Finite Group Databasebeta

← C2×#29978 #6433305

Subgroups — C2×#29978 #6433305

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#29978C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2088C211yesmaximal, characteristic, complemented
9C2×#2086C211yesmaximal, characteristic, complemented
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2059C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1854C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15C2×#1823C211yesmaximal, characteristic
16C2×#1798C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2629—24no—
30#2629C2218yes—
31C2×#398—21no—
32C2×#398—21no—
33C2×#396—21no—
34C2×#396—21no—
35#2316—24no—
36#2316C2218yes—
37C2×#389—21no—
38C2×#389—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#380—21no—
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50#2100C2218yescomplemented
51#2099C2218yescomplemented
52#2088C2218yescomplemented
53#2088C2218yescomplemented
54#2087C2218yescomplemented
55#2086C2218yescomplemented
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#1854C2218yes—
59#1854C2218yes—
60#1824C2218yes—
61#1823C2218yes—
62#1798C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—22no—
84C2×#125—42no—
85C2×#125—42no—
86C2×#125C2314yes—
87C2×#117—22no—
88C2×#117—42no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117D414yes—
93C2×#117C2314yes—
94C2×#117C2314yes—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
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⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4Q818yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#107—42no—
129C2×#107—42no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106—42no—
133C2×#106—42no—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—42no—
137C2×C4⋊C8—42no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C22⋊C8—24no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144#398—28no—
145#398—28no—
146#396—28no—
147#396—28no—
148#389—28no—
149#389—28no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#384—22no—
163#384C2314yes—
164#380—22no—
165#380—22no—
166#380—28no—
167#380—28no—
168#380C2314yes—
169#380C2314yes—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25C2×D411yescharacteristic
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—82no—
208C23×C4—82no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235#125—44no—
236#125—44no—
237#125—48no—
238#117—48no—
239#117—44no—
240#117—44no—
241#117—44no—
242#117—44no—
243#117—48no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q814yes—
296C2×C4⋊C4C2×Q8116yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325#107—44no—
326#107—44no—
327#106—44no—
328#106—44no—
329C4⋊C8—44no—
330C4⋊C8—44no—
331C22⋊C8—28no—
332C22⋊C8—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24C22⋊Q814yes—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
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—816no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C22⋊Q814yes—
411C22×C4C22⋊Q818yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q814yes—
419C22×C4C22×Q812yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4112yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C22×C42−1+412yes—
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8—44no—
442C2×C8—44no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4C22×Q814yes—
479C4⋊C4C22×Q814yes—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C2×C22⋊Q814yes—
506C23C2×C22⋊Q812yes—
507C23#53412yes—
508C23#53612yes—
509C23#57612yes—
510C23#57712yes—
511C23#57712yes—
512C23#57812yes—
513C23C23×D411yescharacteristic
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—216no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
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—816no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4C2×C22⋊Q814yes—
572C2×C4C2×C22⋊Q812yes—
573C2×C4#53412yes—
574C2×C4#53612yes—
575C2×C4#53914yes—
576C2×C4#53914yes—
577C2×C4#54414yes—
578C2×C4#54414yes—
579C2×C4C23×Q811yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C8—42no—
586C8—42no—
587C8—42no—
588C8—42no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—416no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22#262714yes—
598C22#308114yes—
599C22#308214yes—
600C22#320314yes—
601C22#320414yes—
602C22#324114yes—
603C22#324214yes—
604C22C22×C22⋊Q811yescharacteristic
605C22C2×#53411yescharacteristic
606C22C2×#53611yescharacteristic
607C22C2×#57611yescharacteristic
608C22C2×#57711yescharacteristic
609C22C2×#57711yescharacteristic
610C22C2×#57811yescharacteristic
611C4—24no—
612C4—44no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4#262714yes—
625C4C2×#53911yescharacteristic
626C4C2×#53911yescharacteristic
627C4C2×#54411yescharacteristic
628C4C2×#54411yescharacteristic
629C2—44no—
630C2#2997818yesdirect product factor
631C2C2×#262711yescharacteristic
632C2C2×#308111yescharacteristic
633C2C2×#308211yescharacteristic
634C2C2×#320311yescharacteristic
635C2C2×#320411yescharacteristic
636C2C2×#324111yescharacteristic
637C2C2×#324211yescharacteristic
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. #29978C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2099C2semidirect
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
8. C2×#2088C2semidirect
9. C2×#2086C2semidirect
10. C2×#2059C2semidirect
11. C2×#2059C2semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2088C22semidirect
53. #2088C22semidirect
54. #2087C22semidirect
55. #2086C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
630. C2#29978direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23162
#299782
C2×#21002
C2×#20992
C2×#20872
C2×#20882
C2×#20862
C2×#20592
C2×#18542
C2×#18242
C2×#18232
C2×#17982