Finite Group Databasebeta

← C2×#26353 #6439166

Subgroups — C2×#26353 #6439166

Order 512.

2484 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×#428C211yesmaximal, characteristic
3C2×#2440C211yesmaximal, characteristic
4C2×#2362C211yesmaximal, characteristic
5#26353C2116yesmaximal, direct product factor
6C2×#2095C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1948C211yesmaximal, characteristic, complemented
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1894C211yesmaximal, characteristic
13C2×#1872C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic
16C2×#1851C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#475—28no—
25C2×#475C2218yescomplemented
26C2×#428—28no—
27C2×#428C2218yes—
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×#106C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41#2440—24no—
42#2440C2218yes—
43C2×#398—21no—
44C2×#395—21no—
45#2362—24no—
46#2362C2218yes—
47C2×#391C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#387—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2095C2218yescomplemented
61#2086C2218yescomplemented
62#2068C2218yescomplemented
63#2057C2218yescomplemented
64#1948C2218yescomplemented
65#1920C2218yescomplemented
66#1894C2218yes—
67#1872C2218yes—
68#1865C2218yescomplemented
69#1854C2218yes—
70#1851C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×M16—24no—
84C22×M16—24no—
85C22×M16—24no—
86C22×M16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—48no—
99C2×C22⋊Q8Q818yescomplemented
100C2×C22⋊Q8C2314yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—24no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C2×#117—22no—
106C2×#117—24no—
107C2×#117—42no—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—42no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#475—232no—
140#428—232no—
141C2×#107—24no—
142C2×#107—24no—
143C2×#107—42no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106—24no—
148C2×#106—24no—
149C2×#106D418yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8—24no—
154C2×C4⋊C8—24no—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2314yes—
157C2×#103—24no—
158C2×#103—24no—
159C2×#103—24no—
160C2×#103—42no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×C22⋊C8—24no—
166C2×C22⋊C8C2314yes—
167#398—28no—
168#395—28no—
169#391—22no—
170#391C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#388—28no—
178#388—28no—
179#387—28no—
180#385—28no—
181#384—22no—
182#384C2314yes—
183#380—22no—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—28no—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4D414yes—
189C4×C22⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95D411yescharacteristic
198C2×#95D411yescharacteristic
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25C2×D411yescharacteristic
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8—82no—
218C22×Q8C2×Q814yes—
219C22×Q8C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×Q814yes—
243C23×C4#4114yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C2×M16—232no—
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—28no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8C2×D414yes—
261C22×C8#4114yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C22⋊Q8—216no—
270C22⋊Q8—28no—
271C22⋊Q8—432no—
272C22⋊Q8C2×Q8116yescomplemented
273C4×Q8—28no—
274C4×Q8—44no—
275C4×Q8—48no—
276#117—28no—
277#117—44no—
278#117—48no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
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—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
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—24no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×Q814yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
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⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—28no—
353#107—28no—
354#107—44no—
355#106—28no—
356#106—28no—
357#106C2×D4116yes—
358C4⋊C8—28no—
359C4⋊C8—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—44no—
364C22⋊C8—28no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24C22⋊Q814yes—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—84no—
398C2×Q8—816no—
399C2×Q8C22×Q814yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—48no—
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—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—84no—
457C22×C4—816no—
458C22×C4C22⋊Q814yes—
459C22×C4C22⋊Q818yes—
460C22×C4AGL(1,ℤ/8)14yes—
461C22×C4#13714yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477M16—232no—
478C2×C8—28no—
479C2×C8—28no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8C22×D414yes—
493C2×C8C2×#4114yes—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—84no—
521C4⋊C4C22×D414yes—
522C4⋊C4C22×D414yes—
523C4⋊C4C22×Q814yes—
524C22⋊C4—216no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—48no—
536C42—48no—
537C42—48no—
538C23—28no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×C22⋊Q814yes—
552C23C2×C22⋊Q812yes—
553C23D4212yes—
554C23#54712yes—
555C23C2×AGL(1,ℤ/8)12yes—
556C23C2×#13712yes—
557C23#57712yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—24no—
562Q8—42no—
563Q8—42no—
564Q8—48no—
565Q8—816no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—216no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
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—42no—
615C2×C4—42no—
616C2×C4—816no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4C2×C22⋊Q814yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4#52514yes—
624C2×C4#53914yes—
625C2×C4D4212yes—
626C2×C4#54712yes—
627C2×C4D4×Q814yes—
628C2×C4#55414yes—
629C2×C4C2×AGL(1,ℤ/8)14yes—
630C2×C4C2×#13714yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C23×Q811yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C8—24no—
638C8—24no—
639C8—42no—
640C8—42no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—416no—
649C22#244914yes—
650C22#307914yes—
651C22#308214yes—
652C22#326514yes—
653C22#327614yes—
654C22#334414yes—
655C22#335314yes—
656C22C22×C22⋊Q811yescharacteristic
657C22C2×D4211yescharacteristic
658C22C2×#54711yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57811yescharacteristic
663C4—24no—
664C4—24no—
665C4—24no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#244914yes—
677C4C2×#52511yescharacteristic
678C4C2×#53911yescharacteristic
679C4C2×D4×Q811yescharacteristic
680C4C2×#55411yescharacteristic
681C2—44no—
682C2#2635318yesdirect product factor
683C2C2×#244911yescharacteristic
684C2C2×#307911yescharacteristic
685C2C2×#308211yescharacteristic
686C2C2×#326511yescharacteristic
687C2C2×#327611yescharacteristic
688C2C2×#334411yescharacteristic
689C2C2×#335311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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. C22×#475C2semidirect
5. #26353C2direct (N × H)
6. C2×#2095C2semidirect
7. C2×#2086C2semidirect
8. C2×#2068C2semidirect
9. C2×#2057C2semidirect
10. C2×#1948C2semidirect
11. C2×#1920C2semidirect
14. C2×#1865C2semidirect
25. C2×#475C22semidirect
60. #2095C22semidirect
61. #2086C22semidirect
62. #2068C22semidirect
63. #2057C22semidirect
64. #1948C22semidirect
65. #1920C22semidirect
68. #1865C22semidirect
99. C2×C22⋊Q8Q8semidirect
272. C22⋊Q8C2×Q8semidirect
682. C2#26353direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4752
C22×#4282
C2×#24402
C2×#23622
#263532
C2×#20952
C2×#20862
C2×#20682
C2×#20572
C2×#19482
C2×#19202
C2×#18942
C2×#18722
C2×#18652
C2×#18542
C2×#18512