Finite Group Databasebeta

← C2×#31866 #7506492

Subgroups — C2×#31866 #7506492

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2506C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#31866C2116yesmaximal, direct product factor
4C2×#2086C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2067C211yesmaximal, characteristic
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1939C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic
14C2×#1925C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2506—24no—
34#2506C2218yescomplemented
35#2452—24no—
36#2452C2218yes—
37C2×#400—21no—
38C2×#398—21no—
39C2×#396—21no—
40C2×#393—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#385—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2086C2218yescomplemented
55#2086C2218yescomplemented
56#2083C2218yescomplemented
57#2067C2218yes—
58#2067C2218yes—
59#2057C2218yescomplemented
60#1988C2218yescomplemented
61#1942C2218yescomplemented
62#1939C2218yescomplemented
63#1934C2218yes—
64#1925C2218yes—
65#1862C2218yes—
66#1802C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#125—28no—
89C2×#125—48no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—24no—
93C2×C4×D4—22no—
94C2×C4×D4—42no—
95C2×C4×D4C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—24no—
124C2×#107—42no—
125C2×#107—42no—
126C2×#107—42no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8—42no—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102Q818yescomplemented
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C4×C8—42no—
146C2×C4×C8C2314yes—
147#400—28no—
148#398—28no—
149#396—28no—
150#393—28no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#387—28no—
168#385—28no—
169#384—22no—
170#384C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D414yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—24no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—82no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2×Q814yes—
206C22×D4C2412yes—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—82no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2411yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8C2×Q814yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—28no—
242C4⋊Q8—44no—
243C4⋊Q8—48no—
244#125—432no—
245C22⋊Q8—432no—
246C4×D4—28no—
247C4×D4—44no—
248C4×D4—48no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4D818yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×Q814yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—28no—
324#107—44no—
325#107—44no—
326#107—44no—
327C4⋊C8—44no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330#102—44no—
331#102—44no—
332#102—44no—
333#102C2×Q8116yescomplemented
334C4×C8—44no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C24—24no—
345C24—22no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24C22×D411yescharacteristic
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—84no—
356C2×Q8—816no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—28no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4C22×Q814yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—84no—
416C22×C4—816no—
417C22×C4C2×D814yes—
418C22×C4C2×D814yes—
419C22×C4C2×D814yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q812yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C42−1+412yes—
433C22×C4C2511yescharacteristic
434C2×C8—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8C22×Q814yes—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4C2×D8116yes—
480C4⋊C4C22×Q814yes—
481C4⋊C4C2×#4114yes—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—48no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#53712yes—
512C23#54612yes—
513C23D4×Q812yes—
514C23C22×D812yes—
515C23#57512yes—
516C23#57712yes—
517C23#57812yes—
518C23C23×D411yescharacteristic
519Q8—816no—
520D4—24no—
521D4—24no—
522D4—42no—
523D4—48no—
524D4—42no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—816no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#53712yes—
576C2×C4#54114yes—
577C2×C4#54414yes—
578C2×C4#54612yes—
579C2×C4D4×Q812yes—
580C2×C4#55414yes—
581C2×C4#56214yes—
582C2×C4C22×D814yes—
583C2×C4C22×D814yes—
584C2×C4C22×D814yes—
585C2×C4C23×Q811yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C8—24no—
592C8—42no—
593C8—42no—
594C8—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—816no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#270414yes—
604C22#324514yes—
605C22#325814yes—
606C22#329414yes—
607C22#332714yes—
608C22Q8×D814yes—
609C22#338814yes—
610C22C2×#53711yescharacteristic
611C22C2×#54611yescharacteristic
612C22C2×D4×Q811yescharacteristic
613C22C23×D811yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57711yescharacteristic
616C22C2×#57811yescharacteristic
617C4—24no—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#270414yes—
631C4C2×#54111yescharacteristic
632C4C2×#54411yescharacteristic
633C4C2×#55411yescharacteristic
634C4C2×#56211yescharacteristic
635C2—84no—
636C2#3186618yesdirect product factor
637C2C2×#270411yescharacteristic
638C2C2×#324511yescharacteristic
639C2C2×#325811yescharacteristic
640C2C2×#329411yescharacteristic
641C2C2×#332711yescharacteristic
642C2C2×Q8×D811yescharacteristic
643C2C2×#338811yescharacteristic
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
1. C2×#2506C2semidirect
3. #31866C2direct (N × H)
4. C2×#2086C2semidirect
5. C2×#2086C2semidirect
6. C2×#2083C2semidirect
9. C2×#2057C2semidirect
10. C2×#1988C2semidirect
11. C2×#1942C2semidirect
12. C2×#1939C2semidirect
34. #2506C22semidirect
54. #2086C22semidirect
55. #2086C22semidirect
56. #2083C22semidirect
59. #2057C22semidirect
60. #1988C22semidirect
61. #1942C22semidirect
62. #1939C22semidirect
140. C2×#102Q8semidirect
333. #102C2×Q8semidirect
636. C2#31866direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25062
C2×#24522
#318662
C2×#20862
C2×#20832
C2×#20672
C2×#20572
C2×#19882
C2×#19422
C2×#19392
C2×#19342
C2×#19252
C2×#18622
C2×#18022