Finite Group Databasebeta

← C2×#18009 #6344913

Subgroups — C2×#18009 #6344913

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3107C211yesmaximal, characteristic
2C2×#2970C211yesmaximal, characteristic
3C2×#2962C211yesmaximal, characteristic, complemented
4#18009C2116yesmaximal, direct product factor
5C2×#1593C211yesmaximal, characteristic, complemented
6C2×#1591C211yesmaximal, characteristic, complemented
7C2×#1587C211yesmaximal, characteristic, complemented
8C2×#1585C211yesmaximal, characteristic, complemented
9C2×#1584C211yesmaximal, characteristic, complemented
10C2×#1582C211yesmaximal, characteristic, complemented
11C2×SD16⋊C8C211yesmaximal, characteristic, complemented
12C2×C8×SD16C211yesmaximal, characteristic, complemented
13C2×#1573C211yesmaximal, characteristic, complemented
14C2×#1497C211yesmaximal, characteristic, complemented
15C2×#1477C211yesmaximal, characteristic
16C2×#1453C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#3107C418yes—
20#3107C2218yes—
21C2×#485C411yescharacteristic
22C2×#485C411yescharacteristic
23C2×#475C411yescharacteristic
24C2×#474C411yescharacteristic
25C2×#462C411yescharacteristic
26C2×#461C411yescharacteristic
27C2×#452C411yescharacteristic
28C2×#452C411yescharacteristic
29#2970—24no—
30#2970C2218yes—
31#2962—24no—
32#2962C2218yescomplemented
33C2×#446—21no—
34C2×#446—21no—
35C2×#446C2211yescharacteristic
36C2×#446C2211yescharacteristic
37C2×#446C2211yescharacteristic
38C2×#446C2211yescharacteristic
39C2×C8×Q8C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C4×Q16C2211yescharacteristic
43C2×C4×SD16C2211yescharacteristic
44C2×C4×SD16C2211yescharacteristic
45C2×#435—21no—
46C2×#435—21no—
47C2×C8×D4C2211yescharacteristic
48C2×C8×D4C2211yescharacteristic
49C2×#432—21no—
50C2×#432C2211yescharacteristic
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53C2×#424—21no—
54C22×C4⋊C8C2211yescharacteristic
55C2×#405—21no—
56C2×C4×M16C2211yescharacteristic
57C22×#97—21no—
58#1593C2218yescomplemented
59#1591C2218yescomplemented
60#1587C2218yescomplemented
61#1585C2218yescomplemented
62#1584C2218yescomplemented
63#1582C2218yescomplemented
64SD16⋊C8C2218yescomplemented
65C8×SD16C2218yescomplemented
66#1573C2218yescomplemented
67C2×#335C2211yescharacteristic
68C2×#335C2211yescharacteristic
69C2×#334C2211yescharacteristic
70C2×#334C2211yescharacteristic
71#1497C2218yescomplemented
72#1477C2218yes—
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×#41C2×C411yescharacteristic
87C22×Q16C2×C411yescharacteristic
88C22×Q16C2×C411yescharacteristic
89C22×SD16C2×C411yescharacteristic
90C22×SD16C2×C411yescharacteristic
91C22×M16D411yescharacteristic
92C22×M16D411yescharacteristic
93C23×C8—21no—
94C23×C8—21no—
95C23×C8—21no—
96#533—22no—
97#533—22no—
98#533C2×C414yes—
99#533C2314yes—
100C2×C4⋊Q8C2×C411yescharacteristic
101C2×C4⋊Q8C2×C411yescharacteristic
102C2×#125—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2×C411yescharacteristic
105C2×C22⋊Q8C2×C411yescharacteristic
106C4×#41—22no—
107C4×#41—22no—
108C4×#41C2×C414yes—
109C4×#41C2314yes—
110C2×C4×Q8—21no—
111C2×C4×Q8C2×C411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2×C411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117C2×C411yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124#485C2×C418yes—
125#485C2×C418yes—
126#475C2×C418yes—
127#474C2×C418yes—
128#462C2×C418yes—
129#461C2×C418yes—
130#452C2×C418yes—
131#452C2×C418yes—
132#446—28no—
133#446—28no—
134#446—22no—
135#446—22no—
136#446—22no—
137#446—22no—
138#446C2314yes—
139#446C2314yes—
140#446C2314yes—
141#446C2314yes—
142C8×Q8—22no—
143C8×Q8—22no—
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#435—28no—
155#435—28no—
156C8×D4—22no—
157C8×D4—22no—
158C8×D4C2314yes—
159C8×D4C2314yes—
160#432—22no—
161#432—28no—
162#432C2314yes—
163#431—22no—
164#431C2×C414yes—
165#431C2314yes—
166#431C2314yes—
167C2×#107C2×C411yescharacteristic
168C2×#107C2×C411yescharacteristic
169C2×#106C2×C411yescharacteristic
170C2×#106C2×C411yescharacteristic
171#424—28no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8—21no—
178C2×C4⋊C8—21no—
179C2×C4⋊C8—22no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8C2×C411yescharacteristic
182C2×C4⋊C8C2×C411yescharacteristic
183C2×C4⋊C8C2311yescharacteristic
184C2×C4⋊C8C2311yescharacteristic
185C2×C4⋊C8C2311yescharacteristic
186C2×C4⋊C8C2311yescharacteristic
187C2×C4⋊C8C2314yes—
188C2×#103C2×C411yescharacteristic
189C2×#103C2×C411yescharacteristic
190C2×#103C2×C411yescharacteristic
191C2×#103C2×C411yescharacteristic
192C2×#103C2×C411yescharacteristic
193C2×#103C2×C411yescharacteristic
194C2×#102C2×C411yescharacteristic
195C2×#102C2×C411yescharacteristic
196C2×C22⋊C8—21no—
197C2×C22⋊C8—21no—
198C2×C22⋊C8—21no—
199C2×C22⋊C8—21no—
200C2×C22⋊C8C2×C411yescharacteristic
201C2×C22⋊C8C2×C411yescharacteristic
202#405—28no—
203C4×M16D414yes—
204C4×M16C2314yes—
205C2×#97—21no—
206C2×#97—21no—
207C2×#97—21no—
208C2×#97—21no—
209C2×#97—21no—
210C2×#97—28no—
211C2×#97D411yescharacteristic
212C2×#97D411yescharacteristic
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—
237C2×#41—22no—
238C2×#41—24no—
239C2×#41—42no—
240C2×#41C22×C414yes—
241C22×Q8—21no—
242C22×Q8C22×C411yescharacteristic
243C22×Q8C22×C411yescharacteristic
244C22×Q8C22×C411yescharacteristic
245C22×D4—21no—
246C22×D4C22×C411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C22×C411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4#4111yescharacteristic
253C2×Q16—42no—
254C2×Q16—42no—
255C2×Q16C22×C414yes—
256C2×Q16C22×C414yes—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16C22×C414yes—
260C2×SD16C22×C414yes—
261C2×M16—24no—
262C2×M16—42no—
263C2×M16C2×D414yes—
264C2×M16C2×D414yes—
265C22×C8—24no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8—44no—
277C22×C8C22×C411yescharacteristic
278C22×C8C22×C411yescharacteristic
279C22×C8C22×C411yescharacteristic
280C22×C8C22×C411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8C22×C414yes—
288C4⋊Q8C22×C414yes—
289#125—24no—
290#125—24no—
291C22⋊Q8—24no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—24no—
295C22⋊Q8C22×C414yes—
296C22⋊Q8C22×C414yes—
297C4×Q8—24no—
298C4×Q8—22no—
299C4×Q8—24no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8C22×C414yes—
307C4×Q8C22×C412yes—
308C4×Q8C22×C412yes—
309C4×Q8C22×C412yes—
310C4×Q8C2412yes—
311C4×Q8C2412yes—
312C4×Q8C2412yes—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22×C414yes—
321C4×D4C22×C412yes—
322C4×D4C2412yes—
323#117—24no—
324#117—22no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117C22×C414yes—
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×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42C22×C411yescharacteristic
355C2×C42C22×C412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42#4111yescharacteristic
359C2×C42#4112yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362#107C22×C414yes—
363#107C22×C414yes—
364#106C22×C414yes—
365#106C22×C414yes—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—24no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—24no—
378C4⋊C8C22×C414yes—
379C4⋊C8C22×C414yes—
380C4⋊C8C2412yes—
381C4⋊C8C2412yes—
382C4⋊C8C2412yes—
383C4⋊C8C2412yes—
384#103C22×C414yes—
385#103C22×C414yes—
386#103C22×C414yes—
387#103C22×C414yes—
388#103C22×C414yes—
389#103C22×C414yes—
390#102C22×C414yes—
391#102C22×C414yes—
392C22⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—24no—
396C22⋊C8C22×C414yes—
397C22⋊C8C22×C414yes—
398#97—24no—
399#97—24no—
400#97—24no—
401#97—24no—
402#97—24no—
403#97C2×D414yes—
404#97C2×D414yes—
405C4×C8—24no—
406C4×C8—24no—
407C4×C8—21no—
408C4×C8—21no—
409C4×C8C22×C412yes—
410C4×C8C22×C412yes—
411C4×C8C2×D412yes—
412C4×C8C2×D412yes—
413C4×C8C2412yes—
414C4×C8C2412yes—
415C4×C8C2412yes—
416C4×C8C2412yes—
417C4×C8C2412yes—
418C4×C8C2412yes—
419C24—21no—
420C24C4×D411yescharacteristic
421#41—28no—
422#41—44no—
423#41—48no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—82no—
435C2×Q8C23×C412yes—
436C2×Q8C23×C412yes—
437C2×Q8C23×C412yes—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—44no—
442C2×D4#13114yes—
443C2×D4C23×C412yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4C4×D411yescharacteristic
465C22×C4C4×D411yescharacteristic
466C22×C4C4×D411yescharacteristic
467C22×C4C4×D412yes—
468C22×C4C4×D412yes—
469C22×C4C4×D412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4112yes—
476Q16—44no—
477Q16—44no—
478SD16—28no—
479SD16—28no—
480M16—28no—
481M16—44no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—48no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—48no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—48no—
504C2×C8#13714yes—
505C2×C8C23×C412yes—
506C2×C8C23×C412yes—
507C2×C8C23×C412yes—
508C2×C8C23×C412yes—
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C2×C8C22×D412yes—
512C2×C8C22×D412yes—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4C23×C412yes—
530C4⋊C4C23×C412yes—
531C4⋊C4C23×C412yes—
532C4⋊C4C23×C412yes—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42C4×D412yes—
548C42C4×D412yes—
549C42C4×D412yes—
550C42C23×C411yescharacteristic
551C42C23×C412yes—
552C42C22×D411yescharacteristic
553C42C22×D412yes—
554C42C2×#4111yescharacteristic
555C42C2×#4112yes—
556C42C2511yescharacteristic
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—42no—
563C23—42no—
564C23C2×C4×D412yes—
565C23C2×C4×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576D4—24no—
577D4—22no—
578D4—48no—
579D4C2×#13114yes—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—84no—
611C2×C4C2×C4×D412yes—
612C2×C4C2×C4×D412yes—
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×C4×D412yes—
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×#13112yes—
618C2×C4#56812yes—
619C2×C4C2×#13712yes—
620C2×C4#57512yes—
621C2×C4C24×C411yescharacteristic
622C2×C4C23×D411yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C8—22no—
625C8—22no—
626C8—22no—
627C8—22no—
628C8—22no—
629C8—22no—
630C8—22no—
631C8—42no—
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C8—42no—
637C8C2×#13714yes—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—22no—
642C22—42no—
643C22—44no—
644C22—48no—
645C22—42no—
646C22#293312yes—
647C22C4×#13712yes—
648C22#295712yes—
649C22C22×C4×D411yescharacteristic
650C4—21no—
651C4—22no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4#293312yes—
663C4C4×#13712yes—
664C4C22×#13111yescharacteristic
665C4C2×#56811yescharacteristic
666C4C22×#13711yescharacteristic
667C4C2×#57511yescharacteristic
668C2—42no—
669C2—44no—
670C2#1800914yesdirect product factor
671C2C2×#293311yescharacteristic
672C2C2×C4×#13711yescharacteristic
673C2C2×#295711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2962C2semidirect
4. #18009C2direct (N × H)
5. C2×#1593C2semidirect
6. C2×#1591C2semidirect
7. C2×#1587C2semidirect
8. C2×#1585C2semidirect
9. C2×#1584C2semidirect
10. C2×#1582C2semidirect
11. C2×SD16⋊C8C2semidirect
12. C2×C8×SD16C2semidirect
13. C2×#1573C2semidirect
14. C2×#1497C2semidirect
16. C2×#1453C2semidirect
32. #2962C22semidirect
58. #1593C22semidirect
59. #1591C22semidirect
60. #1587C22semidirect
61. #1585C22semidirect
62. #1584C22semidirect
63. #1582C22semidirect
64. SD16⋊C8C22semidirect
65. C8×SD16C22semidirect
66. #1573C22semidirect
71. #1497C22semidirect
81. #1453C22semidirect
670. C2#18009direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31072
C2×#29702
C2×#29622
#180092
C2×#15932
C2×#15912
C2×#15872
C2×#15852
C2×#15842
C2×#15822
C2×SD16⋊C82
C2×C8×SD162
C2×#15732
C2×#14972
C2×#14772
C2×#14532