Finite Group Databasebeta

← C2×#31169 #7503608

Subgroups — C2×#31169 #7503608

Order 512.

2210 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic, complemented
2C22×#464C211yesmaximal, characteristic
3C2×#2448C211yesmaximal, characteristic
4C2×#2451C211yesmaximal, characteristic, complemented
5#31169C2116yesmaximal, direct product factor
6C2×#2086C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2056C211yesmaximal, characteristic, complemented
9C2×#2055C211yesmaximal, characteristic
10C2×#2042C211yesmaximal, characteristic
11C2×#2023C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic
13C2×#1940C211yesmaximal, characteristic, complemented
14C2×#1935C211yesmaximal, characteristic, complemented
15C2×#1878C211yesmaximal, characteristic, complemented
16C2×#1863C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C2×#477—28no—
24C2×#477C2218yescomplemented
25C2×#464—28no—
26C2×#464C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41#2451—24no—
42#2451C2218yescomplemented
43#2448—24no—
44#2448C2218yes—
45C2×#399—21no—
46C2×#398—21no—
47C2×#397—21no—
48C2×#395—21no—
49C2×#391—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#388—21no—
59C2×#388—21no—
60C2×#387—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2086C2218yescomplemented
63#2067C2218yes—
64#2056C2218yescomplemented
65#2055C2218yes—
66#2042C2218yes—
67#2023C2218yes—
68#1973C2218yes—
69#1940C2218yescomplemented
70#1935C2218yescomplemented
71#1878C2218yescomplemented
72#1863C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—42no—
86C22×Q16C2314yes—
87C22×SD16—24no—
88C22×SD16—24no—
89C22×SD16—42no—
90C22×SD16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8C2314yes—
99C2×#124—24no—
100C2×#124—22no—
101C2×#124—42no—
102C2×#124C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—42no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#477—232no—
136#464—232no—
137C2×#107—24no—
138C2×#107—24no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×C4⋊C8—24no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8—42no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103D418yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102—24no—
158C2×#102—42no—
159C2×#102—42no—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#97—42no—
164C2×#97C2314yes—
165#399—28no—
166#398—28no—
167#397—28no—
168#395—28no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—28no—
173#391C2314yes—
174#391C2314yes—
175#391C2314yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—28no—
187#388—28no—
188#387—28no—
189C4×C4⋊C4D414yes—
190C4×C4⋊C4C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—82no—
225C22×Q8C2×D414yes—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—416no—
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8C2×D414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#124—28no—
279#124—44no—
280#124—48no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×Q8—48no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351#107—28no—
352#107—28no—
353C4⋊C8—28no—
354C4⋊C8—28no—
355C4⋊C8—44no—
356#103—28no—
357#103—28no—
358#103—44no—
359#103—44no—
360#103C2×D4116yes—
361#102—28no—
362#102—44no—
363#102—44no—
364#97—44no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95C2×D414yes—
375#95C2×D414yes—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—84no—
412C2×Q8—816no—
413C2×Q8C22×D414yes—
414C2×Q8C2×#4114yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—44no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
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—42no—
456C22×C4—42no—
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×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488Q16—416no—
489SD16—416no—
490C2×C8—28no—
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—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8C22×D414yes—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D414yes—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—48no—
545C42—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23#53712yes—
561C23D4212yes—
562C23#54812yes—
563C23C2×#13712yes—
564C23#57512yes—
565C23#57612yes—
566C23#57812yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—48no—
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—816no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—84no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4#53614yes—
636C2×C4#53712yes—
637C2×C4#53914yes—
638C2×C4D4212yes—
639C2×C4#54614yes—
640C2×C4#54812yes—
641C2×C4#56214yes—
642C2×C4C2×#13714yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C8—24no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—168no—
661C22#268114yes—
662C22#325214yes—
663C22#326214yes—
664C22#326814yes—
665C22#327014yes—
666C22#337514yes—
667C22#337714yes—
668C22C2×#53711yescharacteristic
669C22C2×D4211yescharacteristic
670C22C2×#54811yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57611yescharacteristic
674C22C2×#57811yescharacteristic
675C4—24no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#268114yes—
689C4C2×#53611yescharacteristic
690C4C2×#53911yescharacteristic
691C4C2×#54611yescharacteristic
692C4C2×#56211yescharacteristic
693C2—162no—
694C2#3116918yesdirect product factor
695C2C2×#268111yescharacteristic
696C2C2×#325211yescharacteristic
697C2C2×#326211yescharacteristic
698C2C2×#326811yescharacteristic
699C2C2×#327011yescharacteristic
700C2C2×#337511yescharacteristic
701C2C2×#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
1. C22×#477C2semidirect
4. C2×#2451C2semidirect
5. #31169C2direct (N × H)
6. C2×#2086C2semidirect
8. C2×#2056C2semidirect
13. C2×#1940C2semidirect
14. C2×#1935C2semidirect
15. C2×#1878C2semidirect
16. C2×#1863C2semidirect
24. C2×#477C22semidirect
42. #2451C22semidirect
62. #2086C22semidirect
64. #2056C22semidirect
69. #1940C22semidirect
70. #1935C22semidirect
71. #1878C22semidirect
72. #1863C22semidirect
694. C2#31169direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4772
C22×#4642
C2×#24482
C2×#24512
#311692
C2×#20862
C2×#20672
C2×#20562
C2×#20552
C2×#20422
C2×#20232
C2×#19732
C2×#19402
C2×#19352
C2×#18782
C2×#18632