Finite Group Databasebeta

← C2×#29308 #6461313

Subgroups — C2×#29308 #6461313

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#29308C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2085C211yesmaximal, characteristic, complemented
6C2×#2058C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#1944C211yesmaximal, characteristic, complemented
9C2×#1943C211yesmaximal, characteristic, complemented
10C2×#1942C211yesmaximal, characteristic, complemented
11C2×#1941C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1854C211yesmaximal, characteristic
14C2×#1853C211yesmaximal, characteristic
15C2×#1852C211yesmaximal, characteristic
16C2×#1750C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2628—24no—
34#2628C2218yes—
35C2×#400—21no—
36C2×#399—21no—
37C2×#395—21no—
38C2×#393—21no—
39#2316—24no—
40#2316C2218yes—
41C2×#389—21no—
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#380—21no—
51C2×#380—21no—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54#2085C2218yescomplemented
55#2085C2218yescomplemented
56#2058C2218yescomplemented
57#2057C2218yescomplemented
58#1944C2218yescomplemented
59#1943C2218yescomplemented
60#1942C2218yescomplemented
61#1941C2218yescomplemented
62#1854C2218yes—
63#1854C2218yes—
64#1853C2218yes—
65#1852C2218yes—
66#1750C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2314yes—
88C2×#117—22no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117D414yes—
94C2×#117C2314yes—
95C2×#117C2314yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—24no—
127C2×#107—24no—
128C2×#107—42no—
129C2×#107—42no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106—42no—
137C2×#106—42no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C22⋊C8Q818yescomplemented
143C2×C22⋊C8Q818yescomplemented
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97—42no—
147C2×#97C2314yes—
148C2×C4×C8—42no—
149C2×C4×C8C2314yes—
150#400—28no—
151#399—28no—
152#395—28no—
153#393—28no—
154#389—28no—
155#389—28no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#382—22no—
169#382C2314yes—
170#380—22no—
171#380—22no—
172#380—28no—
173#380—28no—
174#380C2314yes—
175#380C2314yes—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25C2×D411yescharacteristic
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—82no—
200C22×Q8—82no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—82no—
216C23×C4—82no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×Q814yes—
224C23×C4C2×Q814yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8C2×Q814yes—
236C22×C8C2×Q814yes—
237C22×C8C2×Q814yes—
238C22×C8C2×Q814yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—48no—
246#117—48no—
247#117—44no—
248#117—44no—
249#117—44no—
250#117—44no—
251#117—48no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4#4114yes—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#107—28no—
328#107—28no—
329#107—44no—
330#107—44no—
331#106—28no—
332#106—28no—
333#106—44no—
334#106—44no—
335C22⋊C8C2×Q8116yescomplemented
336C22⋊C8C2×Q8116yescomplemented
337#97—44no—
338C4×C8—44no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C4⋊Q814yes—
356C24C22×D412yes—
357C24C22×D411yescharacteristic
358C2×Q8—48no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—816no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—816no—
422C22×C4C4⋊Q814yes—
423C22×C4C4⋊Q818yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q814yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q812yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4C2511yescharacteristic
439C2×C8—28no—
440C2×C8—28no—
441C2×C8—28no—
442C2×C8—28no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8C22×Q814yes—
452C2×C8C22×Q814yes—
453C2×C8C22×Q814yes—
454C2×C8C22×Q814yes—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4C2×#4114yes—
488C4⋊C4C2×#4114yes—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—48no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23#52512yes—
515C23C2×C4⋊Q814yes—
516C23C2×C4⋊Q812yes—
517C23#53412yes—
518C23#57612yes—
519C23#57712yes—
520C23#57712yes—
521C23#57812yes—
522C23C23×D411yescharacteristic
523Q8—816no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—216no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4#52512yes—
582C2×C4C2×C4⋊Q814yes—
583C2×C4C2×C4⋊Q812yes—
584C2×C4#53412yes—
585C2×C4#55414yes—
586C2×C4#55414yes—
587C2×C4#55414yes—
588C2×C4#55414yes—
589C2×C4C23×Q811yescharacteristic
590C2×C4C23×Q811yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C8—24no—
596C8—24no—
597C8—24no—
598C8—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—416no—
605C22—42no—
606C22—42no—
607C22#258314yes—
608C22#311814yes—
609C22#312114yes—
610C22#316314yes—
611C22#316414yes—
612C22#319814yes—
613C22#320114yes—
614C22C2×#52511yescharacteristic
615C22C22×C4⋊Q811yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#57611yescharacteristic
618C22C2×#57711yescharacteristic
619C22C2×#57711yescharacteristic
620C22C2×#57811yescharacteristic
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4#258314yes—
635C4C2×#55411yescharacteristic
636C4C2×#55411yescharacteristic
637C4C2×#55411yescharacteristic
638C4C2×#55411yescharacteristic
639C2—44no—
640C2#2930818yesdirect product factor
641C2C2×#258311yescharacteristic
642C2C2×#311811yescharacteristic
643C2C2×#312111yescharacteristic
644C2C2×#316311yescharacteristic
645C2C2×#316411yescharacteristic
646C2C2×#319811yescharacteristic
647C2C2×#320111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29308C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2085C2semidirect
6. C2×#2058C2semidirect
7. C2×#2057C2semidirect
8. C2×#1944C2semidirect
9. C2×#1943C2semidirect
10. C2×#1942C2semidirect
11. C2×#1941C2semidirect
54. #2085C22semidirect
55. #2085C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #1944C22semidirect
59. #1943C22semidirect
60. #1942C22semidirect
61. #1941C22semidirect
142. C2×C22⋊C8Q8semidirect
143. C2×C22⋊C8Q8semidirect
335. C22⋊C8C2×Q8semidirect
336. C22⋊C8C2×Q8semidirect
640. C2#29308direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23162
#293082
C2×#20852
C2×#20582
C2×#20572
C2×#19442
C2×#19432
C2×#19422
C2×#19412
C2×#18542
C2×#18532
C2×#18522
C2×#17502