Finite Group Databasebeta

← C2×#32104 #7507654

Subgroups — C2×#32104 #7507654

Order 512.

2188 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2544C211yesmaximal, characteristic
2C2×#2506C211yesmaximal, characteristic, complemented
3#32104C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2042C211yesmaximal, characteristic, complemented
7C2×#2042C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic, complemented
9C2×#2041C211yesmaximal, characteristic
10C2×#1982C211yesmaximal, characteristic
11C2×#1959C211yesmaximal, characteristic, complemented
12C2×#1936C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125—21no—
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2544—24no—
39#2544C2218yes—
40#2506—24no—
41#2506C2218yescomplemented
42C2×#400—21no—
43C2×#398—21no—
44C2×#394—21no—
45C2×#394—21no—
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#386—21no—
54C2×#386—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2094C2218yescomplemented
59#2066C2218yescomplemented
60#2042C2218yescomplemented
61#2042C2218yescomplemented
62#2042C2218yescomplemented
63#2041C2218yes—
64#1982C2218yes—
65#1959C2218yescomplemented
66#1936C2218yescomplemented
67#1927C2218yescomplemented
68#1917C2218yescomplemented
69#1864C2218yescomplemented
70#1864C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#127—24no—
88C2×#127—22no—
89C2×#127—42no—
90C2×#127C2314yes—
91C2×#125—28no—
92C2×#125—48no—
93C2×#124—24no—
94C2×#124—22no—
95C2×#124—22no—
96C2×#124—42no—
97C2×#124—42no—
98C2×#124—42no—
99C2×#124C2314yes—
100C2×#124C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#106—42no—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#103—24no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#97—42no—
160C2×#97C2314yes—
161#400—28no—
162#398—28no—
163#394—28no—
164#394—28no—
165#391—22no—
166#391C2314yes—
167#390—22no—
168#390C2314yes—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#387—28no—
178#386—28no—
179#386—28no—
180#384—22no—
181#384C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4D414yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
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#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268#127—28no—
269#127—44no—
270#127—48no—
271#125—432no—
272#124—28no—
273#124—44no—
274#124—44no—
275#124—44no—
276#124—48no—
277#124—48no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
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—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4#4114yes—
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—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#106—44no—
346#106—44no—
347C4⋊C8—44no—
348#103—28no—
349#103—44no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360#97—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C22×D411yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—816no—
432C2×D4—816no—
433C2×D4C2×#4114yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—816no—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—44no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8—44no—
510C4⋊C4—24no—
511C4⋊C4—28no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—48no—
532C4⋊C4C2×#4114yes—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—48no—
550C23—24no—
551C23—42no—
552C23—42no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23#53512yes—
579C23#54612yes—
580C23#54812yes—
581C23C2×AGL(1,ℤ/8)12yes—
582C23#57512yes—
583C23#57712yes—
584C23#57712yes—
585C23C23×D411yescharacteristic
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—84no—
591D4—24no—
592D4—24no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—48no—
598D4—42no—
599D4—42no—
600D4—816no—
601D4—816no—
602D4—84no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—816no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4#53512yes—
652C2×C4#53914yes—
653C2×C4#54014yes—
654C2×C4#54612yes—
655C2×C4#54812yes—
656C2×C4#55314yes—
657C2×C4#56214yes—
658C2×C4C2×AGL(1,ℤ/8)14yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C8—42no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—816no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—168no—
685C22#271114yes—
686C22#321414yes—
687C22#322214yes—
688C22#330614yes—
689C22#332314yes—
690C22#337114yes—
691C22#337314yes—
692C22C2×#53511yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54811yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57711yescharacteristic
699C4—24no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#271114yes—
712C4C2×#53911yescharacteristic
713C4C2×#54011yescharacteristic
714C4C2×#55311yescharacteristic
715C4C2×#56211yescharacteristic
716C2—84no—
717C2—162no—
718C2#3210418yesdirect product factor
719C2C2×#271111yescharacteristic
720C2C2×#321411yescharacteristic
721C2C2×#322211yescharacteristic
722C2C2×#330611yescharacteristic
723C2C2×#332311yescharacteristic
724C2C2×#337311yescharacteristic
725C2C2×#337111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. C2×#2506C2semidirect
3. #32104C2direct (N × H)
4. C2×#2094C2semidirect
5. C2×#2066C2semidirect
6. C2×#2042C2semidirect
7. C2×#2042C2semidirect
8. C2×#2042C2semidirect
11. C2×#1959C2semidirect
12. C2×#1936C2semidirect
13. C2×#1927C2semidirect
14. C2×#1917C2semidirect
15. C2×#1864C2semidirect
16. C2×#1864C2semidirect
41. #2506C22semidirect
58. #2094C22semidirect
59. #2066C22semidirect
60. #2042C22semidirect
61. #2042C22semidirect
62. #2042C22semidirect
65. #1959C22semidirect
66. #1936C22semidirect
67. #1927C22semidirect
68. #1917C22semidirect
69. #1864C22semidirect
70. #1864C22semidirect
718. C2#32104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25442
C2×#25062
#321042
C2×#20942
C2×#20662
C2×#20422
C2×#20412
C2×#19822
C2×#19592
C2×#19362
C2×#19272
C2×#19172
C2×#18642