Finite Group Databasebeta

← C2×#29079 #6460599

Subgroups — C2×#29079 #6460599

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic, complemented
3#29079C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2073C211yesmaximal, characteristic, complemented
6C2×#2054C211yesmaximal, characteristic
7C2×#2023C211yesmaximal, characteristic, complemented
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1972C211yesmaximal, characteristic
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic
15C2×#1770C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2513—24no—
33#2513C2218yescomplemented
34#2452—24no—
35#2452C2218yescomplemented
36C2×#397—21no—
37C2×#397—21no—
38C2×#396—21no—
39C2×#393—21no—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#385—21no—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2088C2218yescomplemented
55#2073C2218yescomplemented
56#2054C2218yes—
57#2023C2218yescomplemented
58#1993C2218yescomplemented
59#1972C2218yes—
60#1936C2218yescomplemented
61#1934C2218yescomplemented
62#1934C2218yescomplemented
63#1920C2218yescomplemented
64#1879C2218yes—
65#1770C2218yescomplemented
66#1762C2218yescomplemented
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
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×D8—24no—
82C22×D8—24no—
83C22×D8—42no—
84C22×D8C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8C2314yes—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—48no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#106—24no—
127C2×#106C2314yes—
128C2×C4⋊C8—24no—
129C2×C4⋊C8C2314yes—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—42no—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#97—42no—
143C2×#97—42no—
144C2×#97C2314yes—
145C2×#97C2314yes—
146C2×C4×C8—24no—
147C2×C4×C8C2314yes—
148#397—28no—
149#397—28no—
150#396—28no—
151#393—28no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#388—28no—
169#388—28no—
170#385—28no—
171#382—22no—
172#382C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D414yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25—21no—
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—82no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4#4114yes—
215C22×D4C2412yes—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×D8—28no—
235C2×D8—28no—
236C2×D8—44no—
237C2×D8—416no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8#4114yes—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C22⋊Q8—432no—
258C4×D4—28no—
259C4×D4—44no—
260C4×D4—48no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
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—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
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—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—82no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333#106—28no—
334C4⋊C8—28no—
335#102—28no—
336#102—28no—
337#102—28no—
338#102—28no—
339#102—28no—
340#102—44no—
341#97—44no—
342#97—44no—
343C4×C8—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—84no—
372C2×Q8—816no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4C2×#4114yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—816no—
444C22×C4AGL(1,ℤ/8)14yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459D8—416no—
460C2×C8—28no—
461C2×C8—28no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8C2×#4114yes—
475C2×C8C2×#4114yes—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C4⋊C4C2×#4114yes—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—84no—
521C23—24no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23#53212yes—
550C23#54612yes—
551C23#54812yes—
552C23C2×AGL(1,ℤ/8)12yes—
553C23#57512yes—
554C23#57612yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—816no—
558D4—24no—
559D4—24no—
560D4—48no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—84no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
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—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
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—816no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4#52914yes—
617C2×C4#53212yes—
618C2×C4#53914yes—
619C2×C4#54612yes—
620C2×C4#54812yes—
621C2×C4#55114yes—
622C2×C4#55614yes—
623C2×C4C2×AGL(1,ℤ/8)14yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—24no—
631C8—24no—
632C8—42no—
633C8—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22—816no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—168no—
650C22#256014yes—
651C22#317914yes—
652C22#318014yes—
653C22#332514yes—
654C22#332614yes—
655C22#337014yes—
656C22#337314yes—
657C22C2×#53211yescharacteristic
658C22C2×#54611yescharacteristic
659C22C2×#54811yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57611yescharacteristic
663C22C2×#57711yescharacteristic
664C4—24no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#256014yes—
677C4C2×#52911yescharacteristic
678C4C2×#53911yescharacteristic
679C4C2×#55111yescharacteristic
680C4C2×#55611yescharacteristic
681C2—84no—
682C2—162no—
683C2#2907918yesdirect product factor
684C2C2×#256011yescharacteristic
685C2C2×#317911yescharacteristic
686C2C2×#318011yescharacteristic
687C2C2×#332511yescharacteristic
688C2C2×#332611yescharacteristic
689C2C2×#337311yescharacteristic
690C2C2×#337011yescharacteristic
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×#2513C2semidirect
2. C2×#2452C2semidirect
3. #29079C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2073C2semidirect
7. C2×#2023C2semidirect
8. C2×#1993C2semidirect
10. C2×#1936C2semidirect
11. C2×#1934C2semidirect
12. C2×#1934C2semidirect
13. C2×#1920C2semidirect
15. C2×#1770C2semidirect
16. C2×#1762C2semidirect
33. #2513C22semidirect
35. #2452C22semidirect
54. #2088C22semidirect
55. #2073C22semidirect
57. #2023C22semidirect
58. #1993C22semidirect
60. #1936C22semidirect
61. #1934C22semidirect
62. #1934C22semidirect
63. #1920C22semidirect
65. #1770C22semidirect
66. #1762C22semidirect
683. C2#29079direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#24522
#290792
C2×#20882
C2×#20732
C2×#20542
C2×#20232
C2×#19932
C2×#19722
C2×#19362
C2×#19342
C2×#19202
C2×#18792
C2×#17702
C2×#17622