Finite Group Databasebeta

← C2×#31379 #7504526

Subgroups — C2×#31379 #7504526

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic
3#31379C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic, complemented
8C2×#2034C211yesmaximal, characteristic
9C2×#2023C211yesmaximal, characteristic, complemented
10C2×#1993C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic, complemented
13C2×#1875C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2439—24no—
32#2439C2218yescomplemented
33#2420—24no—
34#2420C2218yes—
35C2×#397—21no—
36C2×#396—21no—
37C2×#394—21no—
38C2×#393—21no—
39C2×#391—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#386—21no—
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2088C2218yescomplemented
54#2085C2218yescomplemented
55#2075C2218yescomplemented
56#2034C2218yes—
57#2023C2218yescomplemented
58#1993C2218yescomplemented
59#1936C2218yescomplemented
60#1924C2218yescomplemented
61#1875C2218yes—
62#1862C2218yescomplemented
63#1825C2218yescomplemented
64#1804C2218yescomplemented
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—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8C2314yes—
90C2×#121—22no—
91C2×#121—24no—
92C2×#121—42no—
93C2×#121C2314yes—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4C2314yes—
98C2×#117—22no—
99C2×#117—24no—
100C2×#117—42no—
101C2×#117C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×C4⋊C8—42no—
130C2×C4⋊C8C2314yes—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102—42no—
134C2×#102—42no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8—42no—
140C2×C22⋊C8—42no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97—42no—
144C2×#97C2314yes—
145#397—28no—
146#396—28no—
147#394—28no—
148#393—28no—
149#391—28no—
150#390—22no—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#387—28no—
159#386—28no—
160#382—22no—
161#382—22no—
162#382C2314yes—
163#382C2314yes—
164#381—28no—
165#380—22no—
166#380—22no—
167#380C2314yes—
168#380C2314yes—
169C4×C22⋊C4D414yes—
170C4×C22⋊C4C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95D411yescharacteristic
181C2×#95D411yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—82no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—82no—
210C22×D4#4114yes—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—82no—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256#121—28no—
257#121—44no—
258#121—48no—
259C4×D4—28no—
260C4×D4—44no—
261C4×D4—48no—
262#117—28no—
263#117—44no—
264#117—48no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
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—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329#107—44no—
330#106—44no—
331C4⋊C8—44no—
332#102—28no—
333#102—28no—
334#102—44no—
335#102—44no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338#97—44no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D414yes—
349#95C2×D414yes—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D412yes—
366C24C22×D411yescharacteristic
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—84no—
370C2×Q8—816no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—84no—
386C2×D4—816no—
387C2×D4C2×#4114yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C2×#4114yes—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—44no—
528C23—48no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23#53612yes—
547C23#54612yes—
548C23#54612yes—
549C23#57512yes—
550C23#57512yes—
551C23#57612yes—
552C23#57712yes—
553C23C23×D411yescharacteristic
554Q8—816no—
555D4—24no—
556D4—24no—
557D4—42no—
558D4—48no—
559D4—42no—
560D4—816no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
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—816no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#53612yes—
616C2×C4#53914yes—
617C2×C4#54214yes—
618C2×C4#54612yes—
619C2×C4#54612yes—
620C2×C4#55214yes—
621C2×C4#56214yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C8—42no—
629C8—42no—
630C8—42no—
631C8—42no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—88no—
643C22—816no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22#268714yes—
649C22#324014yes—
650C22#324114yes—
651C22#331114yes—
652C22#331914yes—
653C22#332114yes—
654C22#332214yes—
655C22C2×#53611yescharacteristic
656C22C2×#54611yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57711yescharacteristic
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#268714yes—
675C4C2×#53911yescharacteristic
676C4C2×#54211yescharacteristic
677C4C2×#55211yescharacteristic
678C4C2×#56211yescharacteristic
679C2—82no—
680C2—84no—
681C2#3137918yesdirect product factor
682C2C2×#268711yescharacteristic
683C2C2×#324011yescharacteristic
684C2C2×#324111yescharacteristic
685C2C2×#331111yescharacteristic
686C2C2×#331911yescharacteristic
687C2C2×#332111yescharacteristic
688C2C2×#332211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2439C2semidirect
3. #31379C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
7. C2×#2075C2semidirect
9. C2×#2023C2semidirect
10. C2×#1993C2semidirect
11. C2×#1936C2semidirect
12. C2×#1924C2semidirect
14. C2×#1862C2semidirect
15. C2×#1825C2semidirect
16. C2×#1804C2semidirect
32. #2439C22semidirect
52. #2101C22semidirect
53. #2088C22semidirect
54. #2085C22semidirect
55. #2075C22semidirect
57. #2023C22semidirect
58. #1993C22semidirect
59. #1936C22semidirect
60. #1924C22semidirect
62. #1862C22semidirect
63. #1825C22semidirect
64. #1804C22semidirect
681. C2#31379direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#24202
#313792
C2×#21012
C2×#20882
C2×#20852
C2×#20752
C2×#20342
C2×#20232
C2×#19932
C2×#19362
C2×#19242
C2×#18752
C2×#18622
C2×#18252
C2×#18042