Finite Group Databasebeta

← C2×#29672 #6435002

Subgroups — C2×#29672 #6435002

Order 512.

3548 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2C22×#462C211yesmaximal, characteristic, complemented
3C22×#461C211yesmaximal, characteristic, complemented
4C2×#2621C211yesmaximal, characteristic, complemented
5#29672C2116yesmaximal, direct product factor
6C2×#2024C212yesmaximal, complemented
7C2×#2015C212yesmaximal, complemented
8C2×#2001C212yesmaximal, complemented
9C2×#2000C212yesmaximal
10C2×#1805C211yesmaximal, characteristic, complemented
11C2×#1804C211yesmaximal, characteristic, complemented
12C2×#1798C211yesmaximal, characteristic, complemented
13C23×Q16C2212yes—
14C23×SD16C2212yes—
15C2×#542—28no—
16C2×#542—28no—
17C2×#542—24no—
18C2×#542C2218yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126—21no—
22C22×#124—21no—
23C22×#124C2211yescharacteristic
24C22×#123—21no—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×#121C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C2×#462—28no—
33C2×#462C2218yescomplemented
34C2×#461—28no—
35C2×#461C2218yescomplemented
36C22×C4⋊C8C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×#103C2212yes—
41C22×#102C2212yes—
42C22×C22⋊C8C2212yes—
43#2621—24no—
44#2621C2218yescomplemented
45C2×#398—21no—
46C2×#398—21no—
47C2×#397—21no—
48C2×#397—21no—
49C2×#391C2212yes—
50C2×#382C2211yescharacteristic
51#2024C22116yescomplemented
52#2015C22116yescomplemented
53#2001C22116yescomplemented
54#2000C22116yes—
55#1805C2218yescomplemented
56#1804C2218yescomplemented
57#1798C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8C2312yes—
63C23×Q8C2311yescharacteristic
64C23×D4—21no—
65C23×D4C2311yescharacteristic
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C22×Q16—28no—
69C22×Q16—44no—
70C22×Q16—44no—
71C22×Q16C2318yes—
72C22×SD16—28no—
73C22×SD16—44no—
74C22×SD16—44no—
75C22×SD16C2318yes—
76C23×C8C2312yes—
77C23×C8C2312yes—
78#542—264no—
79#542—232no—
80#542—232no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2314yes—
88C2×C4⋊Q8C2314yes—
89C2×#126—216no—
90C2×#126—28no—
91C2×#124—216no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124—28no—
95C2×#124—22no—
96C2×#124C2314yes—
97C2×#123—216no—
98C2×#123—28no—
99C2×C22⋊Q8—216no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—44no—
106C2×C22⋊Q8—44no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2318yes—
109C2×#121—24no—
110C2×#121—22no—
111C2×#121—24no—
112C2×#121C2314yes—
113C2×C4×Q8—24no—
114C2×C4×Q8—22no—
115C2×C4×Q8D418yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—22no—
118C2×C4×D4—24no—
119C2×C4×D4D418yes—
120C2×C4×D4C2314yes—
121C2×#117—24no—
122C2×#117—24no—
123C2×#117D414yes—
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4D411yescharacteristic
143C22×C22⋊C4D411yescharacteristic
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#462—232no—
148#461—232no—
149C2×C4⋊C8—24no—
150C2×C4⋊C8—24no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×#103—28no—
154C2×#103—28no—
155C2×#103—44no—
156C2×#103C2318yes—
157C2×#103C2318yes—
158C2×#103C2318yes—
159C2×#102—44no—
160C2×#102C2318yes—
161C2×C22⋊C8—44no—
162C2×C22⋊C8C2318yes—
163#398—28no—
164#398—28no—
165#397—28no—
166#397—28no—
167#391—24no—
168#391C2318yes—
169#382—22no—
170#382C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—22no—
174C2×#95—22no—
175#340C2318yes—
176#340C2318yes—
177#336C2318yes—
178C25—21no—
179C25C2×D411yescharacteristic
180C22×Q8—28no—
181C22×Q8—22no—
182C22×Q8—21no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—84no—
196C22×Q8C2×D414yes—
197C22×Q8C2414yes—
198C22×Q8C2412yes—
199C22×D4—28no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—84no—
208C22×D4C2×D414yes—
209C22×D4C2412yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—84no—
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C2×Q16—216no—
233C2×Q16—48no—
234C2×Q16—48no—
235C2×Q16—432no—
236C2×SD16—216no—
237C2×SD16—48no—
238C2×SD16—48no—
239C2×SD16—432no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—216no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—48no—
254#126—264no—
255#124—264no—
256#124—216no—
257#124—28no—
258#124—28no—
259#123—264no—
260C22⋊Q8—264no—
261C22⋊Q8—216no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C22⋊Q8—416no—
267#121—216no—
268#121—28no—
269#121—28no—
270C4×Q8—216no—
271C4×Q8—28no—
272C4×Q8C2×D4116yes—
273C4×D4—216no—
274C4×D4—28no—
275C4×D4C2×D4116yes—
276#117—216no—
277#117—28no—
278#117—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C42—28no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C4⋊C8—28no—
334C4⋊C8—28no—
335#103—216no—
336#103—216no—
337#103—48no—
338#102—48no—
339C22⋊C8—48no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—42no—
352C24—42no—
353C24—44no—
354C24—44no—
355C24C22×D412yes—
356C24C22×D411yescharacteristic
357C2×Q8—216no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—88no—
384C2×Q8—816no—
385C2×Q8C22×D414yes—
386C2×D4—216no—
387C2×D4—24no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—88no—
400C2×D4—816no—
401C2×D4C22×D414yes—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—816no—
444C22×C4—88no—
445C22×C4#12014yes—
446C22×C4#12014yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460Q16—432no—
461SD16—432no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—48no—
469C2×C8—48no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4C22×D414yes—
486C4⋊C4C22×D414yes—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—416no—
495C42—216no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42#12018yes—
500C42C22×D414yes—
501C42C22×D414yes—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—44no—
509C23—48no—
510C23—42no—
511C23—42no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—88no—
521C23—88no—
522C23C2×#12012yes—
523C23#53512yes—
524C23#53712yes—
525C23#57714yes—
526C23#57814yes—
527C23C23×D411yescharacteristic
528Q8—24no—
529Q8—24no—
530Q8—48no—
531Q8—42no—
532Q8—42no—
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—816no—
538Q8—88no—
539D4—24no—
540D4—24no—
541D4—48no—
542D4—42no—
543D4—42no—
544D4—816no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—816no—
581C2×C4—88no—
582C2×C4—88no—
583C2×C4C2×#12014yes—
584C2×C4C2×#12014yes—
585C2×C4C2×#12012yes—
586C2×C4#53512yes—
587C2×C4#53514yes—
588C2×C4#53514yes—
589C2×C4#53712yes—
590C2×C4#53714yes—
591C2×C4#53714yes—
592C2×C4C23×D411yescharacteristic
593C2×C4C23×D411yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C8—44no—
599C8—44no—
600C22—42no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—84no—
609C22—88no—
610C22—816no—
611C22—84no—
612C22#261514yes—
613C22#300914yes—
614C22#301114yes—
615C22#322518yes—
616C22#325818yes—
617C22C22×#12011yescharacteristic
618C22C2×#53511yescharacteristic
619C22C2×#53711yescharacteristic
620C22C2×#57712yes—
621C22C2×#57812yes—
622C4—24no—
623C4—24no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—84no—
631C4—84no—
632C4#261514yes—
633C4C2×#53511yescharacteristic
634C4C2×#53511yescharacteristic
635C4C2×#53711yescharacteristic
636C4C2×#53711yescharacteristic
637C2—82no—
638C2—84no—
639C2#2967218yesdirect product factor
640C2C2×#261511yescharacteristic
641C2C2×#300911yescharacteristic
642C2C2×#301111yescharacteristic
643C2C2×#322512yes—
644C2C2×#325812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C22×#462C2semidirect
3. C22×#461C2semidirect
4. C2×#2621C2semidirect
5. #29672C2direct (N × H)
6. C2×#2024C2semidirect
7. C2×#2015C2semidirect
8. C2×#2001C2semidirect
10. C2×#1805C2semidirect
11. C2×#1804C2semidirect
12. C2×#1798C2semidirect
33. C2×#462C22semidirect
35. C2×#461C22semidirect
44. #2621C22semidirect
51. #2024C22semidirect
52. #2015C22semidirect
53. #2001C22semidirect
55. #1805C22semidirect
56. #1804C22semidirect
57. #1798C22semidirect
639. C2#29672direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5422
C22×#4622
C22×#4612
C2×#26212
#296722
C2×#20242
C2×#20152
C2×#20012
C2×#20002
C2×#18052
C2×#18042
C2×#17982