Finite Group Databasebeta

← C2×#30951 #7502420

Subgroups — C2×#30951 #7502420

Order 512.

2064 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×#2552C211yesmaximal, characteristic, complemented
4C2×#2454C211yesmaximal, characteristic
5#30951C2116yesmaximal, direct product factor
6C2×#2094C211yesmaximal, characteristic
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2083C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic
12C2×#1934C211yesmaximal, characteristic
13C2×#1925C211yesmaximal, characteristic
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1916C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#499—28no—
22C2×#499C2218yescomplemented
23C2×#478—28no—
24C2×#478C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2552—24no—
38#2552C2218yescomplemented
39#2454—24no—
40#2454C2218yes—
41C2×#399—21no—
42C2×#398—21no—
43C2×#398—21no—
44C2×#396—21no—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#385—21no—
53C2×#384—21no—
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2094C2218yes—
59#2088C2218yescomplemented
60#2083C2218yescomplemented
61#2059C2218yescomplemented
62#2059C2218yescomplemented
63#2025C2218yes—
64#1934C2218yes—
65#1925C2218yes—
66#1920C2218yescomplemented
67#1916C2218yescomplemented
68#1862C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#125—42no—
92C2×#125—42no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×#125C2314yes—
96C2×#125C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—42no—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
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—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#499—232no—
132#478—232no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106—24no—
140C2×#106—42no—
141C2×#106D418yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102Q818yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C4×C8—24no—
156C2×C4×C8C2314yes—
157#399—28no—
158#398—28no—
159#398—28no—
160#396—28no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—28no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#385—28no—
175#384—28no—
176#384—22no—
177#384C2314yes—
178#380—28no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D414yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
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—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2×Q814yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—82no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
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×C8—22no—
237C22×C8C2×D414yes—
238C22×C8C2×Q814yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243#125—28no—
244#125—28no—
245#125—28no—
246#125—44no—
247#125—44no—
248#125—44no—
249#125—48no—
250#125—48no—
251#125—48no—
252C4×D4—28no—
253C4×D4—44no—
254C4×D4—48no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×Q814yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—28no—
336#107—28no—
337#107—28no—
338#106—28no—
339#106—44no—
340#106C2×D4116yes—
341C4⋊C8—28no—
342#102—28no—
343#102—28no—
344#102—44no—
345#102C2×Q8116yes—
346C4×C8—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—24no—
355C24—22no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24C22×D411yescharacteristic
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4C22×Q814yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
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—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—48no—
404C22×C4—48no—
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—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—84no—
424C22×C4—816no—
425C22×C4#13514yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×Q812yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440C2×C8—28no—
441C2×C8—28no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8C22×D414yes—
455C2×C8C22×Q814yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4C22×D414yes—
489C4⋊C4C22×D414yes—
490C4⋊C4C22×Q814yes—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—48no—
506C23—24no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23#53612yes—
521C23D4212yes—
522C23D4×Q812yes—
523C23C2×#13512yes—
524C23#57512yes—
525C23#57612yes—
526C23#57712yes—
527C23C23×D411yescharacteristic
528D4—24no—
529D4—24no—
530D4—48no—
531D4—42no—
532D4—42no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—816no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#53612yes—
586C2×C4#53614yes—
587C2×C4#54414yes—
588C2×C4D4212yes—
589C2×C4D4×Q812yes—
590C2×C4D4×Q814yes—
591C2×C4#56314yes—
592C2×C4C2×#13514yes—
593C2×C4C23×D411yescharacteristic
594C2×C4C23×Q811yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×2−1+411yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—24no—
600C8—24no—
601C8—42no—
602C8—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—816no—
611C22#267714yes—
612C22#323214yes—
613C22#324114yes—
614C22#326814yes—
615C22#328414yes—
616C22#338014yes—
617C22#338814yes—
618C22C2×#53611yescharacteristic
619C22C2×D4211yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22C22×#13511yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57611yescharacteristic
624C22C2×#57711yescharacteristic
625C4—24no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#267714yes—
639C4C2×#53611yescharacteristic
640C4C2×#54411yescharacteristic
641C4C2×D4×Q811yescharacteristic
642C4C2×#56311yescharacteristic
643C2—84no—
644C2#3095118yesdirect product factor
645C2C2×#267711yescharacteristic
646C2C2×#323211yescharacteristic
647C2C2×#324111yescharacteristic
648C2C2×#326811yescharacteristic
649C2C2×#328411yescharacteristic
650C2C2×#338011yescharacteristic
651C2C2×#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. C22×#499C2semidirect
3. C2×#2552C2semidirect
5. #30951C2direct (N × H)
7. C2×#2088C2semidirect
8. C2×#2083C2semidirect
9. C2×#2059C2semidirect
10. C2×#2059C2semidirect
14. C2×#1920C2semidirect
15. C2×#1916C2semidirect
22. C2×#499C22semidirect
38. #2552C22semidirect
59. #2088C22semidirect
60. #2083C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
66. #1920C22semidirect
67. #1916C22semidirect
644. C2#30951direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4992
C22×#4782
C2×#25522
C2×#24542
#309512
C2×#20942
C2×#20882
C2×#20832
C2×#20592
C2×#20252
C2×#19342
C2×#19252
C2×#19202
C2×#19162
C2×#18622