Finite Group Databasebeta

← C2×#18032 #6344933

Subgroups — C2×#18032 #6344933

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×#2986C211yesmaximal, characteristic
3C2×#2984C211yesmaximal, characteristic
4#18032C2116yesmaximal, direct product factor
5C2×#1593C211yesmaximal, characteristic, complemented
6C2×#1591C211yesmaximal, characteristic, complemented
7C2×#1587C211yesmaximal, characteristic, complemented
8C2×#1587C211yesmaximal, characteristic, complemented
9C2×#1584C211yesmaximal, characteristic, complemented
10C2×#1584C211yesmaximal, characteristic, complemented
11C2×#1579C211yesmaximal, characteristic, complemented
12C2×C8×Q16C211yesmaximal, characteristic, complemented
13C2×#1572C211yesmaximal, characteristic
14C2×#1497C211yesmaximal, characteristic
15C2×#1479C211yesmaximal, characteristic
16C2×#1454C211yesmaximal, characteristic
17C2×#533C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#3104C418yes—
20#3104C2218yes—
21C2×#484C411yescharacteristic
22C2×#484C411yescharacteristic
23C2×#477C411yescharacteristic
24C2×#477C411yescharacteristic
25C2×#462C411yescharacteristic
26C2×#462C411yescharacteristic
27C2×#451C411yescharacteristic
28C2×#451C411yescharacteristic
29#2986—24no—
30#2986C2218yes—
31#2984—24no—
32#2984C2218yes—
33C2×#446—21no—
34C2×#446—21no—
35C2×#446—21no—
36C2×#446C2211yescharacteristic
37C2×#446C2211yescharacteristic
38C2×#446C2211yescharacteristic
39C2×#446C2211yescharacteristic
40C2×C8×Q8—21no—
41C2×C8×Q8C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×C8×Q8C2211yescharacteristic
44C2×C8×Q8C2211yescharacteristic
45C2×C4×Q16C2211yescharacteristic
46C2×C4×Q16C2211yescharacteristic
47C2×C4×Q16C2211yescharacteristic
48C2×C4×Q16C2211yescharacteristic
49C2×#433—21no—
50C2×#433—21no—
51C2×#432—21no—
52C2×#432C2211yescharacteristic
53C2×#431—21no—
54C2×#431C2211yescharacteristic
55C2×#431C2211yescharacteristic
56C2×#431C2211yescharacteristic
57C2×#431C2211yescharacteristic
58#1593C2218yescomplemented
59#1591C2218yescomplemented
60#1587C2218yescomplemented
61#1587C2218yescomplemented
62#1584C2218yescomplemented
63#1584C2218yescomplemented
64#1579C2218yescomplemented
65C8×Q16C2218yescomplemented
66#1572C2218yes—
67C2×#335C2211yescharacteristic
68C2×#335C2211yescharacteristic
69C2×#334C2211yescharacteristic
70C2×#334C2211yescharacteristic
71#1497C2218yes—
72#1479C2218yes—
73C2×Q8⋊C8C2211yescharacteristic
74C2×Q8⋊C8C2211yescharacteristic
75C2×Q8⋊C8C2211yescharacteristic
76C2×Q8⋊C8C2211yescharacteristic
77C2×Q8⋊C8C2211yescharacteristic
78C2×Q8⋊C8C2211yescharacteristic
79C2×Q8⋊C8C2211yescharacteristic
80C2×Q8⋊C8C2211yescharacteristic
81#1454C2218yes—
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C2×C82C2211yescharacteristic
86C23×Q8C2×C411yescharacteristic
87C22×Q16C2×C411yescharacteristic
88C22×Q16C2×C411yescharacteristic
89C22×Q16C2×C411yescharacteristic
90C22×Q16C2×C411yescharacteristic
91#533—22no—
92#533—22no—
93#533C2×C414yes—
94#533C2314yes—
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×C4⋊Q8C2×C411yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2×C411yescharacteristic
100C2×C22⋊Q8C2×C411yescharacteristic
101C2×C4×Q8—21no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8C2×C411yescharacteristic
107C2×C4×Q8C2×C411yescharacteristic
108C2×C4×Q8C2×C414yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119#484C2×C418yes—
120#484C2×C418yes—
121#477C2×C418yes—
122#477C2×C418yes—
123#462C2×C418yes—
124#462C2×C418yes—
125#451C2×C418yes—
126#451C2×C418yes—
127#446—28no—
128#446—28no—
129#446—28no—
130#446—22no—
131#446—22no—
132#446—22no—
133#446—22no—
134#446C2314yes—
135#446C2314yes—
136#446C2314yes—
137#446C2314yes—
138C8×Q8—28no—
139C8×Q8—22no—
140C8×Q8—22no—
141C8×Q8—22no—
142C8×Q8—22no—
143C8×Q8C2314yes—
144C8×Q8C2314yes—
145C8×Q8C2314yes—
146C8×Q8C2314yes—
147C4×Q16C2×C414yes—
148C4×Q16C2×C414yes—
149C4×Q16C2×C414yes—
150C4×Q16C2×C414yes—
151C4×Q16C2314yes—
152C4×Q16C2314yes—
153C4×Q16C2314yes—
154C4×Q16C2314yes—
155#433—28no—
156#433—28no—
157#432—22no—
158#432—28no—
159#432C2314yes—
160#431—22no—
161#431—22no—
162#431—28no—
163#431C2×C414yes—
164#431D414yes—
165#431C2314yes—
166#431C2314yes—
167#431C2314yes—
168#431C2314yes—
169C2×#107C2×C411yescharacteristic
170C2×#107C2×C411yescharacteristic
171C2×#107C2×C411yescharacteristic
172C2×#107C2×C411yescharacteristic
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—21no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8—21no—
182C2×C4⋊C8C2×C411yescharacteristic
183C2×C4⋊C8C2×C411yescharacteristic
184C2×C4⋊C8D411yescharacteristic
185C2×C4⋊C8D411yescharacteristic
186C2×C4⋊C8C2311yescharacteristic
187C2×C4⋊C8C2311yescharacteristic
188C2×C4⋊C8C2311yescharacteristic
189C2×C4⋊C8C2311yescharacteristic
190C2×#103C2×C411yescharacteristic
191C2×#103C2×C411yescharacteristic
192C2×#103C2×C411yescharacteristic
193C2×#103C2×C411yescharacteristic
194C2×#103C2×C411yescharacteristic
195C2×#103C2×C411yescharacteristic
196C2×#103C2×C411yescharacteristic
197C2×#103C2×C411yescharacteristic
198C2×C22⋊C8—21no—
199C2×C22⋊C8—21no—
200C2×C22⋊C8—21no—
201C2×C22⋊C8C2×C411yescharacteristic
202C2×C22⋊C8C2×C411yescharacteristic
203C2×C22⋊C8D411yescharacteristic
204C2×C22⋊C8D411yescharacteristic
205C2×#97—21no—
206C2×#97—21no—
207C2×#97—21no—
208C2×#97—21no—
209C2×#97—21no—
210C2×C4×C8—21no—
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—
231Q8⋊C8C2314yes—
232Q8⋊C8C2314yes—
233#322C2314yes—
234#322C2314yes—
235#322C2314yes—
236C82C2314yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—42no—
242C22×Q8C22×C411yescharacteristic
243C22×Q8C22×C411yescharacteristic
244C22×Q8C22×C411yescharacteristic
245C22×Q8C22×C414yes—
246C22×Q8C22×C411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22×C411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4#4111yescharacteristic
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—42no—
255C2×Q16—42no—
256C2×Q16C22×C414yes—
257C2×Q16C22×C414yes—
258C2×Q16C22×C414yes—
259C2×Q16C22×C414yes—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8C22×C411yescharacteristic
267C22×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C22×C411yescharacteristic
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8C22×C414yes—
277C4⋊Q8C22×C414yes—
278#125—24no—
279#125—24no—
280C22⋊Q8—24no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8C22×C414yes—
285C22⋊Q8C22×C414yes—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—21no—
292C4×Q8—24no—
293C4×Q8—21no—
294C4×Q8—24no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—24no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8C22×C412yes—
303C4×Q8C22×C412yes—
304C4×Q8C22×C414yes—
305C4×Q8C22×C414yes—
306C4×Q8C22×C412yes—
307C4×Q8C22×C412yes—
308C4×Q8C2412yes—
309C4×Q8C2412yes—
310C4×Q8C2412yes—
311C4×Q8C2412yes—
312#117—24no—
313#117—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C22×C411yescharacteristic
330C2×C4⋊C4C22×C411yescharacteristic
331C2×C4⋊C4C22×C414yes—
332C2×C4⋊C4C22×C411yescharacteristic
333C2×C4⋊C4C22×C411yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42C22×C411yescharacteristic
345C2×C42C22×C412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42#4111yescharacteristic
349C2×C42#4112yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352#107C22×C414yes—
353#107C22×C414yes—
354#107C22×C414yes—
355#107C22×C414yes—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368C4⋊C8—24no—
369C4⋊C8C22×C414yes—
370C4⋊C8C22×C414yes—
371C4⋊C8C2×D414yes—
372C4⋊C8C2×D414yes—
373C4⋊C8C2412yes—
374C4⋊C8C2412yes—
375C4⋊C8C2412yes—
376C4⋊C8C2412yes—
377#103C22×C414yes—
378#103C22×C414yes—
379#103C22×C414yes—
380#103C22×C414yes—
381#103C22×C414yes—
382#103C22×C414yes—
383#103C22×C414yes—
384#103C22×C414yes—
385C22⋊C8—24no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388C22⋊C8C22×C414yes—
389C22⋊C8C22×C414yes—
390C22⋊C8C2×D414yes—
391C22⋊C8C2×D414yes—
392#97—24no—
393#97—24no—
394#97—24no—
395#97—24no—
396#97—24no—
397C4×C8—24no—
398C4×C8—24no—
399C4×C8—24no—
400C4×C8—24no—
401C4×C8—24no—
402C4×C8—21no—
403C4×C8—21no—
404C4×C8C22×C412yes—
405C4×C8C22×C412yes—
406C4×C8C2×D412yes—
407C4×C8C2×D412yes—
408C4×C8C2412yes—
409C4×C8C2412yes—
410C4×C8C2412yes—
411C4×C8C2412yes—
412C4×C8C2412yes—
413C4×C8C2412yes—
414C24C4×D411yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8#13114yes—
434C2×Q8C23×C412yes—
435C2×Q8C23×C412yes—
436C2×Q8C23×C412yes—
437C2×Q8C23×C412yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D412yes—
457C22×C4C4×D411yescharacteristic
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—28no—
469Q16—44no—
470Q16—44no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
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—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8C23×C412yes—
492C2×C8C23×C412yes—
493C2×C8C23×C412yes—
494C2×C8C23×C412yes—
495C2×C8C22×D412yes—
496C2×C8C22×D412yes—
497C2×C8C22×D412yes—
498C2×C8C22×D412yes—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4C23×C412yes—
520C4⋊C4C23×C412yes—
521C4⋊C4C23×C412yes—
522C4⋊C4C23×C412yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C42—24no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42C4×D412yes—
536C42C4×D412yes—
537C42C4×D412yes—
538C42C23×C411yescharacteristic
539C42C23×C412yes—
540C42C22×D412yes—
541C42C22×D411yescharacteristic
542C42C2×#4111yescharacteristic
543C42C2×#4112yes—
544C42C2511yescharacteristic
545C23—22no—
546C23—22no—
547C23—22no—
548C23C2×C4×D412yes—
549C23C2×C4×D411yescharacteristic
550Q8—24no—
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—48no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8C2×#13114yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4C2×C4×D412yes—
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D412yes—
597C2×C4C2×C4×D412yes—
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×#13112yes—
601C2×C4#56812yes—
602C2×C4#57812yes—
603C2×C4#57812yes—
604C2×C4C24×C411yescharacteristic
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C8—22no—
608C8—22no—
609C8—22no—
610C8—22no—
611C8—22no—
612C8—22no—
613C8—22no—
614C8—22no—
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—44no—
625C22#293312yes—
626C22#295012yes—
627C22#295512yes—
628C22C22×C4×D411yescharacteristic
629C4—22no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4#293312yes—
643C4#295012yes—
644C4C22×#13111yescharacteristic
645C4C2×#56811yescharacteristic
646C4C2×#57811yescharacteristic
647C4C2×#57811yescharacteristic
648C2—42no—
649C2#1803214yesdirect product factor
650C2C2×#293311yescharacteristic
651C2C2×#295011yescharacteristic
652C2C2×#295511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C2×C4×D4
prime radicalC24C4×D4
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
4. #18032C2direct (N × H)
5. C2×#1593C2semidirect
6. C2×#1591C2semidirect
7. C2×#1587C2semidirect
8. C2×#1587C2semidirect
9. C2×#1584C2semidirect
10. C2×#1584C2semidirect
11. C2×#1579C2semidirect
12. C2×C8×Q16C2semidirect
58. #1593C22semidirect
59. #1591C22semidirect
60. #1587C22semidirect
61. #1587C22semidirect
62. #1584C22semidirect
63. #1584C22semidirect
64. #1579C22semidirect
65. C8×Q16C22semidirect
649. C2#18032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#29862
C2×#29842
#180322
C2×#15932
C2×#15912
C2×#15872
C2×#15842
C2×#15792
C2×C8×Q162
C2×#15722
C2×#14972
C2×#14792
C2×#14542