Finite Group Databasebeta

← C2×#27018 #6445841

Subgroups — C2×#27018 #6445841

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#27018C2116yesmaximal, direct product factor
4C2×#2067C211yesmaximal, characteristic
5C2×#2059C211yesmaximal, characteristic, complemented
6C2×#2057C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1943C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic
11C2×#1925C211yesmaximal, characteristic
12C2×#1921C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2454—24no—
31#2454C2218yes—
32C2×#399—21no—
33C2×#396—21no—
34#2369—24no—
35#2369C2218yescomplemented
36C2×#391—21no—
37C2×#391—21no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46C2×#380—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2067C2218yes—
53#2059C2218yescomplemented
54#2057C2218yescomplemented
55#2023C2218yes—
56#1989C2218yescomplemented
57#1943C2218yescomplemented
58#1926C2218yes—
59#1925C2218yes—
60#1921C2218yescomplemented
61#1918C2218yescomplemented
62#1917C2218yescomplemented
63#1862C2218yes—
64#1762C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
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—24no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2314yes—
88C2×#125—24no—
89C2×#125—22no—
90C2×#125—42no—
91C2×#125C2314yes—
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⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—24no—
124C2×#107—24no—
125C2×#107—24no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—42no—
136C2×#102Q818yescomplemented
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#97—42no—
142C2×#97C2314yes—
143#399—28no—
144#396—28no—
145#391—28no—
146#391—28no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#389—28no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#385—28no—
161#380—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4D414yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×Q8—28no—
194C22×Q8—24no—
195C22×Q8—82no—
196C22×Q8—82no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C2×Q814yes—
204C22×D4C2412yes—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—82no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C22×C8—22no—
224C22×C8—22no—
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×C8C2×Q814yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—216no—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242#125—28no—
243#125—44no—
244#125—48no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—48no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
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—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42—82no—
315C2×C42—82no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—28no—
323#107—28no—
324#107—28no—
325#106—44no—
326C4⋊C8—44no—
327#102—28no—
328#102—28no—
329#102—44no—
330#102C2×Q8116yescomplemented
331#97—44no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—24no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24C22×D411yescharacteristic
350C2×Q8—216no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—816no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—28no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4C22×Q814yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
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—22no—
395C22×C4—48no—
396C22×C4—48no—
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—84no—
415C22×C4—816no—
416C22×C4#13714yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q812yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+412yes—
430C22×C4C2511yescharacteristic
431C2×C8—28no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
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—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8C22×Q814yes—
447C4⋊C4—216no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—28no—
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—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4C22×Q814yes—
474C4⋊C4C2×#4114yes—
475C4⋊C4C2×#4114yes—
476C4⋊C4C2×#4114yes—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C42—84no—
493C42—84no—
494C23—24no—
495C23—48no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23#52512yes—
509C23#54612yes—
510C23D4×Q812yes—
511C23C2×#13712yes—
512C23#57512yes—
513C23#57612yes—
514C23#57612yes—
515C23C23×D411yescharacteristic
516Q8—816no—
517D4—24no—
518D4—24no—
519D4—42no—
520D4—42no—
521D4—48no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
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—42no—
570C2×C4—816no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4#52512yes—
575C2×C4#53314yes—
576C2×C4#54214yes—
577C2×C4#54612yes—
578C2×C4D4×Q812yes—
579C2×C4#55414yes—
580C2×C4#55614yes—
581C2×C4C2×#13714yes—
582C2×C4C23×Q811yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C8—24no—
589C8—42no—
590C8—42no—
591C8—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—82no—
596C22—82no—
597C22—816no—
598C22—82no—
599C22—82no—
600C22#246514yes—
601C22#311714yes—
602C22#312214yes—
603C22#331914yes—
604C22#333314yes—
605C22#338514yes—
606C22#338914yes—
607C22C2×#52511yescharacteristic
608C22C2×#54611yescharacteristic
609C22C2×D4×Q811yescharacteristic
610C22C22×#13711yescharacteristic
611C22C2×#57511yescharacteristic
612C22C2×#57611yescharacteristic
613C22C2×#57611yescharacteristic
614C4—24no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#246514yes—
628C4C2×#53311yescharacteristic
629C4C2×#54211yescharacteristic
630C4C2×#55411yescharacteristic
631C4C2×#55611yescharacteristic
632C2—84no—
633C2#2701818yesdirect product factor
634C2C2×#246511yescharacteristic
635C2C2×#311711yescharacteristic
636C2C2×#312211yescharacteristic
637C2C2×#331911yescharacteristic
638C2C2×#333311yescharacteristic
639C2C2×#338511yescharacteristic
640C2C2×#338911yescharacteristic
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
2. C2×#2369C2semidirect
3. #27018C2direct (N × H)
5. C2×#2059C2semidirect
6. C2×#2057C2semidirect
8. C2×#1989C2semidirect
9. C2×#1943C2semidirect
12. C2×#1921C2semidirect
13. C2×#1918C2semidirect
14. C2×#1917C2semidirect
35. #2369C22semidirect
53. #2059C22semidirect
54. #2057C22semidirect
56. #1989C22semidirect
57. #1943C22semidirect
60. #1921C22semidirect
61. #1918C22semidirect
62. #1917C22semidirect
136. C2×#102Q8semidirect
330. #102C2×Q8semidirect
633. C2#27018direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#23692
#270182
C2×#20672
C2×#20592
C2×#20572
C2×#20232
C2×#19892
C2×#19432
C2×#19262
C2×#19252
C2×#19212
C2×#19182
C2×#19172
C2×#18622
C2×#17622