Finite Group Databasebeta

← C2×#18027 #6344934

Subgroups — C2×#18027 #6344934

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3102C211yesmaximal, characteristic
2C2×#2994C211yesmaximal, characteristic, complemented
3C2×#2990C211yesmaximal, characteristic, complemented
4#18027C2116yesmaximal, direct product factor
5C2×#1592C211yesmaximal, characteristic, complemented
6C2×#1591C211yesmaximal, characteristic, complemented
7C2×#1585C211yesmaximal, characteristic, complemented
8C2×#1585C211yesmaximal, characteristic, complemented
9C2×#1584C211yesmaximal, characteristic, complemented
10C2×#1581C211yesmaximal, characteristic, complemented
11C2×SD16⋊C8C211yesmaximal, characteristic, complemented
12C2×C8×D8C211yesmaximal, characteristic, complemented
13C2×#1571C211yesmaximal, characteristic, complemented
14C2×#1495C211yesmaximal, characteristic, complemented
15C2×#1477C211yesmaximal, characteristic, complemented
16C2×#1454C211yesmaximal, characteristic, complemented
17C2×#529C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#3102C418yes—
20#3102C2218yes—
21C2×#485C411yescharacteristic
22C2×#482C411yescharacteristic
23C2×#479C411yescharacteristic
24C2×#478C411yescharacteristic
25C2×#464C411yescharacteristic
26C2×#463C411yescharacteristic
27C2×#452C411yescharacteristic
28C2×#449C411yescharacteristic
29#2994—24no—
30#2994C2218yescomplemented
31#2990—24no—
32#2990C2218yescomplemented
33C2×#446C2211yescharacteristic
34C2×#446C2211yescharacteristic
35C2×#446C2211yescharacteristic
36C2×C8×Q8C2211yescharacteristic
37C2×C4×Q16C2211yescharacteristic
38C2×C4×SD16C2211yescharacteristic
39C2×C4×SD16C2211yescharacteristic
40C2×C4×D8C2211yescharacteristic
41C2×#436C2211yescharacteristic
42C2×#435—21no—
43C2×#435—21no—
44C2×#435—21no—
45C2×C8×D4—21no—
46C2×C8×D4C2211yescharacteristic
47C2×C8×D4C2211yescharacteristic
48C2×C8×D4C2211yescharacteristic
49C2×#433—21no—
50C2×#432—21no—
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53C2×#431C2211yescharacteristic
54C2×#424—21no—
55C2×#423C2211yescharacteristic
56C22×C4⋊C8—21no—
57C22×C4⋊C8C2211yescharacteristic
58#1592C2218yescomplemented
59#1591C2218yescomplemented
60#1585C2218yescomplemented
61#1585C2218yescomplemented
62#1584C2218yescomplemented
63#1581C2218yescomplemented
64SD16⋊C8C2218yescomplemented
65C8×D8C2218yescomplemented
66#1571C2218yescomplemented
67C2×#335C2211yescharacteristic
68C2×#335C2211yescharacteristic
69C2×#335C2211yescharacteristic
70C2×#335C2211yescharacteristic
71#1495C2218yescomplemented
72#1477C2218yescomplemented
73C2×Q8⋊C8C2211yescharacteristic
74C2×Q8⋊C8C2211yescharacteristic
75C2×Q8⋊C8C2211yescharacteristic
76C2×Q8⋊C8C2211yescharacteristic
77C2×D4⋊C8C2211yescharacteristic
78C2×D4⋊C8C2211yescharacteristic
79C2×D4⋊C8C2211yescharacteristic
80C2×D4⋊C8C2211yescharacteristic
81#1454C2218yescomplemented
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C2×C82C2211yescharacteristic
86C22×#41C2×C411yescharacteristic
87C22×Q16C2×C411yescharacteristic
88C22×SD16C2×C411yescharacteristic
89C22×SD16C2×C411yescharacteristic
90C22×D8C2×C411yescharacteristic
91C22×M16—21no—
92C22×M16—21no—
93C23×C8—21no—
94C23×C8—21no—
95C23×C8—21no—
96C23×C8—21no—
97#529—22no—
98#529—22no—
99#529C2×C414yes—
100#529C2314yes—
101C2×#126—21no—
102C2×#125C2×C411yescharacteristic
103C2×#124C2×C411yescharacteristic
104C2×#123—21no—
105C2×C22⋊Q8C2×C411yescharacteristic
106C2×#121C2×C411yescharacteristic
107C4×#41—22no—
108C4×#41—22no—
109C4×#41C2×C414yes—
110C4×#41C2314yes—
111C2×C4×Q8C2×C411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2×C411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—21no—
122C2×#117C2×C411yescharacteristic
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125#485C2×C418yes—
126#482C2×C418yes—
127#479C2×C418yes—
128#478C2×C418yes—
129#464C2×C418yes—
130#463C2×C418yes—
131#452C2×C418yes—
132#449C2×C418yes—
133#446—22no—
134#446—22no—
135#446—22no—
136#446C2314yes—
137#446C2314yes—
138#446C2314yes—
139C8×Q8—22no—
140C8×Q8C2314yes—
141C4×Q16C2×C414yes—
142C4×Q16C2314yes—
143C4×SD16C2×C414yes—
144C4×SD16C2×C414yes—
145C4×SD16C2314yes—
146C4×SD16C2314yes—
147C4×D8C2×C414yes—
148C4×D8C2314yes—
149#436—22no—
150#436C2314yes—
151#435—28no—
152#435—28no—
153#435—28no—
154C8×D4—28no—
155C8×D4—22no—
156C8×D4—22no—
157C8×D4—22no—
158C8×D4C2314yes—
159C8×D4C2314yes—
160C8×D4C2314yes—
161#433—28no—
162#432—28no—
163#431—22no—
164#431C2×C414yes—
165#431D414yes—
166#431C2314yes—
167#431C2314yes—
168#431C2314yes—
169C2×#107C2×C411yescharacteristic
170C2×#107C2×C411yescharacteristic
171C2×#106C2×C411yescharacteristic
172C2×#106C2×C411yescharacteristic
173#424—28no—
174#423—22no—
175#423C2314yes—
176C2×C4⋊C8—21no—
177C2×C4⋊C8—21no—
178C2×C4⋊C8—21no—
179C2×C4⋊C8—21no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8—21no—
182C2×C4⋊C8—22no—
183C2×C4⋊C8—21no—
184C2×C4⋊C8—28no—
185C2×C4⋊C8C2×C411yescharacteristic
186C2×C4⋊C8C2×C411yescharacteristic
187C2×C4⋊C8D411yescharacteristic
188C2×C4⋊C8D411yescharacteristic
189C2×C4⋊C8C2311yescharacteristic
190C2×C4⋊C8C2311yescharacteristic
191C2×C4⋊C8C2311yescharacteristic
192C2×C4⋊C8C2311yescharacteristic
193C2×C4⋊C8C2314yes—
194C2×#103C2×C411yescharacteristic
195C2×#103C2×C411yescharacteristic
196C2×#103C2×C411yescharacteristic
197C2×#103C2×C411yescharacteristic
198C2×#102C2×C411yescharacteristic
199C2×#102C2×C411yescharacteristic
200C2×#102C2×C411yescharacteristic
201C2×#102C2×C411yescharacteristic
202C2×C22⋊C8—21no—
203C2×C22⋊C8—21no—
204C2×C22⋊C8—21no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—21no—
207C2×C22⋊C8C2×C411yescharacteristic
208C2×C22⋊C8C2×C411yescharacteristic
209C2×C22⋊C8D411yescharacteristic
210C2×C22⋊C8D411yescharacteristic
211C2×#97—21no—
212C2×#97—21no—
213C2×#97—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#335C2314yes—
224#335C2314yes—
225Q8⋊C8C2314yes—
226Q8⋊C8C2314yes—
227Q8⋊C8C2314yes—
228Q8⋊C8C2314yes—
229D4⋊C8C2314yes—
230D4⋊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×Q8C22×C411yescharacteristic
242C22×Q8C22×C411yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C22×C411yescharacteristic
246C22×D4C22×C411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C22×C411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4#4111yescharacteristic
254C2×Q16—42no—
255C2×Q16C22×C414yes—
256C2×SD16—24no—
257C2×SD16—42no—
258C2×SD16C22×C414yes—
259C2×SD16C22×C414yes—
260C2×D8—24no—
261C2×D8C22×C414yes—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—82no—
265C2×M16—82no—
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—24no—
275C22×C8—24no—
276C22×C8—82no—
277C22×C8—82no—
278C22×C8—82no—
279C22×C8—82no—
280C22×C8C22×C411yescharacteristic
281C22×C8C22×C411yescharacteristic
282C22×C8C22×C411yescharacteristic
283C22×C8C22×C411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288#126—24no—
289#126—24no—
290#125—22no—
291#125C22×C414yes—
292#124—22no—
293#124C22×C414yes—
294#123—24no—
295#123—24no—
296C22⋊Q8—22no—
297C22⋊Q8C22×C414yes—
298#121—22no—
299#121C22×C414yes—
300C4×Q8—22no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8C22×C414yes—
306C4×Q8C22×C412yes—
307C4×Q8C22×C412yes—
308C4×Q8C2412yes—
309C4×Q8C2412yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—22no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4C22×C414yes—
322C4×D4C22×C412yes—
323C4×D4C22×C412yes—
324C4×D4C2412yes—
325C4×D4C2412yes—
326#117—22no—
327#117—24no—
328#117—24no—
329#117—24no—
330#117—24no—
331#117C22×C414yes—
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×C22⋊C4—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42C22×C411yescharacteristic
357C2×C42C22×C412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42#4111yescharacteristic
361C2×C42#4112yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#107C22×C414yes—
365#107C22×C414yes—
366#106C22×C414yes—
367#106C22×C414yes—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—24no—
379C4⋊C8C22×C414yes—
380C4⋊C8C22×C414yes—
381C4⋊C8C2×D414yes—
382C4⋊C8C2×D414yes—
383C4⋊C8C2412yes—
384C4⋊C8C2412yes—
385C4⋊C8C2412yes—
386C4⋊C8C2412yes—
387#103C22×C414yes—
388#103C22×C414yes—
389#103C22×C414yes—
390#103C22×C414yes—
391#102C22×C414yes—
392#102C22×C414yes—
393#102C22×C414yes—
394#102C22×C414yes—
395C22⋊C8—24no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400C22⋊C8C22×C414yes—
401C22⋊C8C22×C414yes—
402C22⋊C8C2×D414yes—
403C22⋊C8C2×D414yes—
404#97—24no—
405#97—24no—
406#97—24no—
407C4×C8—24no—
408C4×C8—21no—
409C4×C8—21no—
410C4×C8C22×C412yes—
411C4×C8C22×C412yes—
412C4×C8C2×D412yes—
413C4×C8C2×D412yes—
414C4×C8C2412yes—
415C4×C8C2412yes—
416C4×C8C2412yes—
417C4×C8C2412yes—
418C4×C8C2412yes—
419C4×C8C2412yes—
420C24—21no—
421C24—21no—
422C24C4×D411yescharacteristic
423#41—28no—
424#41—44no—
425#41—48no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C23×C412yes—
433C2×Q8C23×C412yes—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4#13114yes—
444C2×D4C23×C412yes—
445C2×D4C23×C412yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D412yes—
473C22×C4C4×D412yes—
474C22×C4C4×D412yes—
475C22×C4C23×C412yes—
476C22×C4C23×C411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4112yes—
481Q16—44no—
482SD16—28no—
483SD16—44no—
484D8—28no—
485M16—84no—
486M16—84no—
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—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—84no—
508C2×C8—84no—
509C2×C8—84no—
510C2×C8—84no—
511C2×C8C23×C412yes—
512C2×C8C23×C412yes—
513C2×C8C23×C412yes—
514C2×C8C23×C412yes—
515C2×C8C22×D412yes—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4C23×C412yes—
532C4⋊C4C23×C412yes—
533C4⋊C4C23×C412yes—
534C4⋊C4C23×C412yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42C4×D412yes—
552C42C4×D412yes—
553C42C4×D412yes—
554C42C23×C411yescharacteristic
555C42C23×C412yes—
556C42C22×D411yescharacteristic
557C42C22×D412yes—
558C42C2×#4111yescharacteristic
559C42C2×#4112yes—
560C42C2511yescharacteristic
561C23—22no—
562C23—22no—
563C23—24no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23C2×C4×D412yes—
573C23C2×C4×D411yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—42no—
579Q8—42no—
580D4—24no—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—48no—
585D4—48no—
586D4—42no—
587D4C2×#13114yes—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4C2×C4×D412yes—
621C2×C4C2×C4×D412yes—
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×D412yes—
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×#13112yes—
627C2×C4#56812yes—
628C2×C4#57612yes—
629C2×C4#57812yes—
630C2×C4C24×C411yescharacteristic
631C2×C4C23×D411yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C8—22no—
634C8—22no—
635C8—22no—
636C8—22no—
637C8—22no—
638C8—22no—
639C8—22no—
640C8—22no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—48no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—84no—
659C22#293312yes—
660C22#295112yes—
661C22#295512yes—
662C22C22×C4×D411yescharacteristic
663C4—22no—
664C4—21no—
665C4—22no—
666C4—21no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4#293312yes—
675C4#295112yes—
676C4C22×#13111yescharacteristic
677C4C2×#56811yescharacteristic
678C4C2×#57611yescharacteristic
679C4C2×#57811yescharacteristic
680C2—42no—
681C2—44no—
682C2—82no—
683C2#1802714yesdirect product factor
684C2C2×#293311yescharacteristic
685C2C2×#295111yescharacteristic
686C2C2×#295511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C2×C4×D4
prime radicalC2×#449C4
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
2. C2×#2994C2semidirect
3. C2×#2990C2semidirect
4. #18027C2direct (N × H)
5. C2×#1592C2semidirect
6. C2×#1591C2semidirect
7. C2×#1585C2semidirect
8. C2×#1585C2semidirect
9. C2×#1584C2semidirect
10. C2×#1581C2semidirect
11. C2×SD16⋊C8C2semidirect
12. C2×C8×D8C2semidirect
13. C2×#1571C2semidirect
14. C2×#1495C2semidirect
15. C2×#1477C2semidirect
16. C2×#1454C2semidirect
30. #2994C22semidirect
32. #2990C22semidirect
58. #1592C22semidirect
59. #1591C22semidirect
60. #1585C22semidirect
61. #1585C22semidirect
62. #1584C22semidirect
63. #1581C22semidirect
64. SD16⋊C8C22semidirect
65. C8×D8C22semidirect
66. #1571C22semidirect
71. #1495C22semidirect
72. #1477C22semidirect
81. #1454C22semidirect
683. C2#18027direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31022
C2×#29942
C2×#29902
#180272
C2×#15922
C2×#15912
C2×#15852
C2×#15842
C2×#15812
C2×SD16⋊C82
C2×C8×D82
C2×#15712
C2×#14952
C2×#14772
C2×#14542