Finite Group Databasebeta

← C2×#31802 #7506300

Subgroups — C2×#31802 #7506300

Order 512.

2172 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic, complemented
3#31802C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2054C211yesmaximal, characteristic
8C2×#2054C211yesmaximal, characteristic
9C2×#2023C211yesmaximal, characteristic, complemented
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1939C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic, complemented
14C2×#1925C211yesmaximal, characteristic, complemented
15C2×#1879C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2512—24no—
34#2512C2218yescomplemented
35#2452—24no—
36#2452C2218yescomplemented
37C2×#397—21no—
38C2×#396—21no—
39C2×#396—21no—
40C2×#393—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#385—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2218yescomplemented
55#2086C2218yescomplemented
56#2067C2218yescomplemented
57#2054C2218yes—
58#2054C2218yes—
59#2023C2218yescomplemented
60#1988C2218yescomplemented
61#1939C2218yescomplemented
62#1934C2218yescomplemented
63#1934C2218yescomplemented
64#1925C2218yescomplemented
65#1879C2218yes—
66#1802C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×D8—24no—
82C22×D8—42no—
83C22×D8—42no—
84C22×D8C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8C2314yes—
93C2×#126—28no—
94C2×#126—48no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—42no—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
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×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C2×#107—42no—
131C2×#107C2314yes—
132C2×C4⋊C8—42no—
133C2×C4⋊C8—42no—
134C2×C4⋊C8—42no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C4×C8—42no—
151C2×C4×C8C2314yes—
152#397—28no—
153#396—28no—
154#396—28no—
155#393—28no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#390C2314yes—
164#390C2314yes—
165#390C2314yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#388—28no—
173#385—28no—
174#384—22no—
175#384C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D414yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25—21no—
198C22×Q8—24no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8—82no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4#4114yes—
216C22×D4C2412yes—
217C22×D4C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×D8—28no—
236C2×D8—44no—
237C2×D8—44no—
238C2×D8—416no—
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×C8—22no—
250C22×C8—22no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258#126—432no—
259C22⋊Q8—432no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—48no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
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—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4D818yescomplemented
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4#4114yes—
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—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4—82no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—44no—
340C4⋊C8—44no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343#102—28no—
344#102—28no—
345#102—28no—
346#102—44no—
347#102—44no—
348#102—44no—
349C4×C8—44no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95C2×D414yes—
355#95C2×D414yes—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—84no—
376C2×Q8—816no—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×#4114yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
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—48no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4C2×D814yes—
447C22×C4C2×D814yes—
448C22×C4C2×D814yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463D8—416no—
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—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—44no—
480C4⋊C4—24no—
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—22no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4C2×D8116yescomplemented
505C4⋊C4C2×#4114yes—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—48no—
527C23—24no—
528C23—48no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23#53712yes—
556C23#54612yes—
557C23#54812yes—
558C23C22×D812yes—
559C23#57512yes—
560C23#57712yes—
561C23#57712yes—
562C23C23×D411yescharacteristic
563Q8—816no—
564D4—24no—
565D4—24no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—48no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—84no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
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#53712yes—
621C2×C4#53814yes—
622C2×C4#54214yes—
623C2×C4#54612yes—
624C2×C4#54812yes—
625C2×C4#55214yes—
626C2×C4#56314yes—
627C2×C4C22×D814yes—
628C2×C4C22×D814yes—
629C2×C4C22×D814yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—816no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#270214yes—
657C22#324514yes—
658C22#326314yes—
659C22#329414yes—
660C22#332614yes—
661C22#335814yes—
662C22#337614yes—
663C22C2×#53711yescharacteristic
664C22C2×#54611yescharacteristic
665C22C2×#54811yescharacteristic
666C22C23×D811yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57711yescharacteristic
670C4—24no—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#270214yes—
683C4C2×#53811yescharacteristic
684C4C2×#54211yescharacteristic
685C4C2×#55211yescharacteristic
686C4C2×#56311yescharacteristic
687C2—84no—
688C2—162no—
689C2#3180218yesdirect product factor
690C2C2×#270211yescharacteristic
691C2C2×#324511yescharacteristic
692C2C2×#326311yescharacteristic
693C2C2×#329411yescharacteristic
694C2C2×#332611yescharacteristic
695C2C2×#335811yescharacteristic
696C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. C2×#2512C2semidirect
2. C2×#2452C2semidirect
3. #31802C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2086C2semidirect
6. C2×#2067C2semidirect
9. C2×#2023C2semidirect
10. C2×#1988C2semidirect
11. C2×#1939C2semidirect
12. C2×#1934C2semidirect
13. C2×#1934C2semidirect
14. C2×#1925C2semidirect
16. C2×#1802C2semidirect
34. #2512C22semidirect
36. #2452C22semidirect
54. #2099C22semidirect
55. #2086C22semidirect
56. #2067C22semidirect
59. #2023C22semidirect
60. #1988C22semidirect
61. #1939C22semidirect
62. #1934C22semidirect
63. #1934C22semidirect
64. #1925C22semidirect
66. #1802C22semidirect
293. C2×C4⋊C4D8semidirect
504. C4⋊C4C2×D8semidirect
689. C2#31802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#24522
#318022
C2×#20992
C2×#20862
C2×#20672
C2×#20542
C2×#20232
C2×#19882
C2×#19392
C2×#19342
C2×#19252
C2×#18792
C2×#18022