Finite Group Databasebeta

← C2×#29975 #6433304

Subgroups — C2×#29975 #6433304

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#29975C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2088C211yesmaximal, characteristic, complemented
9C2×#2086C211yesmaximal, characteristic, complemented
10C2×#2058C211yesmaximal, characteristic, complemented
11C2×#2057C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1854C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15C2×#1823C211yesmaximal, characteristic
16C2×#1798C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
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#2628—24no—
30#2628C2218yes—
31C2×#400—21no—
32C2×#399—21no—
33C2×#395—21no—
34C2×#393—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×#382C2211yescharacteristic
46C2×#380—21no—
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50#2101C2218yescomplemented
51#2101C2218yescomplemented
52#2088C2218yescomplemented
53#2088C2218yescomplemented
54#2087C2218yescomplemented
55#2086C2218yescomplemented
56#2058C2218yescomplemented
57#2057C2218yescomplemented
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
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—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⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4Q818yescomplemented
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
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—42no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—42no—
131C2×#106—42no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—42no—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C22⋊C8—24no—
139C2×C22⋊C8—24no—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142#400—28no—
143#399—28no—
144#395—28no—
145#393—28no—
146#389—28no—
147#389—28no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#382—22no—
161#382C2314yes—
162#380—22no—
163#380—22no—
164#380—28no—
165#380—28no—
166#380C2314yes—
167#380C2314yes—
168C2×#95—21no—
169C2×#95—21no—
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
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8—82no—
196C22×Q8—82no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—82no—
212C23×C4—82no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×Q814yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
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×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241C4⋊Q8—48no—
242#117—48no—
243#117—44no—
244#117—44no—
245#117—44no—
246#117—44no—
247#117—48no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
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—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4C2×Q814yes—
294C2×C4⋊C4C2×Q8116yescomplemented
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—44no—
324#107—44no—
325#106—44no—
326#106—44no—
327C4⋊C8—44no—
328C4⋊C8—44no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331#95—24no—
332#95—24no—
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
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—816no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
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—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
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—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—816no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C22⋊Q814yes—
415C22×C4C22⋊Q818yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429C22×C42−1+412yes—
430C22×C4C2511yescharacteristic
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—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—216no—
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—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C22×Q814yes—
480C4⋊C4C22×Q814yes—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—48no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×C22⋊Q814yes—
507C23C2×C22⋊Q812yes—
508C23#53412yes—
509C23#53612yes—
510C23#57612yes—
511C23#57712yes—
512C23#57712yes—
513C23#57812yes—
514C23C23×D411yescharacteristic
515Q8—816no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
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—22no—
537C2×C4—22no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
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—42no—
565C2×C4—42no—
566C2×C4—816no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4C2×C22⋊Q814yes—
574C2×C4C2×C22⋊Q812yes—
575C2×C4#53412yes—
576C2×C4#53612yes—
577C2×C4#53914yes—
578C2×C4#53914yes—
579C2×C4#54414yes—
580C2×C4#54414yes—
581C2×C4C23×Q811yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×H(5,𝔽2)11yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22#262714yes—
600C22#308114yes—
601C22#308214yes—
602C22#319814yes—
603C22#320114yes—
604C22#324314yes—
605C22#324414yes—
606C22C22×C22⋊Q811yescharacteristic
607C22C2×#53411yescharacteristic
608C22C2×#53611yescharacteristic
609C22C2×#57611yescharacteristic
610C22C2×#57711yescharacteristic
611C22C2×#57711yescharacteristic
612C22C2×#57811yescharacteristic
613C4—24no—
614C4—44no—
615C4—44no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#262714yes—
627C4C2×#53911yescharacteristic
628C4C2×#53911yescharacteristic
629C4C2×#54411yescharacteristic
630C4C2×#54411yescharacteristic
631C2—44no—
632C2#2997518yesdirect product factor
633C2C2×#262711yescharacteristic
634C2C2×#308111yescharacteristic
635C2C2×#308211yescharacteristic
636C2C2×#319811yescharacteristic
637C2C2×#320111yescharacteristic
638C2C2×#324311yescharacteristic
639C2C2×#324411yescharacteristic
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. #29975C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2101C2semidirect
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
8. C2×#2088C2semidirect
9. C2×#2086C2semidirect
10. C2×#2058C2semidirect
11. C2×#2057C2semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2088C22semidirect
53. #2088C22semidirect
54. #2087C22semidirect
55. #2086C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
108. C22×C4⋊C4Q8semidirect
294. C2×C4⋊C4C2×Q8semidirect
632. C2#29975direct (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
#299752
C2×#21012
C2×#20872
C2×#20882
C2×#20862
C2×#20582
C2×#20572
C2×#18542
C2×#18242
C2×#18232
C2×#17982