Finite Group Databasebeta

← C2×#18079 #6345121

Subgroups — C2×#18079 #6345121

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C211yesmaximal, characteristic
2C2×#2969C211yesmaximal, characteristic
3C2×#2962C211yesmaximal, characteristic, complemented
4#18079C2116yesmaximal, direct product factor
5C2×#1593C211yesmaximal, characteristic, complemented
6C2×#1590C211yesmaximal, characteristic, complemented
7C2×#1587C211yesmaximal, characteristic, complemented
8C2×#1585C211yesmaximal, characteristic, complemented
9C2×#1584C211yesmaximal, characteristic, complemented
10C2×#1583C211yesmaximal, characteristic, complemented
11C2×SD16⋊C8C211yesmaximal, characteristic, complemented
12C2×C8×Q16C211yesmaximal, characteristic, complemented
13C2×#1565C211yesmaximal, characteristic, complemented
14C2×#1483C211yesmaximal, characteristic
15C2×#1485C211yesmaximal, characteristic, complemented
16C2×#1455C211yesmaximal, 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#2969—24no—
30#2969C2218yes—
31#2962—24no—
32#2962C2218yescomplemented
33C2×#446—21no—
34C2×#446—21no—
35C2×#446C2211yescharacteristic
36C2×#446C2211yescharacteristic
37C2×#446C2211yescharacteristic
38C2×C8×Q8C2211yescharacteristic
39C2×C8×Q8C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C4×Q16C2211yescharacteristic
43C2×C4×SD16C2211yescharacteristic
44C2×C4×SD16C2211yescharacteristic
45C2×#436—21no—
46C2×#436C2211yescharacteristic
47C2×#435—21no—
48C2×C8×D4C2211yescharacteristic
49C2×#433—21no—
50C2×#431C2211yescharacteristic
51C2×#424—21no—
52C22×C4⋊C8C2211yescharacteristic
53C2×#405—21no—
54C2×#405—21no—
55C2×C4×M16C2211yescharacteristic
56C2×C4×M16C2211yescharacteristic
57C2×C4×M16C2211yescharacteristic
58#1593C2218yescomplemented
59#1590C2218yescomplemented
60#1587C2218yescomplemented
61#1585C2218yescomplemented
62#1584C2218yescomplemented
63#1583C2218yescomplemented
64SD16⋊C8C2218yescomplemented
65C8×Q16C2218yescomplemented
66#1565C2218yescomplemented
67C2×#335C2211yescharacteristic
68C2×#335C2211yescharacteristic
69C2×#334C2211yescharacteristic
70C2×#334C2211yescharacteristic
71#1485C2218yescomplemented
72#1483C2218yes—
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#1455C2218yescomplemented
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C2×C82C2211yescharacteristic
86C22×Q16C2×C411yescharacteristic
87C22×Q16C2×C411yescharacteristic
88C22×SD16C2×C411yescharacteristic
89C22×SD16C2×C411yescharacteristic
90C22×M16—21no—
91C22×M16—21no—
92C22×M16C2×C411yescharacteristic
93C22×M16C2×C411yescharacteristic
94C22×M16D411yescharacteristic
95C22×M16D411yescharacteristic
96C23×C8—21no—
97C23×C8—21no—
98#533—22no—
99#533—22no—
100#533C2×C414yes—
101#533C2314yes—
102C2×C4⋊Q8C2×C411yescharacteristic
103C2×C4⋊Q8C2×C411yescharacteristic
104#529—22no—
105#529—22no—
106#529C2×C414yes—
107#529C2314yes—
108C2×#126—21no—
109C2×#125—21no—
110C2×#125C2×C411yescharacteristic
111C2×#124C2×C411yescharacteristic
112C2×#123—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2×C411yescharacteristic
115C2×C22⋊Q8C2×C411yescharacteristic
116C2×C22⋊Q8C2×C411yescharacteristic
117C2×#121C2×C411yescharacteristic
118C2×C4×Q8—21no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C22×C42C2311yescharacteristic
127#501C2×C418yes—
128#497C2×C418yes—
129#491C2×C418yes—
130#489C2×C418yes—
131#470C2×C418yes—
132#470C2×C418yes—
133#468C2×C418yes—
134#468C2×C418yes—
135#446—28no—
136#446—28no—
137#446—22no—
138#446—22no—
139#446—22no—
140#446C2314yes—
141#446C2314yes—
142#446C2314yes—
143C8×Q8—22no—
144C8×Q8—22no—
145C8×Q8—22no—
146C8×Q8C2314yes—
147C8×Q8C2314yes—
148C8×Q8C2314yes—
149C4×Q16C2×C414yes—
150C4×Q16C2×C414yes—
151C4×Q16C2314yes—
152C4×Q16C2314yes—
153C4×SD16C2×C414yes—
154C4×SD16C2×C414yes—
155C4×SD16C2314yes—
156C4×SD16C2314yes—
157#436—28no—
158#436—22no—
159#436C2314yes—
160#435—28no—
161C8×D4—22no—
162C8×D4C2314yes—
163#433—28no—
164#431—22no—
165#431C2314yes—
166C2×#107C2×C411yescharacteristic
167C2×#107C2×C411yescharacteristic
168C2×#106C2×C411yescharacteristic
169C2×#106C2×C411yescharacteristic
170#424—28no—
171C2×C4⋊C8—21no—
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—22no—
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×C4⋊C8C2311yescharacteristic
183C2×C4⋊C8C2314yes—
184C2×#103C2×C411yescharacteristic
185C2×#103C2×C411yescharacteristic
186C2×#103C2×C411yescharacteristic
187C2×#103C2×C411yescharacteristic
188C2×#103C2×C411yescharacteristic
189C2×#103C2×C411yescharacteristic
190C2×#102C2×C411yescharacteristic
191C2×#102C2×C411yescharacteristic
192C2×C22⋊C8—21no—
193C2×C22⋊C8—21no—
194C2×C22⋊C8—21no—
195#405—28no—
196#405—28no—
197C4×M16—22no—
198C4×M16C2×C414yes—
199C4×M16D414yes—
200C4×M16C2314yes—
201C4×M16C2314yes—
202C4×M16C2314yes—
203C2×#97—21no—
204C2×#97—21no—
205C2×#97—21no—
206C2×#97—21no—
207C2×#97C2×C411yescharacteristic
208C2×#97C2×C411yescharacteristic
209C2×#97D411yescharacteristic
210C2×#97D411yescharacteristic
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—24no—
258C2×M16—24no—
259C2×M16—42no—
260C2×M16—42no—
261C2×M16—44no—
262C2×M16—44no—
263C2×M16C22×C414yes—
264C2×M16C22×C414yes—
265C2×M16C2×D414yes—
266C2×M16C2×D414yes—
267C22×C8—21no—
268C22×C8—24no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—24no—
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#126—24no—
290#126—24no—
291#125—24no—
292#125—22no—
293#125—24no—
294#125C22×C414yes—
295#124—22no—
296#124C22×C414yes—
297#123—24no—
298#123—24no—
299C22⋊Q8—24no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—24no—
304C22⋊Q8C22×C414yes—
305C22⋊Q8C22×C414yes—
306C22⋊Q8C22×C414yes—
307#121—22no—
308#121C22×C414yes—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8C22×C412yes—
318C4×Q8C22×C412yes—
319C4×Q8C22×C412yes—
320C4×Q8C2412yes—
321C4×Q8C2412yes—
322C4×Q8C2412yes—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22×C412yes—
328C4×D4C2412yes—
329#117—24no—
330#117—24no—
331#117—24no—
332#117—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4C22×C411yescharacteristic
345C2×C4⋊C4C22×C411yescharacteristic
346C2×C4⋊C4C22×C411yescharacteristic
347C2×C4⋊C4C22×C411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42C22×C411yescharacteristic
359C2×C42C22×C412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42#4111yescharacteristic
363C2×C42#4112yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366#107C22×C414yes—
367#107C22×C414yes—
368#106C22×C414yes—
369#106C22×C414yes—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—21no—
377C4⋊C8—24no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8C2412yes—
382C4⋊C8C2412yes—
383C4⋊C8C2412yes—
384C4⋊C8C2412yes—
385#103C22×C414yes—
386#103C22×C414yes—
387#103C22×C414yes—
388#103C22×C414yes—
389#103C22×C414yes—
390#103C22×C414yes—
391#102C22×C414yes—
392#102C22×C414yes—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—24no—
396#97—24no—
397#97—24no—
398#97—24no—
399#97—24no—
400#97C22×C414yes—
401#97C22×C414yes—
402#97C2×D414yes—
403#97C2×D414yes—
404C4×C8—24no—
405C4×C8—24no—
406C4×C8—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—
421C24C4×D411yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—82no—
433C2×Q8C23×C412yes—
434C2×Q8C23×C412yes—
435C2×Q8C23×C412yes—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—82no—
441C2×D4C23×C412yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D411yescharacteristic
460C22×C4C4×D412yes—
461C22×C4C4×D411yescharacteristic
462C22×C4C4×D412yes—
463C22×C4C4×D412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4112yes—
470Q16—28no—
471Q16—44no—
472SD16—28no—
473SD16—44no—
474M16—28no—
475M16—28no—
476M16—44no—
477M16—44no—
478M16—48no—
479M16—48no—
480C2×C8—24no—
481C2×C8—22no—
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—48no—
494C2×C8—48no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8#13114yes—
501C2×C8#13714yes—
502C2×C8C23×C412yes—
503C2×C8C23×C412yes—
504C2×C8C23×C412yes—
505C2×C8C23×C412yes—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C2×C8C22×D412yes—
509C2×C8C22×D412yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4C23×C412yes—
529C4⋊C4C23×C412yes—
530C4⋊C4C23×C412yes—
531C4⋊C4C23×C412yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42C4×D412yes—
547C42C4×D412yes—
548C42C4×D412yes—
549C42C23×C411yescharacteristic
550C42C23×C412yes—
551C42C22×D412yes—
552C42C22×D411yescharacteristic
553C42C2×#4111yescharacteristic
554C42C2×#4112yes—
555C42C2511yescharacteristic
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C2×C4×D412yes—
564C23C2×C4×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—84no—
575D4—22no—
576D4—22no—
577D4—42no—
578D4—84no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
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—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D412yes—
610C2×C4C2×C4×D412yes—
611C2×C4C2×C4×D412yes—
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×#13112yes—
615C2×C4#56812yes—
616C2×C4C2×#13712yes—
617C2×C4#57512yes—
618C2×C4C24×C411yescharacteristic
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C8—24no—
622C8—22no—
623C8—22no—
624C8—22no—
625C8—22no—
626C8—22no—
627C8—22no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—42no—
632C8—42no—
633C8C2×#13114yes—
634C8C2×#13714yes—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—84no—
643C22#293512yes—
644C22C4×#13712yes—
645C22#295612yes—
646C22C22×C4×D411yescharacteristic
647C4—21no—
648C4—21no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4#293512yes—
660C4C4×#13712yes—
661C4C22×#13111yescharacteristic
662C4C2×#56811yescharacteristic
663C4C22×#13711yescharacteristic
664C4C2×#57511yescharacteristic
665C2—42no—
666C2—82no—
667C2#1807914yesdirect product factor
668C2C2×#293511yescharacteristic
669C2C2×C4×#13711yescharacteristic
670C2C2×#295611yescharacteristic
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×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. #18079C2direct (N × H)
5. C2×#1593C2semidirect
6. C2×#1590C2semidirect
7. C2×#1587C2semidirect
8. C2×#1585C2semidirect
9. C2×#1584C2semidirect
10. C2×#1583C2semidirect
11. C2×SD16⋊C8C2semidirect
12. C2×C8×Q16C2semidirect
13. C2×#1565C2semidirect
15. C2×#1485C2semidirect
16. C2×#1455C2semidirect
32. #2962C22semidirect
58. #1593C22semidirect
59. #1590C22semidirect
60. #1587C22semidirect
61. #1585C22semidirect
62. #1584C22semidirect
63. #1583C22semidirect
64. SD16⋊C8C22semidirect
65. C8×Q16C22semidirect
66. #1565C22semidirect
71. #1485C22semidirect
81. #1455C22semidirect
667. C2#18079direct (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×#29692
C2×#29622
#180792
C2×#15932
C2×#15902
C2×#15872
C2×#15852
C2×#15842
C2×#15832
C2×SD16⋊C82
C2×C8×Q162
C2×#15652
C2×#14832
C2×#14852
C2×#14552