Finite Group Databasebeta

← C2×#24348 #6445458

Subgroups — C2×#24348 #6445458

Order 512.

3210 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic
3C22×#441C211yesmaximal, characteristic
4C2×#2582C211yesmaximal, characteristic
5#24348C2116yesmaximal, direct product factor
6C2×#2059C211yesmaximal, characteristic
7C2×#2058C211yesmaximal, characteristic
8C2×#2026C211yesmaximal, characteristic
9C2×#2024C211yesmaximal, characteristic
10C2×#1992C211yesmaximal, characteristic
11C2×#1985C211yesmaximal, characteristic
12C2×#1878C211yesmaximal, characteristic
13C2×#1878C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic
15C2×#1863C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C2×#557—28no—
19C2×#557—28no—
20C2×#557—24no—
21C2×#557C2218yes—
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33C22×C4×Q8C2211yescharacteristic
34C2×#479—28no—
35C2×#479C2218yes—
36C2×#441—28no—
37C2×#441C2218yes—
38C22×#107C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C22×C4⋊C8C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C22×#97C2211yescharacteristic
49#2582—24no—
50#2582C2218yes—
51C2×#398—21no—
52C2×#398—21no—
53C2×#391—21no—
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#389—21no—
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#384C2211yescharacteristic
60#2059C2218yes—
61#2058C2218yes—
62#2026C2218yes—
63#2024C2218yes—
64#1992C2218yes—
65#1985C2218yes—
66#1878C2218yes—
67#1878C2218yes—
68#1865C2218yes—
69#1863C2218yes—
70#1803C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C2×#336C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C23×Q8—21no—
82C23×Q8D411yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C22×Q16—24no—
86C22×Q16—24no—
87C22×Q16Q818yescomplemented
88C22×Q16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93#557—264no—
94#557—232no—
95#557—232no—
96C2×C4⋊Q8—216no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×#125—216no—
103C2×#125—216no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—22no—
109C2×#125—22no—
110C2×#125—24no—
111C2×#125D418yes—
112C2×#125C2314yes—
113C2×#125C2314yes—
114C2×C4×Q8—216no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—28no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—22no—
127C2×C4×Q8—24no—
128C2×C4×Q8—24no—
129C2×C4×Q8D418yes—
130C2×C4×Q8D414yes—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2314yes—
133C2×C4×Q8C2314yes—
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2314yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4D411yescharacteristic
150C22×C4⋊C4D411yescharacteristic
151C22×C4⋊C4D411yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C42—21no—
159C22×C42—21no—
160C22×C42—21no—
161C22×C42—21no—
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165#479—232no—
166#441—232no—
167C2×#107—24no—
168C2×#107C2314yes—
169C2×#106—24no—
170C2×#106—24no—
171C2×#106C2314yes—
172C2×#106C2314yes—
173C2×C4⋊C8—24no—
174C2×C4⋊C8C2314yes—
175C2×#103—24no—
176C2×#103—24no—
177C2×#103—24no—
178C2×#103—24no—
179C2×#103—24no—
180C2×#103—24no—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#97—24no—
188C2×#97C2314yes—
189#398—28no—
190#398—28no—
191#391—28no—
192#391—22no—
193#391—22no—
194#391C2314yes—
195#391C2314yes—
196#389—28no—
197#389—22no—
198#389—22no—
199#389C2314yes—
200#389C2314yes—
201#384—22no—
202#384C2314yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219C22×Q8—28no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—42no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8C2×D414yes—
237C22×Q8C2×D414yes—
238C22×Q8C2×Q814yes—
239C22×Q8C2×Q814yes—
240C22×Q8#4114yes—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×Q16—232no—
259C2×Q16—28no—
260C2×Q16—28no—
261C2×Q16C2×Q8116yescomplemented
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2×Q814yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—264no—
279C4⋊Q8—216no—
280C4⋊Q8—28no—
281C4⋊Q8—28no—
282#125—264no—
283#125—264no—
284#125—216no—
285#125—28no—
286#125—28no—
287#125—216no—
288#125—28no—
289#125C2×D4116yes—
290C4×Q8—216no—
291C4×Q8—264no—
292C4×Q8—216no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—216no—
298C4×Q8—216no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—48no—
305C4×Q8C2×D4116yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—82no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4—82no—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C42—28no—
363C2×C42—28no—
364C2×C42—28no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—44no—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382#107—28no—
383#106—28no—
384#106—28no—
385C4⋊C8—28no—
386#103—28no—
387#103—28no—
388#103—28no—
389#103—28no—
390#103—28no—
391#103—28no—
392#97—28no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399C24C22×D411yescharacteristic
400C2×Q8—216no—
401C2×Q8—24no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—24no—
406C2×Q8—28no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—28no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—28no—
415C2×Q8—28no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—44no—
421C2×Q8—416no—
422C2×Q8—432no—
423C2×Q8—48no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—48no—
429C2×Q8C22⋊Q8116yes—
430C2×Q8C22×D414yes—
431C2×Q8C22×Q814yes—
432C2×Q8C22×Q814yes—
433C2×Q8C2×#4114yes—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4#12014yes—
482C22×C4#12014yes—
483C22×C4#13714yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4112yes—
495C22×C42−1+412yes—
496C22×C42−1+412yes—
497C22×C4C2511yescharacteristic
498Q16—232no—
499C2×C8—28no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8C22×Q814yes—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—216no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—48no—
539C4⋊C4—42no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4C22×D414yes—
547C4⋊C4C22×D414yes—
548C4⋊C4C22×D414yes—
549C42—216no—
550C42—216no—
551C42—216no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C42#12018yes—
560C42C22×D414yes—
561C42C22×D414yes—
562C23C2×#12012yes—
563C23#54812yes—
564C23D4×Q812yes—
565C23C2×#13712yes—
566C23#57512yes—
567C23#57712yes—
568C23#57812yes—
569C23C23×D411yescharacteristic
570Q8—216no—
571Q8—24no—
572Q8—24no—
573Q8—24no—
574Q8—24no—
575Q8—24no—
576Q8—416no—
577Q8—432no—
578Q8—48no—
579Q8—42no—
580Q8—42no—
581Q8C2×C22⋊Q818yes—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—216no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12014yes—
636C2×C4C2×#12014yes—
637C2×C4C2×#12012yes—
638C2×C4C2×C22⋊Q814yes—
639C2×C4#53614yes—
640C2×C4#54812yes—
641C2×C4#54814yes—
642C2×C4D4×Q812yes—
643C2×C4D4×Q814yes—
644C2×C4C2×#13714yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C23×D411yescharacteristic
647C2×C4C23×Q811yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22#240914yes—
656C22#300714yes—
657C22#301314yes—
658C22#337114yes—
659C22#337414yes—
660C22#338714yes—
661C22#339014yes—
662C22C22×#12011yescharacteristic
663C22C2×#54811yescharacteristic
664C22C2×D4×Q811yescharacteristic
665C22C22×#13711yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57811yescharacteristic
669C4—28no—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#240914yes—
684C4C22×C22⋊Q811yescharacteristic
685C4C2×#53611yescharacteristic
686C4C2×#54811yescharacteristic
687C4C2×D4×Q811yescharacteristic
688C2#2434818yesdirect product factor
689C2C2×#240911yescharacteristic
690C2C2×#300711yescharacteristic
691C2C2×#301311yescharacteristic
692C2C2×#337411yescharacteristic
693C2C2×#337111yescharacteristic
694C2C2×#338711yescharacteristic
695C2C2×#339011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #24348C2direct (N × H)
87. C22×Q16Q8semidirect
261. C2×Q16C2×Q8semidirect
688. C2#24348direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C22×#4792
C22×#4412
C2×#25822
#243482
C2×#20592
C2×#20582
C2×#20262
C2×#20242
C2×#19922
C2×#19852
C2×#18782
C2×#18652
C2×#18632
C2×#18032