Finite Group Databasebeta

← C2×#31538 #7504985

Subgroups — C2×#31538 #7504985

Order 512.

2340 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic, complemented
3#31538C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic, complemented
5C2×#2084C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2058C211yesmaximal, characteristic, complemented
8C2×#2025C211yesmaximal, characteristic, complemented
9C2×#2023C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1972C211yesmaximal, characteristic
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1879C211yesmaximal, characteristic
14C2×#1879C211yesmaximal, characteristic
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2542—24no—
36#2542C2218yescomplemented
37#2454—24no—
38#2454C2218yescomplemented
39C2×#399—21no—
40C2×#399—21no—
41C2×#399—21no—
42C2×#396—21no—
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#385—21no—
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2094C2218yescomplemented
55#2084C2218yescomplemented
56#2065C2218yescomplemented
57#2058C2218yescomplemented
58#2025C2218yescomplemented
59#2023C2218yescomplemented
60#1992C2218yescomplemented
61#1972C2218yes—
62#1917C2218yescomplemented
63#1879C2218yes—
64#1879C2218yes—
65#1864C2218yescomplemented
66#1802C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4—21no—
82C22×D8—42no—
83C22×D8—42no—
84C22×D8Q818yescomplemented
85C22×D8C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8C2314yes—
94C2×#125—24no—
95C2×#125—22no—
96C2×#125—42no—
97C2×#125C2314yes—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C2×C22⋊Q8—48no—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—42no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C2×#106—42no—
141C2×#106—42no—
142C2×#106—42no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—42no—
147C2×C4⋊C8C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102—42no—
153C2×#102—42no—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#97—42no—
161C2×#97C2314yes—
162#399—28no—
163#399—28no—
164#399—28no—
165#396—28no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#385—28no—
179#384—22no—
180#384—22no—
181#384C2314yes—
182#384C2314yes—
183#380—28no—
184C4×C4⋊C4D414yes—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25—21no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2×Q814yes—
219C22×D4C2×Q814yes—
220C22×D4C2412yes—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×D8—416no—
244C2×D8—44no—
245C2×D8—44no—
246C2×D8C2×Q8116yescomplemented
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×C8—22no—
258C22×C8C2×Q814yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266#125—28no—
267#125—44no—
268#125—48no—
269C22⋊Q8—432no—
270C22⋊Q8—432no—
271C4×D4—28no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C4×D4—48no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
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—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
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—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4#4114yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#106—44no—
356#106—44no—
357#106—44no—
358C4⋊C8—44no—
359#102—28no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#97—44no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
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—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—816no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4C22×Q814yes—
416C2×D4C22×Q814yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
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—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—84no—
463C22×C4—816no—
464C22×C4—816no—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481D8—416no—
482C2×C8—28no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8C22×Q814yes—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C2×#4114yes—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C23—24no—
547C23—24no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23#53512yes—
575C23#54612yes—
576C23D4×Q812yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23C2×AGL(1,ℤ/8)12yes—
579C23#57712yes—
580C23#57712yes—
581C23C23×D411yescharacteristic
582Q8—816no—
583D4—24no—
584D4—24no—
585D4—24no—
586D4—24no—
587D4—42no—
588D4—42no—
589D4—48no—
590D4—48no—
591D4—42no—
592D4—42no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—816no—
635C2×C4—816no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#53512yes—
642C2×C4#54214yes—
643C2×C4#54314yes—
644C2×C4#54612yes—
645C2×C4D4×Q812yes—
646C2×C4#55414yes—
647C2×C4#56214yes—
648C2×C4C2×AGL(1,ℤ/8)14yes—
649C2×C4C2×AGL(1,ℤ/8)14yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C8—24no—
657C8—42no—
658C8—42no—
659C8—42no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—816no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#269214yes—
677C22#321714yes—
678C22#322114yes—
679C22#330614yes—
680C22#333214yes—
681C22#338414yes—
682C22#338414yes—
683C22C2×#53511yescharacteristic
684C22C2×#54611yescharacteristic
685C22C2×D4×Q811yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C2×#57711yescharacteristic
689C22C2×#57711yescharacteristic
690C4—24no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#269214yes—
703C4C2×#54211yescharacteristic
704C4C2×#54311yescharacteristic
705C4C2×#55411yescharacteristic
706C4C2×#56211yescharacteristic
707C2—84no—
708C2—84no—
709C2#3153818yesdirect product factor
710C2C2×#269211yescharacteristic
711C2C2×#321711yescharacteristic
712C2C2×#322111yescharacteristic
713C2C2×#330611yescharacteristic
714C2C2×#333211yescharacteristic
715C2C2×#338411yescharacteristic
716C2C2×#338411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. C2×#2542C2semidirect
2. C2×#2454C2semidirect
3. #31538C2direct (N × H)
4. C2×#2094C2semidirect
5. C2×#2084C2semidirect
6. C2×#2065C2semidirect
7. C2×#2058C2semidirect
8. C2×#2025C2semidirect
9. C2×#2023C2semidirect
10. C2×#1992C2semidirect
12. C2×#1917C2semidirect
15. C2×#1864C2semidirect
16. C2×#1802C2semidirect
36. #2542C22semidirect
38. #2454C22semidirect
54. #2094C22semidirect
55. #2084C22semidirect
56. #2065C22semidirect
57. #2058C22semidirect
58. #2025C22semidirect
59. #2023C22semidirect
60. #1992C22semidirect
62. #1917C22semidirect
65. #1864C22semidirect
66. #1802C22semidirect
84. C22×D8Q8semidirect
246. C2×D8C2×Q8semidirect
709. C2#31538direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24542
#315382
C2×#20942
C2×#20842
C2×#20652
C2×#20582
C2×#20252
C2×#20232
C2×#19922
C2×#19722
C2×#19172
C2×#18792
C2×#18642
C2×#18022