Finite Group Databasebeta

← C2×#31235 #7503868

Subgroups — C2×#31235 #7503868

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3C2×#2454C211yesmaximal, characteristic
4C2×#2451C211yesmaximal, characteristic, complemented
5#31235C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2068C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2058C211yesmaximal, characteristic, complemented
12C2×#1968C211yesmaximal, characteristic
13C2×#1942C211yesmaximal, characteristic, complemented
14C2×#1939C211yesmaximal, characteristic, complemented
15C2×#1880C211yesmaximal, characteristic
16C2×#1877C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#479—28no—
23C2×#479C2218yescomplemented
24C2×#474—28no—
25C2×#474C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2454—24no—
42#2454C2218yes—
43#2451—24no—
44#2451C2218yescomplemented
45C2×#399—21no—
46C2×#398—21no—
47C2×#396—21no—
48C2×#395—21no—
49C2×#391—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#387—21no—
59C2×#385—21no—
60C2×#380—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2068C2218yescomplemented
64#2067C2218yes—
65#2065C2218yes—
66#2059C2218yescomplemented
67#2058C2218yescomplemented
68#1968C2218yes—
69#1942C2218yescomplemented
70#1939C2218yescomplemented
71#1880C2218yes—
72#1877C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16Q818yescomplemented
86C22×SD16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×#125—24no—
96C2×#125—22no—
97C2×#125—42no—
98C2×#125C2314yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—24no—
101C2×C4×Q8—42no—
102C2×C4×Q8C2314yes—
103C2×C4×D4—24no—
104C2×C4×D4—22no—
105C2×C4×D4—42no—
106C2×C4×D4C2314yes—
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⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#479—232no—
133#474—232no—
134C2×#107—24no—
135C2×#107—42no—
136C2×#107D418yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—42no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—42no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—42no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—42no—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C4×C8—42no—
163C2×C4×C8C2314yes—
164#399—28no—
165#398—28no—
166#396—28no—
167#395—28no—
168#391—22no—
169#391—22no—
170#391—28no—
171#391C2314yes—
172#391C2314yes—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#387—28no—
186#385—28no—
187#380—28no—
188C4×C4⋊C4D414yes—
189C4×C4⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—82no—
217C22×Q8C2×Q814yes—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2×Q814yes—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—416no—
246C2×SD16—44no—
247C2×SD16—44no—
248C2×SD16C2×Q8116yescomplemented
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×C8—22no—
259C22×C8C2×D414yes—
260C22×C8C2×Q814yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268#125—28no—
269#125—44no—
270#125—48no—
271C4×Q8—28no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×D4—28no—
275C4×D4—44no—
276C4×D4—48no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
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—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
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—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—82no—
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#107—28no—
346#107—44no—
347#107C2×D4116yes—
348#106—28no—
349#106—44no—
350C4⋊C8—28no—
351C4⋊C8—28no—
352C4⋊C8—44no—
353#103—28no—
354#103—28no—
355#103—44no—
356#102—28no—
357#102—28no—
358#102—44no—
359C4×C8—44no—
360#95—24no—
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—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—84no—
394C2×Q8—816no—
395C2×Q8C22×Q814yes—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4C22×Q814yes—
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—24no—
418C22×C4—24no—
419C22×C4—24no—
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—22no—
434C22×C4—22no—
435C22×C4—48no—
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—42no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—816no—
457C22×C4#13514yes—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×Q812yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473SD16—416no—
474C2×C8—28no—
475C2×C8—28no—
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—22no—
485C2×C8—22no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8C22×D414yes—
489C2×C8C22×Q814yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
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—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D414yes—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—48no—
533C23—24no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23#53712yes—
548C23D4212yes—
549C23D4×Q812yes—
550C23C2×#13512yes—
551C23C2×#13712yes—
552C23#57612yes—
553C23#57712yes—
554C23C23×D411yescharacteristic
555Q8—24no—
556Q8—24no—
557Q8—42no—
558Q8—48no—
559Q8—42no—
560Q8—816no—
561D4—24no—
562D4—24no—
563D4—42no—
564D4—48no—
565D4—42no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
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—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—816no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4#53714yes—
618C2×C4#53712yes—
619C2×C4#54314yes—
620C2×C4D4212yes—
621C2×C4D4×Q812yes—
622C2×C4D4×Q814yes—
623C2×C4#56314yes—
624C2×C4C2×#13514yes—
625C2×C4C2×#13714yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C23×Q811yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—24no—
633C8—24no—
634C8—42no—
635C8—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—816no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22#268314yes—
645C22#324814yes—
646C22#325714yes—
647C22#327814yes—
648C22#328014yes—
649C22#338114yes—
650C22#338514yes—
651C22C2×#53711yescharacteristic
652C22C2×D4211yescharacteristic
653C22C2×D4×Q811yescharacteristic
654C22C22×#13511yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57611yescharacteristic
657C22C2×#57711yescharacteristic
658C4—24no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#268314yes—
672C4C2×#53711yescharacteristic
673C4C2×#54311yescharacteristic
674C4C2×D4×Q811yescharacteristic
675C4C2×#56311yescharacteristic
676C2—84no—
677C2#3123518yesdirect product factor
678C2C2×#268311yescharacteristic
679C2C2×#324811yescharacteristic
680C2C2×#325711yescharacteristic
681C2C2×#327811yescharacteristic
682C2C2×#328011yescharacteristic
683C2C2×#338111yescharacteristic
684C2C2×#338511yescharacteristic
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. C22×#479C2semidirect
4. C2×#2451C2semidirect
5. #31235C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2068C2semidirect
10. C2×#2059C2semidirect
11. C2×#2058C2semidirect
13. C2×#1942C2semidirect
14. C2×#1939C2semidirect
23. C2×#479C22semidirect
44. #2451C22semidirect
62. #2068C22semidirect
63. #2068C22semidirect
66. #2059C22semidirect
67. #2058C22semidirect
69. #1942C22semidirect
70. #1939C22semidirect
85. C22×SD16Q8semidirect
248. C2×SD16C2×Q8semidirect
677. C2#31235direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4742
C2×#24542
C2×#24512
#312352
C2×#20682
C2×#20672
C2×#20652
C2×#20592
C2×#20582
C2×#19682
C2×#19422
C2×#19392
C2×#18802
C2×#18772