Finite Group Databasebeta

← C2×#26918 #6439655

Subgroups — C2×#26918 #6439655

Order 512.

2412 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#440C211yesmaximal, characteristic
3C2×#2454C211yesmaximal, characteristic
4C2×#2365C211yesmaximal, characteristic, complemented
5#26918C2116yesmaximal, direct product factor
6C2×#2055C211yesmaximal, characteristic
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2026C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10C2×#1935C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic
12C2×#1921C211yesmaximal, characteristic, complemented
13C2×#1917C211yesmaximal, characteristic, complemented
14C2×#1880C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic
16C2×#1865C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#479—28no—
25C2×#479C2218yescomplemented
26C2×#440—28no—
27C2×#440C2218yes—
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2454—24no—
41#2454C2218yes—
42C2×#399—21no—
43C2×#396—21no—
44#2365—24no—
45#2365C2218yescomplemented
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#387—21no—
54C2×#387—21no—
55C2×#385—21no—
56C2×#380—21no—
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2055C2218yes—
61#2043C2218yescomplemented
62#2026C2218yescomplemented
63#2013C2218yes—
64#1935C2218yescomplemented
65#1926C2218yes—
66#1921C2218yescomplemented
67#1917C2218yescomplemented
68#1880C2218yes—
69#1877C2218yes—
70#1865C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—42no—
89C22×SD16C2314yes—
90C22×SD16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2314yes—
99C2×#125—24no—
100C2×#125—22no—
101C2×#125—42no—
102C2×#125C2314yes—
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—
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—42no—
114C2×C4×D4C2314yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#479—232no—
142#440—232no—
143C2×#107D418yes—
144C2×#107C2314yes—
145C2×#106—24no—
146C2×#106C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—42no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102—24no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97—24no—
166C2×#97C2314yes—
167#399—28no—
168#396—28no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389—28no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#387—28no—
183#387—28no—
184#385—28no—
185#380—28no—
186C4×C4⋊C4—28no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D414yes—
189C4×C4⋊C4C2314yes—
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×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—22no—
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#4114yes—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4#4114yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—232no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—28no—
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—216no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278#125—28no—
279#125—44no—
280#125—48no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×Q8—48no—
287C4×D4—28no—
288C4×D4—44no—
289C4×D4—48no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42—44no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358#107C2×D4116yes—
359#106—28no—
360C4⋊C8—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—44no—
367#102—28no—
368#102—28no—
369#97—28no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95C2×D414yes—
378#95C2×D414yes—
379C24—24no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8C2×#4114yes—
413C2×Q8C2×#4114yes—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4C2×#4114yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
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—48no—
456C22×C4—48no—
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—42no—
474C22×C4—84no—
475C22×C4—816no—
476C22×C4#13714yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491SD16—232no—
492SD16—416no—
493C2×C8—28no—
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—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8—44no—
507C2×C8—44no—
508C2×C8C22×D414yes—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—48no—
549C42—48no—
550C42—48no—
551C23—24no—
552C23—48no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23#52512yes—
566C23D4212yes—
567C23#54812yes—
568C23C2×#13712yes—
569C23#57512yes—
570C23#57712yes—
571C23#57712yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—48no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—48no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
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—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4#52514yes—
641C2×C4#52512yes—
642C2×C4#53914yes—
643C2×C4D4212yes—
644C2×C4#54814yes—
645C2×C4#54812yes—
646C2×C4#55114yes—
647C2×C4C2×#13714yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C8—24no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—816no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22#246314yes—
667C22#311714yes—
668C22#312614yes—
669C22#326714yes—
670C22#327814yes—
671C22#337214yes—
672C22#337614yes—
673C22C2×#52511yescharacteristic
674C22C2×D4211yescharacteristic
675C22C2×#54811yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57711yescharacteristic
679C22C2×#57711yescharacteristic
680C4—24no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#246314yes—
694C4C2×#52511yescharacteristic
695C4C2×#53911yescharacteristic
696C4C2×#54811yescharacteristic
697C4C2×#55111yescharacteristic
698C2—84no—
699C2#2691818yesdirect product factor
700C2C2×#246311yescharacteristic
701C2C2×#311711yescharacteristic
702C2C2×#312611yescharacteristic
703C2C2×#326711yescharacteristic
704C2C2×#327811yescharacteristic
705C2C2×#337211yescharacteristic
706C2C2×#337611yescharacteristic
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×#479C2semidirect
4. C2×#2365C2semidirect
5. #26918C2direct (N × H)
7. C2×#2043C2semidirect
8. C2×#2026C2semidirect
10. C2×#1935C2semidirect
12. C2×#1921C2semidirect
13. C2×#1917C2semidirect
16. C2×#1865C2semidirect
25. C2×#479C22semidirect
45. #2365C22semidirect
61. #2043C22semidirect
62. #2026C22semidirect
64. #1935C22semidirect
66. #1921C22semidirect
67. #1917C22semidirect
70. #1865C22semidirect
699. C2#26918direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4402
C2×#24542
C2×#23652
#269182
C2×#20552
C2×#20432
C2×#20262
C2×#20132
C2×#19352
C2×#19262
C2×#19212
C2×#19172
C2×#18802
C2×#18772
C2×#18652