Finite Group Databasebeta

← C2×#18083 #6345126

Subgroups — C2×#18083 #6345126

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C211yesmaximal, characteristic
2C2×#2994C211yesmaximal, characteristic
3C2×#2986C211yesmaximal, characteristic, complemented
4#18083C2116yesmaximal, direct product factor
5C2×#1593C211yesmaximal, characteristic, complemented
6C2×#1590C211yesmaximal, characteristic, complemented
7C2×#1588C211yesmaximal, characteristic, complemented
8C2×#1587C211yesmaximal, characteristic, complemented
9C2×#1584C211yesmaximal, characteristic, complemented
10C2×#1582C211yesmaximal, characteristic, complemented
11C2×SD16⋊C8C211yesmaximal, characteristic, complemented
12C2×C8×Q16C211yesmaximal, characteristic, complemented
13C2×#1565C211yesmaximal, characteristic, complemented
14C2×#1482C211yesmaximal, characteristic
15C2×#1481C211yesmaximal, characteristic, complemented
16C2×#1453C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#529C2211yescharacteristic
19#3172C418yes—
20#3172C2218yes—
21C2×#501C411yescharacteristic
22C2×#497C411yescharacteristic
23C2×#491C411yescharacteristic
24C2×#489C411yescharacteristic
25C2×#470C411yescharacteristic
26C2×#470C411yescharacteristic
27C2×#468C411yescharacteristic
28C2×#468C411yescharacteristic
29#2994—24no—
30#2994C2218yes—
31#2986—24no—
32#2986C2218yescomplemented
33C2×#446—21no—
34C2×#446C2211yescharacteristic
35C2×#446C2211yescharacteristic
36C2×#446C2211yescharacteristic
37C2×C8×Q8—21no—
38C2×C8×Q8C2211yescharacteristic
39C2×C8×Q8C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C4×Q16C2211yescharacteristic
43C2×C4×SD16C2211yescharacteristic
44C2×C4×SD16C2211yescharacteristic
45C2×#436C2211yescharacteristic
46C2×#435—21no—
47C2×C8×D4—21no—
48C2×C8×D4C2211yescharacteristic
49C2×#433—21no—
50C2×#433—21no—
51C2×#433—21no—
52C2×#431C2211yescharacteristic
53C2×#431C2211yescharacteristic
54C2×#431C2211yescharacteristic
55C2×#424—21no—
56C22×C4⋊C8C2211yescharacteristic
57C2×C4×M16C2211yescharacteristic
58#1593C2218yescomplemented
59#1590C2218yescomplemented
60#1588C2218yescomplemented
61#1587C2218yescomplemented
62#1584C2218yescomplemented
63#1582C2218yescomplemented
64SD16⋊C8C2218yescomplemented
65C8×Q16C2218yescomplemented
66#1565C2218yescomplemented
67C2×#335C2211yescharacteristic
68C2×#335C2211yescharacteristic
69C2×#334C2211yescharacteristic
70C2×#334C2211yescharacteristic
71#1482C2218yes—
72#1481C2218yescomplemented
73C2×Q8⋊C8C2211yescharacteristic
74C2×Q8⋊C8C2211yescharacteristic
75C2×Q8⋊C8C2211yescharacteristic
76C2×Q8⋊C8C2211yescharacteristic
77C2×Q8⋊C8C2211yescharacteristic
78C2×Q8⋊C8C2211yescharacteristic
79C2×D4⋊C8C2211yescharacteristic
80C2×D4⋊C8C2211yescharacteristic
81#1453C2218yescomplemented
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C2×C82C2211yescharacteristic
86C22×Q16C2×C411yescharacteristic
87C22×Q16C2×C411yescharacteristic
88C22×SD16C2×C411yescharacteristic
89C22×SD16C2×C411yescharacteristic
90C22×M16—21no—
91C22×M16C2×C411yescharacteristic
92C22×M16C2×C411yescharacteristic
93C23×C8—21no—
94C23×C8—21no—
95#533—22no—
96#533—22no—
97#533C2×C414yes—
98#533C2314yes—
99C2×C4⋊Q8C2×C411yescharacteristic
100C2×C4⋊Q8C2×C411yescharacteristic
101#529—22no—
102#529—22no—
103#529C2×C414yes—
104#529C2314yes—
105C2×#126—21no—
106C2×#125—21no—
107C2×#125C2×C411yescharacteristic
108C2×#124C2×C411yescharacteristic
109C2×#123—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2×C411yescharacteristic
112C2×C22⋊Q8C2×C411yescharacteristic
113C2×C22⋊Q8C2×C411yescharacteristic
114C2×#121C2×C411yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—21no—
123C22×C42C2311yescharacteristic
124#501C2×C418yes—
125#497C2×C418yes—
126#491C2×C418yes—
127#489C2×C418yes—
128#470C2×C418yes—
129#470C2×C418yes—
130#468C2×C418yes—
131#468C2×C418yes—
132#446—28no—
133#446—22no—
134#446—22no—
135#446—22no—
136#446C2314yes—
137#446C2314yes—
138#446C2314yes—
139C8×Q8—28no—
140C8×Q8—22no—
141C8×Q8—22no—
142C8×Q8—22no—
143C8×Q8C2314yes—
144C8×Q8C2314yes—
145C8×Q8C2314yes—
146C4×Q16C2×C414yes—
147C4×Q16C2×C414yes—
148C4×Q16C2314yes—
149C4×Q16C2314yes—
150C4×SD16C2×C414yes—
151C4×SD16C2×C414yes—
152C4×SD16C2314yes—
153C4×SD16C2314yes—
154#436—22no—
155#436C2314yes—
156#435—28no—
157C8×D4—28no—
158C8×D4—22no—
159C8×D4C2314yes—
160#433—28no—
161#433—28no—
162#433—28no—
163#431—22no—
164#431—22no—
165#431D414yes—
166#431C2314yes—
167#431C2314yes—
168#431C2314yes—
169C2×#107C2×C411yescharacteristic
170C2×#107C2×C411yescharacteristic
171C2×#106C2×C411yescharacteristic
172C2×#106C2×C411yescharacteristic
173#424—28no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8—21no—
178C2×C4⋊C8—21no—
179C2×C4⋊C8—21no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8—22no—
182C2×C4⋊C8—21no—
183C2×C4⋊C8D411yescharacteristic
184C2×C4⋊C8D411yescharacteristic
185C2×C4⋊C8C2311yescharacteristic
186C2×C4⋊C8C2311yescharacteristic
187C2×C4⋊C8C2311yescharacteristic
188C2×C4⋊C8C2311yescharacteristic
189C2×C4⋊C8C2314yes—
190C2×#103C2×C411yescharacteristic
191C2×#103C2×C411yescharacteristic
192C2×#103C2×C411yescharacteristic
193C2×#103C2×C411yescharacteristic
194C2×#103C2×C411yescharacteristic
195C2×#103C2×C411yescharacteristic
196C2×#102C2×C411yescharacteristic
197C2×#102C2×C411yescharacteristic
198C2×C22⋊C8—21no—
199C2×C22⋊C8—21no—
200C2×C22⋊C8—21no—
201C2×C22⋊C8—21no—
202C2×C22⋊C8D411yescharacteristic
203C2×C22⋊C8D411yescharacteristic
204C4×M16C2×C414yes—
205C4×M16C2314yes—
206C2×#97—21no—
207C2×#97—21no—
208C2×#97—21no—
209C2×#97C2×C411yescharacteristic
210C2×#97C2×C411yescharacteristic
211C2×C4×C8—21no—
212C2×C4×C8—21no—
213C2×C4×C8—21no—
214C2×C4×C8—21no—
215C2×C4×C8C2311yescharacteristic
216C2×C4×C8C2311yescharacteristic
217C2×C4×C8C2311yescharacteristic
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2311yescharacteristic
220C2×C4×C8C2311yescharacteristic
221#335C2314yes—
222#335C2314yes—
223#334C2314yes—
224#334C2314yes—
225Q8⋊C8C2314yes—
226Q8⋊C8C2314yes—
227Q8⋊C8C2314yes—
228Q8⋊C8C2314yes—
229Q8⋊C8C2314yes—
230Q8⋊C8C2314yes—
231D4⋊C8C2314yes—
232D4⋊C8C2314yes—
233#322C2314yes—
234#322C2314yes—
235#322C2314yes—
236C82C2314yes—
237C22×Q8—21no—
238C22×Q8C22×C411yescharacteristic
239C22×Q8C22×C411yescharacteristic
240C22×Q8C22×C411yescharacteristic
241C22×D4—21no—
242C22×D4C22×C411yescharacteristic
243C23×C4—21no—
244C23×C4C22×C411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4#4111yescharacteristic
247C2×Q16—24no—
248C2×Q16—42no—
249C2×Q16C22×C414yes—
250C2×Q16C22×C414yes—
251C2×SD16—24no—
252C2×SD16—42no—
253C2×SD16C22×C414yes—
254C2×SD16C22×C414yes—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—42no—
258C2×M16—82no—
259C2×M16C22×C414yes—
260C2×M16C22×C414yes—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—82no—
270C22×C8—82no—
271C22×C8C22×C411yescharacteristic
272C22×C8C22×C411yescharacteristic
273C22×C8C22×C411yescharacteristic
274C22×C8C22×C411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8C22×C414yes—
282C4⋊Q8C22×C414yes—
283#126—24no—
284#126—24no—
285#125—24no—
286#125—22no—
287#125—24no—
288#125C22×C414yes—
289#124—22no—
290#124C22×C414yes—
291#123—24no—
292#123—24no—
293C22⋊Q8—24no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—24no—
298C22⋊Q8C22×C414yes—
299C22⋊Q8C22×C414yes—
300C22⋊Q8C22×C414yes—
301#121—22no—
302#121C22×C414yes—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8C22×C412yes—
312C4×Q8C22×C412yes—
313C4×Q8C22×C412yes—
314C4×Q8C2412yes—
315C4×Q8C2412yes—
316C4×Q8C2412yes—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4C22×C412yes—
322C4×D4C2412yes—
323#117—24no—
324#117—24no—
325#117—24no—
326#117—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C22×C411yescharacteristic
339C2×C4⋊C4C22×C411yescharacteristic
340C2×C4⋊C4C22×C411yescharacteristic
341C2×C4⋊C4C22×C411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42C22×C411yescharacteristic
353C2×C42C22×C412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42#4111yescharacteristic
357C2×C42#4112yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107C22×C414yes—
361#107C22×C414yes—
362#106C22×C414yes—
363#106C22×C414yes—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—24no—
376C4⋊C8C2×D414yes—
377C4⋊C8C2×D414yes—
378C4⋊C8C2412yes—
379C4⋊C8C2412yes—
380C4⋊C8C2412yes—
381C4⋊C8C2412yes—
382#103C22×C414yes—
383#103C22×C414yes—
384#103C22×C414yes—
385#103C22×C414yes—
386#103C22×C414yes—
387#103C22×C414yes—
388#102C22×C414yes—
389#102C22×C414yes—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8C2×D414yes—
395C22⋊C8C2×D414yes—
396#97—24no—
397#97—24no—
398#97—24no—
399#97C22×C414yes—
400#97C22×C414yes—
401C4×C8—24no—
402C4×C8—24no—
403C4×C8—24no—
404C4×C8—24no—
405C4×C8—21no—
406C4×C8—21no—
407C4×C8C22×C412yes—
408C4×C8C22×C412yes—
409C4×C8C2×D412yes—
410C4×C8C2×D412yes—
411C4×C8C2412yes—
412C4×C8C2412yes—
413C4×C8C2412yes—
414C4×C8C2412yes—
415C4×C8C2412yes—
416C4×C8C2412yes—
417C24—21no—
418C24C4×D411yescharacteristic
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8C23×C412yes—
431C2×Q8C23×C412yes—
432C2×Q8C23×C412yes—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—44no—
438C2×D4C23×C412yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D412yes—
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D412yes—
460C22×C4C4×D412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4C2×#4112yes—
467Q16—28no—
468Q16—44no—
469SD16—28no—
470SD16—44no—
471M16—28no—
472M16—44no—
473M16—84no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—84no—
494C2×C8—84no—
495C2×C8#13114yes—
496C2×C8C23×C412yes—
497C2×C8C23×C412yes—
498C2×C8C23×C412yes—
499C2×C8C23×C412yes—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C2×C8C22×D412yes—
503C2×C8C22×D412yes—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4C23×C412yes—
523C4⋊C4C23×C412yes—
524C4⋊C4C23×C412yes—
525C4⋊C4C23×C412yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42C4×D412yes—
541C42C4×D412yes—
542C42C4×D412yes—
543C42C23×C411yescharacteristic
544C42C23×C412yes—
545C42C22×D412yes—
546C42C22×D411yescharacteristic
547C42C2×#4111yescharacteristic
548C42C2×#4112yes—
549C42C2511yescharacteristic
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23C2×C4×D412yes—
558C23C2×C4×D411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—48no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569D4—22no—
570D4—22no—
571D4—48no—
572D4—42no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
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—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D412yes—
604C2×C4C2×C4×D412yes—
605C2×C4C2×C4×D412yes—
606C2×C4C2×C4×D411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×#13112yes—
609C2×C4#56812yes—
610C2×C4#57712yes—
611C2×C4#57812yes—
612C2×C4C24×C411yescharacteristic
613C2×C4C23×D411yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C8—22no—
616C8—22no—
617C8—22no—
618C8—22no—
619C8—22no—
620C8—22no—
621C8—22no—
622C8—42no—
623C8—42no—
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8C2×#13114yes—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—84no—
637C22#293512yes—
638C22#294712yes—
639C22#295512yes—
640C22C22×C4×D411yescharacteristic
641C4—22no—
642C4—21no—
643C4—21no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4#293512yes—
654C4#294712yes—
655C4C22×#13111yescharacteristic
656C4C2×#56811yescharacteristic
657C4C2×#57711yescharacteristic
658C4C2×#57811yescharacteristic
659C2—42no—
660C2—82no—
661C2#1808314yesdirect product factor
662C2C2×#293511yescharacteristic
663C2C2×#294711yescharacteristic
664C2C2×#295511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2986C2semidirect
4. #18083C2direct (N × H)
5. C2×#1593C2semidirect
6. C2×#1590C2semidirect
7. C2×#1588C2semidirect
8. C2×#1587C2semidirect
9. C2×#1584C2semidirect
10. C2×#1582C2semidirect
11. C2×SD16⋊C8C2semidirect
12. C2×C8×Q16C2semidirect
13. C2×#1565C2semidirect
15. C2×#1481C2semidirect
16. C2×#1453C2semidirect
32. #2986C22semidirect
58. #1593C22semidirect
59. #1590C22semidirect
60. #1588C22semidirect
61. #1587C22semidirect
62. #1584C22semidirect
63. #1582C22semidirect
64. SD16⋊C8C22semidirect
65. C8×Q16C22semidirect
66. #1565C22semidirect
72. #1481C22semidirect
81. #1453C22semidirect
661. C2#18083direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31722
C2×#29942
C2×#29862
#180832
C2×#15932
C2×#15902
C2×#15882
C2×#15872
C2×#15842
C2×#15822
C2×SD16⋊C82
C2×C8×Q162
C2×#15652
C2×#14822
C2×#14812
C2×#14532