Finite Group Databasebeta

← C2×#33219 #6630133

Subgroups — C2×#33219 #6630133

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#405C211yesmaximal, characteristic
3C2×#2629C211yesmaximal, characteristic
4C2×#2628C211yesmaximal, characteristic
5#33219C2116yesmaximal, direct product factor
6C2×#1991C211yesmaximal, characteristic, complemented
7C2×#1990C211yesmaximal, characteristic, complemented
8C2×#1944C211yesmaximal, characteristic, complemented
9C2×#1943C211yesmaximal, characteristic, complemented
10C2×#1942C211yesmaximal, characteristic, complemented
11C2×#1941C211yesmaximal, characteristic, complemented
12C2×#1847C211yesmaximal, characteristic, complemented
13C2×#1809C211yesmaximal, characteristic
14C2×#1809C211yesmaximal, characteristic
15C2×#1808C211yesmaximal, characteristic
16C2×#1807C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C2×#501—28no—
23C2×#501C2218yescomplemented
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C2×#405—28no—
33C2×#405C2218yes—
34C22×#97C2211yescharacteristic
35C22×#97C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2629—24no—
39#2629C2218yes—
40#2628—24no—
41#2628C2218yes—
42C2×#400—21no—
43C2×#399—21no—
44C2×#398—21no—
45C2×#398—21no—
46C2×#396—21no—
47C2×#396—21no—
48C2×#395—21no—
49C2×#393—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×#380C2211yescharacteristic
60#1991C2218yescomplemented
61#1990C2218yescomplemented
62#1944C2218yescomplemented
63#1943C2218yescomplemented
64#1942C2218yescomplemented
65#1941C2218yescomplemented
66#1847C2218yescomplemented
67#1809C2218yes—
68#1809C2218yes—
69#1808C2218yes—
70#1807C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8—21no—
80C24×C4C2311yescharacteristic
81C22×M16—24no—
82C22×M16—24no—
83C22×M16Q818yescomplemented
84C22×M16C2314yes—
85C23×C8—24no—
86C23×C8—24no—
87C23×C8Q818yescomplemented
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8C2314yes—
97C2×#125—22no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125C2314yes—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117D414yes—
104C2×#117C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#501—232no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—24no—
136C2×#107—24no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106—24no—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149#405—232no—
150C2×#97—24no—
151C2×#97D418yes—
152C2×#97C2314yes—
153C2×#97C2314yes—
154C2×C4×C8—24no—
155C2×C4×C8—24no—
156C2×C4×C8C2314yes—
157C2×C4×C8C2314yes—
158#400—28no—
159#399—28no—
160#398—28no—
161#398—28no—
162#396—28no—
163#396—28no—
164#395—28no—
165#393—28no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#384—22no—
175#384C2314yes—
176#382—22no—
177#382C2314yes—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185#380C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204C25C2×D411yescharacteristic
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—82no—
208C22×Q8—82no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×Q814yes—
228C23×C4C2×Q814yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C2×M16—232no—
232C2×M16—28no—
233C2×M16—28no—
234C2×M16C2×Q8116yescomplemented
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—232no—
240C22×C8—28no—
241C22×C8—28no—
242C22×C8C2×D414yes—
243C22×C8C2×D414yes—
244C22×C8C2×Q814yes—
245C22×C8C2×Q814yes—
246C22×C8C2×Q814yes—
247C22×C8C2×Q814yes—
248C22×C8C2×Q8116yescomplemented
249C22×C8#4114yes—
250C22×C8#4114yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—48no—
258#125—28no—
259#125—28no—
260#125—48no—
261#117—216no—
262#117—28no—
263#117—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D414yes—
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107—28no—
339#107—28no—
340#107—28no—
341#107—28no—
342#106—28no—
343#106—28no—
344#106—28no—
345#106—28no—
346#97—28no—
347#97C2×D4116yes—
348C4×C8—28no—
349C4×C8—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24C4⋊Q814yes—
367C24C22×D412yes—
368C24C22×D411yescharacteristic
369C2×Q8—48no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×Q8—816no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4#12114yes—
428C22×C4#12114yes—
429C22×C4C4⋊Q814yes—
430C22×C4C4⋊Q818yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446M16—232no—
447C2×C8—28no—
448C2×C8—28no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—232no—
454C2×C8C22⋊Q8116yes—
455C2×C8C22⋊Q8116yes—
456C2×C8C22×D414yes—
457C2×C8C22×D414yes—
458C2×C8C22×Q814yes—
459C2×C8C22×Q814yes—
460C2×C8C22×Q814yes—
461C2×C8C22×Q814yes—
462C2×C8C2×#4114yes—
463C2×C8C2×#4114yes—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42#12118yes—
509C42C22×D414yes—
510C42C2×#4114yes—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C2×#12112yes—
525C23C2×C4⋊Q814yes—
526C23C2×C4⋊Q812yes—
527C23#53212yes—
528C23#57612yes—
529C23#57712yes—
530C23#57712yes—
531C23#57812yes—
532C23C23×D411yescharacteristic
533Q8—816no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—216no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C2×#12114yes—
589C2×C4C2×#12114yes—
590C2×C4C2×#12112yes—
591C2×C4C2×C22⋊Q814yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×C4⋊Q814yes—
594C2×C4C2×C4⋊Q812yes—
595C2×C4#53212yes—
596C2×C4#53314yes—
597C2×C4#53314yes—
598C2×C4C23×D411yescharacteristic
599C2×C4C23×Q811yescharacteristic
600C2×C4C23×Q811yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C8—24no—
605C8—24no—
606C8C2×C22⋊Q818yes—
607C8C2×C22⋊Q818yes—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—416no—
616C22#285714yes—
617C22#306214yes—
618C22#306314yes—
619C22#316314yes—
620C22#316414yes—
621C22#317814yes—
622C22#317914yes—
623C22C22×#12111yescharacteristic
624C22C22×C4⋊Q811yescharacteristic
625C22C2×#53211yescharacteristic
626C22C2×#57611yescharacteristic
627C22C2×#57711yescharacteristic
628C22C2×#57711yescharacteristic
629C22C2×#57811yescharacteristic
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#285714yes—
644C4C22×C22⋊Q811yescharacteristic
645C4C22×C22⋊Q811yescharacteristic
646C4C2×#53311yescharacteristic
647C4C2×#53311yescharacteristic
648C2—44no—
649C2#3321918yesdirect product factor
650C2C2×#285711yescharacteristic
651C2C2×#306211yescharacteristic
652C2C2×#306311yescharacteristic
653C2C2×#316311yescharacteristic
654C2C2×#316411yescharacteristic
655C2C2×#317811yescharacteristic
656C2C2×#317911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#501C2semidirect
5. #33219C2direct (N × H)
6. C2×#1991C2semidirect
7. C2×#1990C2semidirect
8. C2×#1944C2semidirect
9. C2×#1943C2semidirect
10. C2×#1942C2semidirect
11. C2×#1941C2semidirect
12. C2×#1847C2semidirect
23. C2×#501C22semidirect
60. #1991C22semidirect
61. #1990C22semidirect
62. #1944C22semidirect
63. #1943C22semidirect
64. #1942C22semidirect
65. #1941C22semidirect
66. #1847C22semidirect
83. C22×M16Q8semidirect
87. C23×C8Q8semidirect
234. C2×M16C2×Q8semidirect
248. C22×C8C2×Q8semidirect
649. C2#33219direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4052
C2×#26292
C2×#26282
#332192
C2×#19912
C2×#19902
C2×#19442
C2×#19432
C2×#19422
C2×#19412
C2×#18472
C2×#18092
C2×#18082
C2×#18072