ℤ/8)14yes—
623C2×C4C2×#13714yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—24no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#267414yes—
643C22#323814yes—
644C22#324414yes—
645C22#326614yes—
646C22#327214yes—
647C22#336914yes—
648C22#337314yes—
649C22C2×#53611yescharacteristic
650C22C2×D4211yescharacteristic
651C22C2×#54811yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57711yescharacteristic
655C22C2×#57711yescharacteristic
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#267414yes—
670C4C2×#53611yescharacteristic
671C4C2×#54211yescharacteristic
672C4C2×#54711yescharacteristic
673C4C2×#56311yescharacteristic
674C2—84no—
675C2#3082918yesdirect product factor
676C2C2×#267411yescharacteristic
677C2C2×#323811yescharacteristic
678C2C2×#324411yescharacteristic
679C2C2×#326611yescharacteristic
680C2C2×#327211yescharacteristic
681C2C2×#337311yescharacteristic
682C2C2×#336911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2551C2semidirect
5. #30829C2direct (N × H)
6. C2×#2096C2semidirect
8. C2×#2068C2semidirect
9. C2×#2058C2semidirect
11. C2×#2024C2semidirect
12. C2×#1943C2semidirect
13. C2×#1927C2semidirect
15. C2×#1917C2semidirect
41. #2551C22semidirect
60. #2096C22semidirect
62. #2068C22semidirect
63. #2058C22semidirect
65. #2024C22semidirect
66. #1943C22semidirect
67. #1927C22semidirect
69. #1917C22semidirect
675. C2#30829direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4762
C2×#25512
C2×#24472
#308292
C2×#20962
C2×#20712
C2×#20682
C2×#20582
C2×#20422
C2×#20242
C2×#19432
C2×#19272
C2×#19252
C2×#19172
C2×#18642