Finite Group Databasebeta

← C2×#32109 #7507664

Subgroups — C2×#32109 #7507664

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#2507C211yesmaximal, characteristic, complemented
3#32109C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2067C211yesmaximal, characteristic, complemented
6C2×#2042C211yesmaximal, characteristic, complemented
7C2×#2042C211yesmaximal, characteristic, complemented
8C2×#2041C211yesmaximal, characteristic
9C2×#2041C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic, complemented
11C2×#1959C211yesmaximal, characteristic, complemented
12C2×#1936C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1863C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2548—24no—
38#2548C2218yes—
39#2507—24no—
40#2507C2218yescomplemented
41C2×#398—21no—
42C2×#398—21no—
43C2×#394—21no—
44C2×#394—21no—
45C2×#390—21no—
46C2×#390—21no—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#387—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2093C2218yescomplemented
59#2067C2218yescomplemented
60#2042C2218yescomplemented
61#2042C2218yescomplemented
62#2041C2218yes—
63#2041C2218yes—
64#1983C2218yescomplemented
65#1959C2218yescomplemented
66#1936C2218yescomplemented
67#1925C2218yescomplemented
68#1917C2218yescomplemented
69#1863C2218yescomplemented
70#1863C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#127—22no—
87C2×#127—42no—
88C2×#127—42no—
89C2×#127C2314yes—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—22no—
93C2×#124—22no—
94C2×#124—42no—
95C2×#124—42no—
96C2×#124C2314yes—
97C2×#124C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—42no—
100C2×C4×Q8—42no—
101C2×C4×Q8C2314yes—
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⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#107—42no—
130C2×#107—42no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#103—24no—
136C2×#103—24no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102—42no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#97—42no—
160C2×#97C2314yes—
161#398—28no—
162#398—28no—
163#394—28no—
164#394—28no—
165#390—28no—
166#390—28no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—28no—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#387—28no—
181#384—22no—
182#384C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D414yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8#4114yes—
209C22×Q8C2412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
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×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263#127—44no—
264#127—44no—
265#127—48no—
266#124—28no—
267#124—28no—
268#124—44no—
269#124—44no—
270#124—48no—
271#124—48no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
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—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
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—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—82no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#107—44no—
347C4⋊C8—44no—
348#103—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360#97—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—22no—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×#4114yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—816no—
431C2×D4—816no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4#13714yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—44no—
502C2×C8—44no—
503C2×C8—44no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4C2×#4114yes—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—84no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#53512yes—
576C23#54612yes—
577C23#54812yes—
578C23C2×#13712yes—
579C23#57512yes—
580C23#57612yes—
581C23#57612yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—24no—
585Q8—42no—
586Q8—48no—
587Q8—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—816no—
597D4—816no—
598D4—84no—
599D4—84no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#53512yes—
648C2×C4#53914yes—
649C2×C4#54014yes—
650C2×C4#54612yes—
651C2×C4#54812yes—
652C2×C4#55314yes—
653C2×C4#56214yes—
654C2×C4C2×#13714yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—42no—
662C8—42no—
663C8—42no—
664C8—42no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—168no—
680C22—168no—
681C22#271114yes—
682C22#321514yes—
683C22#322014yes—
684C22#330914yes—
685C22#333314yes—
686C22#337014yes—
687C22#337514yes—
688C22C2×#53511yescharacteristic
689C22C2×#54611yescharacteristic
690C22C2×#54811yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57611yescharacteristic
694C22C2×#57611yescharacteristic
695C4—24no—
696C4—42no—
697C4—44no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#271114yes—
708C4C2×#53911yescharacteristic
709C4C2×#54011yescharacteristic
710C4C2×#55311yescharacteristic
711C4C2×#56211yescharacteristic
712C2—162no—
713C2—162no—
714C2#3210918yesdirect product factor
715C2C2×#271111yescharacteristic
716C2C2×#321511yescharacteristic
717C2C2×#322011yescharacteristic
718C2C2×#330911yescharacteristic
719C2C2×#333311yescharacteristic
720C2C2×#337511yescharacteristic
721C2C2×#337011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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×#2507C2semidirect
3. #32109C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2067C2semidirect
6. C2×#2042C2semidirect
7. C2×#2042C2semidirect
10. C2×#1983C2semidirect
11. C2×#1959C2semidirect
12. C2×#1936C2semidirect
13. C2×#1925C2semidirect
14. C2×#1917C2semidirect
15. C2×#1863C2semidirect
16. C2×#1863C2semidirect
40. #2507C22semidirect
58. #2093C22semidirect
59. #2067C22semidirect
60. #2042C22semidirect
61. #2042C22semidirect
64. #1983C22semidirect
65. #1959C22semidirect
66. #1936C22semidirect
67. #1925C22semidirect
68. #1917C22semidirect
69. #1863C22semidirect
70. #1863C22semidirect
714. C2#32109direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#25072
#321092
C2×#20932
C2×#20672
C2×#20422
C2×#20412
C2×#19832
C2×#19592
C2×#19362
C2×#19252
C2×#19172
C2×#18632