Finite Group Databasebeta

← C2×#26476 #6444601

Subgroups — C2×#26476 #6444601

Order 512.

2452 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2367C211yesmaximal, characteristic
5#26476C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2023C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1959C211yesmaximal, characteristic
12C2×#1948C211yesmaximal, characteristic, complemented
13C2×#1940C211yesmaximal, characteristic, complemented
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#475—28no—
24C2×#475C2218yescomplemented
25C2×#476—28no—
26C2×#476C2218yes—
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#97C2211yescharacteristic
43#2551—24no—
44#2551C2218yescomplemented
45C2×#395—21no—
46C2×#395—21no—
47#2367—24no—
48#2367C2218yes—
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#388—21no—
55C2×#388—21no—
56C2×#385—21no—
57C2×#384C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4—21no—
62#2068C2218yescomplemented
63#2067C2218yes—
64#2024C2218yescomplemented
65#2023C2218yes—
66#1983C2218yes—
67#1959C2218yes—
68#1948C2218yescomplemented
69#1940C2218yescomplemented
70#1865C2218yescomplemented
71#1864C2218yes—
72#1776C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8—48no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2314yes—
98C2×#124—22no—
99C2×#124—24no—
100C2×#124Q818yescomplemented
101C2×#124C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
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×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#476—232no—
137#475—232no—
138C2×#107—24no—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106D418yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—42no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—24no—
162C2×#102—24no—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#97—24no—
169C2×#97C2314yes—
170#395—28no—
171#395—28no—
172#391—22no—
173#391C2314yes—
174#390—22no—
175#390C2314yes—
176#389—28no—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#388—28no—
182#388—28no—
183#385—28no—
184#384—22no—
185#384C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D414yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—28no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25—21no—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8—82no—
221C22×Q8—82no—
222C22×Q8C2×Q814yes—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4C2×Q814yes—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
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×C8—22no—
261C22×C8C2×D414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—432no—
273#124—216no—
274#124—28no—
275#124C2×Q8116yescomplemented
276C4×Q8—28no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×D4—28no—
280C4×D4—44no—
281C4×D4—48no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
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—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42—82no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×Q814yes—
348C2×C42#4114yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—28no—
353#106—28no—
354#106—28no—
355#106C2×D4116yes—
356C4⋊C8—28no—
357C4⋊C8—28no—
358C4⋊C8—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367#97—28no—
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—24no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—816no—
404C2×Q8—816no—
405C2×Q8C22×Q814yes—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C22×Q814yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—48no—
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—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—84no—
464C22×C4—816no—
465C22×C4C22⋊Q814yes—
466C22×C4C22⋊Q814yes—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4#13714yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483C2×C8—28no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8C22×D414yes—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—84no—
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D414yes—
525C4⋊C4C2×#4114yes—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—84no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—48no—
538C42—48no—
539C42—84no—
540C42C22⋊Q818yes—
541C42C22×Q814yes—
542C42C2×#4114yes—
543C23—24no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—48no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23C2×C22⋊Q812yes—
558C23D4212yes—
559C23#54812yes—
560C23C2×AGL(1,ℤ/8)12yes—
561C23C2×#13712yes—
562C23#57712yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—24no—
566Q8—24no—
567Q8—42no—
568Q8—42no—
569Q8—48no—
570Q8—816no—
571Q8—816no—
572D4—24no—
573D4—24no—
574D4—42no—
575D4—42no—
576D4—48no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
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—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—816no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4C2×C22⋊Q812yes—
629C2×C4#53214yes—
630C2×C4#54214yes—
631C2×C4D4212yes—
632C2×C4#54812yes—
633C2×C4D4×Q814yes—
634C2×C4#55414yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×#13714yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C23×Q811yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C8—24no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22#245214yes—
656C22#307914yes—
657C22#308414yes—
658C22#326514yes—
659C22#326614yes—
660C22#336914yes—
661C22#337314yes—
662C22C22×C22⋊Q811yescharacteristic
663C22C2×D4211yescharacteristic
664C22C2×#54811yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C22×#13711yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57711yescharacteristic
669C4—24no—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#245214yes—
683C4C2×#53211yescharacteristic
684C4C2×#54211yescharacteristic
685C4C2×D4×Q811yescharacteristic
686C4C2×#55411yescharacteristic
687C2—84no—
688C2#2647618yesdirect product factor
689C2C2×#245211yescharacteristic
690C2C2×#307911yescharacteristic
691C2C2×#308411yescharacteristic
692C2C2×#326511yescharacteristic
693C2C2×#326611yescharacteristic
694C2C2×#337311yescharacteristic
695C2C2×#336911yescharacteristic
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×#475C2semidirect
3. C2×#2551C2semidirect
5. #26476C2direct (N × H)
6. C2×#2068C2semidirect
8. C2×#2024C2semidirect
12. C2×#1948C2semidirect
13. C2×#1940C2semidirect
14. C2×#1865C2semidirect
16. C2×#1776C2semidirect
24. C2×#475C22semidirect
44. #2551C22semidirect
62. #2068C22semidirect
64. #2024C22semidirect
68. #1948C22semidirect
69. #1940C22semidirect
70. #1865C22semidirect
72. #1776C22semidirect
100. C2×#124Q8semidirect
275. #124C2×Q8semidirect
688. C2#26476direct (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×#4762
C2×#25512
C2×#23672
#264762
C2×#20682
C2×#20672
C2×#20242
C2×#20232
C2×#19832
C2×#19592
C2×#19482
C2×#19402
C2×#18652
C2×#18642
C2×#17762