Finite Group Databasebeta

← C2×#26795 #6443752

Subgroups — C2×#26795 #6443752

Order 512.

2294 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3C2×#2492C211yesmaximal, characteristic
4C2×#2362C211yesmaximal, characteristic
5#26795C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#2035C211yesmaximal, characteristic
9C2×#2001C211yesmaximal, characteristic
10C2×#1969C211yesmaximal, characteristic, complemented
11C2×#1928C211yesmaximal, characteristic, complemented
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic
16C2×C4×#106C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#485—28no—
24C2×#485C2218yes—
25C2×#475—28no—
26C2×#475C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2492—24no—
43#2492C2218yes—
44C2×#400—21no—
45C2×#393—21no—
46#2362—24no—
47#2362C2218yes—
48C2×#391—21no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#388—21no—
56C2×#387—21no—
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2066C2218yescomplemented
63#2057C2218yescomplemented
64#2035C2218yes—
65#2001C2218yes—
66#1969C2218yescomplemented
67#1928C2218yescomplemented
68#1927C2218yescomplemented
69#1923C2218yes—
70#1878C2218yescomplemented
71#1854C2218yes—
72C4×#106C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—42no—
86C22×Q16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—42no—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—42no—
106C2×C4×Q8C2314yes—
107C2×#117—22no—
108C2×#117—24no—
109C2×#117—42no—
110C2×#117C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#485—232no—
135#475—232no—
136C2×#107—24no—
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106D418yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×C22⋊C8—24no—
161C2×C22⋊C8—24no—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2314yes—
166#400—28no—
167#393—28no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—28no—
172#391C2314yes—
173#391C2314yes—
174#391C2314yes—
175#389—22no—
176#389C2314yes—
177#388—28no—
178#388—28no—
179#388—28no—
180#387—28no—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—28no—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4D414yes—
189C4×C22⋊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×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25C2×D411yescharacteristic
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—82no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—416no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8C2×D414yes—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274C22⋊Q8—216no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—44no—
279C22⋊Q8—48no—
280C4×Q8—28no—
281C4×Q8—44no—
282C4×Q8—48no—
283#117—28no—
284#117—44no—
285#117—48no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4#4114yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42—82no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—28no—
349#107—44no—
350#106—28no—
351#106—28no—
352#106C2×D4116yes—
353C4⋊C8—28no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—28no—
359#103—28no—
360C22⋊C8—28no—
361C22⋊C8—28no—
362C4×C8—28no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C22×D412yes—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—48no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—84no—
412C2×Q8—816no—
413C2×Q8C2×#4114yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
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—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
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—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4#13514yes—
473C22×C4#13714yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487Q16—416no—
488C2×C8—28no—
489C2×C8—28no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—44no—
502C2×C8C22×D414yes—
503C2×C8C2×#4114yes—
504C4⋊C4—216no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4C22×D414yes—
525C4⋊C4C22×D414yes—
526C4⋊C4C2×#4114yes—
527C22⋊C4—216no—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—48no—
540C42—48no—
541C42—48no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23#52512yes—
558C23D4212yes—
559C23#54712yes—
560C23C2×#13512yes—
561C23C2×#13712yes—
562C23#57712yes—
563C23#57812yes—
564C23C23×D411yescharacteristic
565Q8—24no—
566Q8—24no—
567Q8—42no—
568Q8—42no—
569Q8—48no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—816no—
575Q8—84no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
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—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
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—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4#52512yes—
632C2×C4#53214yes—
633C2×C4#54214yes—
634C2×C4D4212yes—
635C2×C4#54712yes—
636C2×C4#54814yes—
637C2×C4#55114yes—
638C2×C4C2×#13514yes—
639C2×C4C2×#13714yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C8—24no—
647C8—24no—
648C8—42no—
649C8—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—88no—
658C22#246114yes—
659C22#310714yes—
660C22#312114yes—
661C22#327614yes—
662C22#328214yes—
663C22#334214yes—
664C22#334414yes—
665C22C2×#52511yescharacteristic
666C22C2×D4211yescharacteristic
667C22C2×#54711yescharacteristic
668C22C22×#13511yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57811yescharacteristic
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#246114yes—
686C4C2×#53211yescharacteristic
687C4C2×#54211yescharacteristic
688C4C2×#54811yescharacteristic
689C4C2×#55111yescharacteristic
690C2—82no—
691C2#2679518yesdirect product factor
692C2C2×#246111yescharacteristic
693C2C2×#310711yescharacteristic
694C2C2×#312111yescharacteristic
695C2C2×#327611yescharacteristic
696C2C2×#328211yescharacteristic
697C2C2×#334211yescharacteristic
698C2C2×#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. #26795C2direct (N × H)
6. C2×#2066C2semidirect
7. C2×#2057C2semidirect
10. C2×#1969C2semidirect
11. C2×#1928C2semidirect
12. C2×#1927C2semidirect
14. C2×#1878C2semidirect
16. C2×C4×#106C2semidirect
26. C2×#475C22semidirect
62. #2066C22semidirect
63. #2057C22semidirect
66. #1969C22semidirect
67. #1928C22semidirect
68. #1927C22semidirect
70. #1878C22semidirect
72. C4×#106C22semidirect
691. C2#26795direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4752
C2×#24922
C2×#23622
#267952
C2×#20662
C2×#20572
C2×#20352
C2×#20012
C2×#19692
C2×#19282
C2×#19272
C2×#19232
C2×#18782
C2×#18542
C2×C4×#1062