Finite Group Databasebeta

← C2×#8382 #185621

Subgroups — C2×#8382 #185621

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#1823C212yesmaximal, complemented
3C2×#1799C211yesmaximal, characteristic
4C2×#1520C212yesmaximal, complemented
5C2×#1459C211yesmaximal, characteristic
6#8382C218yesmaximal, direct product factor
7C2×#552—24no—
8C2×#552—24no—
9C2×#552—22no—
10C2×#552—22no—
11C2×#552—24no—
12C2×#552—24no—
13C2×#552C414yescomplemented
14C2×#552C2214yes—
15C22×#126—21no—
16C22×#125—21no—
17C22×#125C411yescharacteristic, complemented
18C22×#123—21no—
19C22×C22⋊Q8C411yescharacteristic, complemented
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C411yescharacteristic, complemented
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C22×#117C411yescharacteristic, complemented
25C23×C4⋊C4C2211yescharacteristic
26C22×#99—21no—
27C22×C22⋊C8C2212yes—
28C2×#380C2212yes—
29#1823—24no—
30#1823C2218yescomplemented
31#1799—22no—
32#1799C2214yes—
33C2×#342—21no—
34C2×#340—24no—
35#1520—24no—
36#1520C2218yescomplemented
37C2×#328—24no—
38#1459—22no—
39#1459C2214yes—
40C2×#324—21no—
41C2×#323—21no—
42C23×Q8C2×C411yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48#552—216no—
49#552—216no—
50#552—216no—
51#552—216no—
52#552—216no—
53#552—216no—
54#552—216no—
55#552—216no—
56C2×#126—28no—
57C2×#126—28no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#125—28no—
61C2×#125—28no—
62C2×#125—24no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—24no—
68C2×#125—22no—
69C2×#125C2×C414yescomplemented
70C2×#123—28no—
71C2×#123—28no—
72C2×#123—24no—
73C2×#123—24no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8C2×C412yes—
84C2×C22⋊Q8C2×C414yescomplemented
85C2×C22⋊Q8D414yes—
86C2×C22⋊Q8D414yes—
87C2×C22⋊Q8C2312yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2×C414yescomplemented
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2×C412yes—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117C2×C414yescomplemented
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2×C411yescharacteristic
124C22×C4⋊C4C2×C411yescharacteristic
125C22×C4⋊C4C2×C412yes—
126C22×C4⋊C4C2×C411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D414yes—
129C22×C4⋊C4D414yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2×C411yescharacteristic
140C2×#99—24no—
141C2×#99—44no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8—22no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8C2314yes—
148#380—22no—
149#380—22no—
150#380—22no—
151#380C2314yes—
152C2×#95—22no—
153#342—24no—
154#340—216no—
155#328—216no—
156#324—24no—
157#323—24no—
158C25C22⋊C411yescharacteristic
159C25C2×D411yescharacteristic
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—44no—
164C22×Q8C22⋊C412yes—
165C22×Q8C22⋊C412yes—
166C22×Q8C22×C412yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4C22⋊C412yes—
173C22×D4C2×D412yes—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C412yes—
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8—24no—
208C22×C8—48no—
209C22×C8—42no—
210C22×C8—42no—
211#126—232no—
212#126—232no—
213#125—232no—
214#125—232no—
215#125—28no—
216#125—28no—
217#125—28no—
218#125—28no—
219#123—232no—
220#123—232no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8C22⋊C418yes—
232C22⋊Q8C2×D418yes—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—28no—
236C4×Q8—28no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245#117—28no—
246#117—28no—
247#117—232no—
248#117—232no—
249#117—28no—
250#117—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4C22⋊C412yes—
292C2×C4⋊C4C22⋊C412yes—
293C2×C4⋊C4C22⋊C412yes—
294C2×C4⋊C4C22⋊C412yes—
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C418yes—
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D418yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4C22⋊C412yes—
323C2×C22⋊C4C22⋊C412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42C22⋊C412yes—
337C2×C42C22×C412yes—
338#99—48no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343#95—24no—
344#95—42no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C412yes—
355C24C2×C22⋊C411yescharacteristic
356C24#12012yes—
357C24#12012yes—
358C24#13514yes—
359C24AGL(1,ℤ/8)12yes—
360C24#13712yes—
361C24C22×D411yescharacteristic
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8C2×C22⋊C414yes—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C412yes—
453C22×C4C2×C22⋊C412yes—
454C22×C4C2×C22⋊C412yes—
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4#12012yes—
458C22×C4#12012yes—
459C22×C4#12012yes—
460C22×C4#12012yes—
461C22×C4#12012yes—
462C22×C4#12012yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C2×C8—48no—
468C2×C8—44no—
469C2×C8—84no—
470C2×C8—84no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4C2×C22⋊C414yes—
493C4⋊C4C2×C22⋊C414yes—
494C4⋊C4C2×C22⋊C414yes—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C42—216no—
507C42—216no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—48no—
525C23—44no—
526C23—44no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23#37912yes—
539C23#37911yescharacteristic
540C23#37911yescharacteristic
541C23#41612yes—
542C23#41612yes—
543C23#41712yes—
544C23#41712yes—
545C23#44914yes—
546C23#44914yes—
547C23#45214yes—
548C23#45214yes—
549C23C22×C22⋊C411yescharacteristic
550C23C2×#12012yes—
551C23C2×#13512yes—
552C23C2×AGL(1,ℤ/8)11yescharacteristic
553C23C2×#13711yescharacteristic
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558D4—44no—
559D4—44no—
560D4—48no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—416no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—168no—
620C2×C4#37911yescharacteristic
621C2×C4#37912yes—
622C2×C4#37911yescharacteristic
623C2×C4#41014yes—
624C2×C4C22×C22⋊C411yescharacteristic
625C2×C4C22×C22⋊C411yescharacteristic
626C2×C4C2×#12012yes—
627C8—84no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—22no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22#178212yes—
655C22#178812yes—
656C22#178814yes—
657C22#178812yes—
658C22#179612yes—
659C22C2×#37911yescharacteristic
660C22C2×#41611yescharacteristic
661C22C2×#41711yescharacteristic
662C22C2×#44912yes—
663C22C2×#45212yes—
664C4—22no—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—42no—
674C4—44no—
675C4—164no—
676C4#178212yes—
677C4C2×#41012yes—
678C2—22no—
679C2—21no—
680C2—21no—
681C2—22no—
682C2—22no—
683C2—24no—
684C2—82no—
685C2#838214yesdirect product factor
686C2C2×#178211yescharacteristic
687C2C2×#178811yescharacteristic
688C2C2×#179611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1823C2semidirect
4. C2×#1520C2semidirect
6. #8382C2direct (N × H)
13. C2×#552C4semidirect
17. C22×#125C4semidirect
19. C22×C22⋊Q8C4semidirect
21. C22×C4×Q8C4semidirect
24. C22×#117C4semidirect
30. #1823C22semidirect
36. #1520C22semidirect
69. C2×#125C2×C4semidirect
84. C2×C22⋊Q8C2×C4semidirect
93. C2×C4×Q8C2×C4semidirect
111. C2×#117C2×C4semidirect
685. C2#8382direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#18232
C2×#17992
C2×#15202
C2×#14592
#83822