Finite Group Databasebeta

← C2×#32103 #7507658

Subgroups — C2×#32103 #7507658

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2506C211yesmaximal, characteristic, complemented
3#32103C2116yesmaximal, direct product factor
4C2×#2095C211yesmaximal, characteristic, complemented
5C2×#2068C211yesmaximal, characteristic, complemented
6C2×#2043C211yesmaximal, characteristic, complemented
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic
9C2×#2042C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1959C211yesmaximal, characteristic
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1928C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1862C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2547—24no—
39#2547C2218yes—
40#2506—24no—
41#2506C2218yescomplemented
42C2×#400—21no—
43C2×#398—21no—
44C2×#397—21no—
45C2×#397—21no—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#384C2211yescharacteristic
54C2×#380—21no—
55C2×#380—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2095C2218yescomplemented
59#2068C2218yescomplemented
60#2043C2218yescomplemented
61#2043C2218yescomplemented
62#2042C2218yes—
63#2042C2218yes—
64#1983C2218yes—
65#1959C2218yes—
66#1935C2218yescomplemented
67#1928C2218yescomplemented
68#1917C2218yescomplemented
69#1862C2218yes—
70#1862C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8C2314yes—
89C2×#125—28no—
90C2×#125—48no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—22no—
94C2×#124—22no—
95C2×#124—42no—
96C2×#124—42no—
97C2×#124C2314yes—
98C2×#124C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
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⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#107—42no—
128C2×#107—42no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102—42no—
152C2×#102—42no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97—42no—
158C2×#97C2314yes—
159#400—28no—
160#398—28no—
161#397—28no—
162#397—28no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175#387—28no—
176#384—22no—
177#384C2314yes—
178#380—28no—
179#380—28no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D414yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8C2412yes—
217C22×Q8C2412yes—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4#4114yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263#125—432no—
264#124—28no—
265#124—28no—
266#124—44no—
267#124—44no—
268#124—48no—
269#124—48no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—48no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—44no—
338#107—44no—
339C4⋊C8—44no—
340#103—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352#97—44no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—24no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—816no—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—44no—
489C2×C8—44no—
490C2×C8—44no—
491C2×C8—44no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4C2×#4114yes—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C42—24no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—48no—
532C23—24no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—48no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23#53512yes—
547C23#54612yes—
548C23#54812yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23#57512yes—
551C23#57812yes—
552C23#57812yes—
553C23C23×D411yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—84no—
563Q8—84no—
564D4—24no—
565D4—24no—
566D4—42no—
567D4—48no—
568D4—42no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—816no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#53512yes—
622C2×C4#53914yes—
623C2×C4#54014yes—
624C2×C4#54612yes—
625C2×C4#54812yes—
626C2×C4#55314yes—
627C2×C4#56214yes—
628C2×C4C2×AGL(1,ℤ/8)14yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—816no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22#271114yes—
648C22#321314yes—
649C22#322314yes—
650C22#331514yes—
651C22#333014yes—
652C22#336714yes—
653C22#337714yes—
654C22C2×#53511yescharacteristic
655C22C2×#54611yescharacteristic
656C22C2×#54811yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57811yescharacteristic
660C22C2×#57811yescharacteristic
661C4—24no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#271114yes—
675C4C2×#53911yescharacteristic
676C4C2×#54011yescharacteristic
677C4C2×#55311yescharacteristic
678C4C2×#56211yescharacteristic
679C2—84no—
680C2#3210318yesdirect product factor
681C2C2×#271111yescharacteristic
682C2C2×#321311yescharacteristic
683C2C2×#322311yescharacteristic
684C2C2×#331511yescharacteristic
685C2C2×#333011yescharacteristic
686C2C2×#336711yescharacteristic
687C2C2×#337711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2506C2semidirect
3. #32103C2direct (N × H)
4. C2×#2095C2semidirect
5. C2×#2068C2semidirect
6. C2×#2043C2semidirect
7. C2×#2043C2semidirect
12. C2×#1935C2semidirect
13. C2×#1928C2semidirect
14. C2×#1917C2semidirect
41. #2506C22semidirect
58. #2095C22semidirect
59. #2068C22semidirect
60. #2043C22semidirect
61. #2043C22semidirect
66. #1935C22semidirect
67. #1928C22semidirect
68. #1917C22semidirect
680. C2#32103direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25062
#321032
C2×#20952
C2×#20682
C2×#20432
C2×#20422
C2×#19832
C2×#19592
C2×#19352
C2×#19282
C2×#19172
C2×#18622