Finite Group Databasebeta

← C2×#31089 #7503390

Subgroups — C2×#31089 #7503390

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3C2×#2423C211yesmaximal, characteristic
4C2×#2440C211yesmaximal, characteristic
5#31089C2116yesmaximal, direct product factor
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2084C211yesmaximal, characteristic, complemented
8C2×#2077C211yesmaximal, characteristic
9C2×#2072C211yesmaximal, characteristic, complemented
10C2×#2058C211yesmaximal, characteristic, complemented
11C2×#2035C211yesmaximal, characteristic
12C2×#1933C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1853C211yesmaximal, characteristic
16C2×#1844C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#483—28no—
23C2×#483C2218yes—
24C2×#475—28no—
25C2×#475C2218yescomplemented
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2440—24no—
41#2440C2218yes—
42#2423—24no—
43#2423C2218yes—
44C2×#399—21no—
45C2×#398—21no—
46C2×#398—21no—
47C2×#395—21no—
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#387—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2088C2218yescomplemented
61#2084C2218yescomplemented
62#2077C2218yes—
63#2072C2218yescomplemented
64#2058C2218yescomplemented
65#2035C2218yes—
66#1933C2218yescomplemented
67#1923C2218yes—
68#1865C2218yescomplemented
69#1853C2218yes—
70#1844C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2314yes—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—24no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#483—232no—
132#475—232no—
133C2×#106—24no—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106—24no—
137C2×#106—42no—
138C2×#106—42no—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2314yes—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103D418yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161#399—28no—
162#398—28no—
163#398—28no—
164#395—28no—
165#391—22no—
166#391C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#387—28no—
175#385—28no—
176#384—22no—
177#384—22no—
178#384C2314yes—
179#384C2314yes—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184C4×C22⋊C4D414yes—
185C4×C22⋊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#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25C2×D411yescharacteristic
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—82no—
214C22×Q8C2×D414yes—
215C22×Q8C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8C2×D414yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264C22⋊Q8—432no—
265C4×Q8—28no—
266C4×Q8—44no—
267C4×Q8—48no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4—44no—
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⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4SD1618yes—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341#106—28no—
342#106—28no—
343#106—28no—
344#106—28no—
345#106—44no—
346#106—44no—
347C4⋊C8—28no—
348#103—28no—
349#103—28no—
350#103—28no—
351#103C2×D4116yes—
352C22⋊C8—28no—
353C22⋊C8—28no—
354C4×C8—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24C2×SD1614yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—84no—
388C2×Q8—816no—
389C2×Q8C22×D414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
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—84no—
450C22×C4—816no—
451C22×C4C2×SD1614yes—
452C22×C4C2×SD1614yes—
453C22×C4#13714yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8C22×D414yes—
482C2×C8C2×#4114yes—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×D414yes—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4C2×SD16116yes—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—48no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23#53612yes—
541C23D4212yes—
542C23#54712yes—
543C23C22×SD1614yes—
544C23C22×SD1612yes—
545C23C2×#13712yes—
546C23#57712yes—
547C23#57712yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554Q8—816no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
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—816no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4#53614yes—
611C2×C4#53612yes—
612C2×C4#53914yes—
613C2×C4D4212yes—
614C2×C4#54712yes—
615C2×C4#54714yes—
616C2×C4#56314yes—
617C2×C4C22×SD1614yes—
618C2×C4C22×SD1614yes—
619C2×C4C2×#13714yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4C2×2−1+411yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—24no—
627C8—24no—
628C8—42no—
629C8—42no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—416no—
636C22—42no—
637C22—42no—
638C22#268014yes—
639C22#322914yes—
640C22#324414yes—
641C22#327814yes—
642C22D4×SD1614yes—
643C22#333514yes—
644C22#334414yes—
645C22C2×#53611yescharacteristic
646C22C2×D4211yescharacteristic
647C22C2×#54711yescharacteristic
648C22C23×SD1611yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57711yescharacteristic
651C22C2×#57711yescharacteristic
652C4—24no—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#268014yes—
666C4C2×#53611yescharacteristic
667C4C2×#53911yescharacteristic
668C4C2×#54711yescharacteristic
669C4C2×#56311yescharacteristic
670C2—44no—
671C2#3108918yesdirect product factor
672C2C2×#268011yescharacteristic
673C2C2×#322911yescharacteristic
674C2C2×#324411yescharacteristic
675C2C2×#327811yescharacteristic
676C2C2×D4×SD1611yescharacteristic
677C2C2×#333511yescharacteristic
678C2C2×#334411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#475C2semidirect
5. #31089C2direct (N × H)
6. C2×#2088C2semidirect
7. C2×#2084C2semidirect
9. C2×#2072C2semidirect
10. C2×#2058C2semidirect
12. C2×#1933C2semidirect
14. C2×#1865C2semidirect
16. C2×#1844C2semidirect
25. C2×#475C22semidirect
60. #2088C22semidirect
61. #2084C22semidirect
63. #2072C22semidirect
64. #2058C22semidirect
66. #1933C22semidirect
68. #1865C22semidirect
70. #1844C22semidirect
671. C2#31089direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4752
C2×#24232
C2×#24402
#310892
C2×#20882
C2×#20842
C2×#20772
C2×#20722
C2×#20582
C2×#20352
C2×#19332
C2×#19232
C2×#18652
C2×#18532
C2×#18442