Finite Group Databasebeta

← C2×#26169 #6438402

Subgroups — C2×#26169 #6438402

Order 512.

2390 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic, complemented
3C2×#2448C211yesmaximal, characteristic
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#26169C2116yesmaximal, direct product factor
6C2×#2093C211yesmaximal, characteristic
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2072C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic
10C2×#1934C211yesmaximal, characteristic
11C2×#1916C211yesmaximal, characteristic, complemented
12C2×#1880C211yesmaximal, characteristic
13C2×#1880C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1863C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C2×#464—28no—
25C2×#464C2218yes—
26C2×C4×Q16—28no—
27C2×C4×Q16C2218yescomplemented
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2448—24no—
41#2448C2218yes—
42C2×#399—21no—
43C2×#397—21no—
44Q8×C4⋊C4—24no—
45Q8×C4⋊C4C2218yescomplemented
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389—21no—
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#388—21no—
55C2×#388—21no—
56C2×#384—21no—
57C2×#384C2211yescharacteristic
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60#2093C2218yes—
61#2086C2218yescomplemented
62#2072C2218yescomplemented
63#2042C2218yes—
64#1934C2218yes—
65#1916C2218yescomplemented
66#1880C2218yes—
67#1880C2218yes—
68#1878C2218yescomplemented
69#1863C2218yescomplemented
70#1863C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C22×Q16—24no—
83C22×Q16—24no—
84C22×Q16D418yes—
85C22×Q16C2314yes—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—24no—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#124—24no—
95C2×#124—22no—
96C2×#124—42no—
97C2×#124C2314yes—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—42no—
108C2×C4×Q8—42no—
109C2×C4×Q8—42no—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2314yes—
114C22×C4⋊C4—21no—
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×C22⋊C4—21no—
136C22×C42—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#464—232no—
144C4×Q16—232no—
145C2×#107—24no—
146C2×#107—42no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×C4⋊C8—24no—
150C2×C4⋊C8C2314yes—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102—24no—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C4×C8—24no—
168C2×C4×C8C2314yes—
169#399—28no—
170#397—28no—
171#391—22no—
172#391C2314yes—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—28no—
181#389—28no—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#388—28no—
186#388—28no—
187#384—28no—
188#384—22no—
189#384C2314yes—
190C4×C4⋊C4—28no—
191C4×C4⋊C4D414yes—
192C4×C4⋊C4C2314yes—
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×Q8C2×D414yes—
220C22×Q8C2×D414yes—
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—22no—
232C22×D4C2412yes—
233C23×C4—21no—
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
247C2×Q16—232no—
248C2×Q16—28no—
249C2×Q16—28no—
250C2×Q16C2×D4116yes—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—416no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8C2×D414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#124—28no—
272#124—44no—
273#124—48no—
274C4×Q8—216no—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×Q8—48no—
285C4×Q8—48no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
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—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C42—28no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—44no—
352C2×C42—44no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—28no—
361#107—44no—
362C4⋊C8—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#102—28no—
369#102—28no—
370#102—44no—
371C4×C8—28no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95C2×D414yes—
378#95C2×D414yes—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—28no—
393C2×Q8—24no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8#123116yes—
410C2×Q8C22×D414yes—
411C2×Q8C22×D414yes—
412C2×Q8C2×#4114yes—
413C2×Q8C2×#4114yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—44no—
427C22×C4—22no—
428C22×C4—24no—
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—22no—
438C22×C4—22no—
439C22×C4—22no—
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—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#13514yes—
472C22×C4#13714yes—
473C22×C4C22×D412yes—
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×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487Q16—232no—
488SD16—416no—
489C2×C8—28no—
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—22no—
501C2×C8—44no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8C22×D414yes—
505C4⋊C4—216no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—28no—
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—22no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—48no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C2×#4114yes—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—216no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—48no—
548C42—48no—
549C42—48no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23#52512yes—
565C23D4212yes—
566C23#54812yes—
567C23C2×#13512yes—
568C23C2×#13712yes—
569C23#57612yes—
570C23#57812yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—24no—
574Q8—216no—
575Q8—24no—
576Q8—24no—
577Q8—24no—
578Q8—42no—
579Q8—48no—
580Q8—42no—
581Q8C2×#12318yes—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—84no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—216no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
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—42no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4C2×#12314yes—
638C2×C4#52512yes—
639C2×C4#53914yes—
640C2×C4D4212yes—
641C2×C4#54714yes—
642C2×C4#54812yes—
643C2×C4#55314yes—
644C2×C4C2×#13514yes—
645C2×C4C2×#13714yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C8—24no—
653C8—42no—
654C8—42no—
655C8—42no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—168no—
664C22#244314yes—
665C22#310814yes—
666C22#312414yes—
667C22#327014yes—
668C22#328714yes—
669C22#336414yes—
670C22#337514yes—
671C22C2×#52511yescharacteristic
672C22C2×D4211yescharacteristic
673C22C2×#54811yescharacteristic
674C22C22×#13511yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57811yescharacteristic
678C4—28no—
679C4—24no—
680C4—24no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#244314yes—
692C4C22×#12311yescharacteristic
693C4C2×#53911yescharacteristic
694C4C2×#54711yescharacteristic
695C4C2×#55311yescharacteristic
696C2—162no—
697C2#2616918yesdirect product factor
698C2C2×#244311yescharacteristic
699C2C2×#310811yescharacteristic
700C2C2×#312411yescharacteristic
701C2C2×#327011yescharacteristic
702C2C2×#328711yescharacteristic
703C2C2×#336411yescharacteristic
704C2C2×#337511yescharacteristic
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
2. C22×C4×Q16C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. #26169C2direct (N × H)
7. C2×#2086C2semidirect
8. C2×#2072C2semidirect
11. C2×#1916C2semidirect
14. C2×#1878C2semidirect
15. C2×#1863C2semidirect
16. C2×#1863C2semidirect
27. C2×C4×Q16C22semidirect
45. Q8×C4⋊C4C22semidirect
61. #2086C22semidirect
62. #2072C22semidirect
65. #1916C22semidirect
68. #1878C22semidirect
69. #1863C22semidirect
70. #1863C22semidirect
697. C2#26169direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4642
C22×C4×Q162
C2×#24482
C2×Q8×C4⋊C42
#261692
C2×#20932
C2×#20862
C2×#20722
C2×#20422
C2×#19342
C2×#19162
C2×#18802
C2×#18782
C2×#18632