Finite Group Databasebeta

← C2×#62394 #10578403

Subgroups — C2×#62394 #10578403

Order 512.

6234 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
1C2×#3522C211yesmaximal, characteristic
2C2×#3502C211yesmaximal, characteristic, complemented
3#62394C2132yesmaximal, direct product factor
4C2×#3400C213yesmaximal, complemented
5C2×#3392C211yesmaximal, characteristic, complemented
6C2×#3385C213yesmaximal, complemented
7C2×Q8×Q16C211yesmaximal, characteristic, complemented
8C2×#3375C213yesmaximal, complemented
9C2×#3372C213yesmaximal, complemented
10C2×#3363C211yesmaximal, characteristic, complemented
11C2×#3362C211yesmaximal, characteristic, complemented
12C2×#3254C213yesmaximal, complemented
13C2×#3252C213yesmaximal, complemented
14C2×#3182C213yesmaximal, complemented
15C2×#3172C213yesmaximal, complemented
16C2×Q8×M16C211yesmaximal, characteristic, complemented
17#3522—28no—
18#3522C22116yes—
19#3502—28no—
20#3502C22116yescomplemented
21C2×#559C2213yes—
22C2×Q82C2211yescharacteristic
23C2×Q82C2213yes—
24C2×Q82C2211yescharacteristic
25C2×#557—21no—
26C2×#557—23no—
27C2×#557C2211yescharacteristic
28C2×#557C2213yes—
29C2×#554C2213yes—
30C2×#550C2211yescharacteristic
31C2×#548—21no—
32C2×#547—23no—
33C2×#544—23no—
34C2×#542—23no—
35C2×#533C2213yes—
36C2×#533C2213yes—
37C2×#532C2213yes—
38C2×#529C2213yes—
39C2×#520—21no—
40C2×#519—21no—
41C22×C4×Q8C2211yescharacteristic
42#3400C22148yescomplemented
43#3392C22116yescomplemented
44#3385C22148yescomplemented
45Q8×Q16C22116yescomplemented
46#3375C22148yescomplemented
47#3372C22148yescomplemented
48#3363C22116yescomplemented
49#3362C22116yescomplemented
50#3254C22148yescomplemented
51#3252C22148yescomplemented
52#3182C22148yescomplemented
53#3172C22148yescomplemented
54C2×#501C2213yes—
55C2×#500C2213yes—
56C2×#497C2213yes—
57C2×#492C2213yes—
58C2×#491C2213yes—
59C2×#489C2213yes—
60C2×#487C2213yes—
61C2×#486C2213yes—
62C2×#479C2216yes—
63C2×#477C2213yes—
64C2×#477C2213yes—
65C2×#477C2213yes—
66C2×#476C2213yes—
67C2×#474C2216yes—
68C2×#470C2211yescharacteristic
69C2×#470C2213yes—
70C2×#468C2213yes—
71C2×#468C2213yes—
72C2×#468C2216yes—
73C2×#464C2216yes—
74C2×#462C2213yes—
75C2×#460C2213yes—
76C2×#460C2213yes—
77C2×#461C2213yes—
78C2×#461C2216yes—
79Q8×M16C22116yescomplemented
80C2×#446C2213yes—
81C2×#446C2213yes—
82C2×C8×Q8C2211yescharacteristic
83C2×C8×Q8C2211yescharacteristic
84C2×#440C2216yes—
85C2×#440C2216yes—
86C2×C4×Q16C2213yes—
87C2×C4×Q16C2213yes—
88C2×C4×SD16C2213yes—
89C2×C4×SD16C2213yes—
90C2×#423C2213yes—
91C2×C4×M16C2213yes—
92C22×#41—21no—
93C22×#41—23no—
94C23×Q8D411yescharacteristic
95C22×Q16C2311yescharacteristic
96C22×Q16C2313yes—
97C22×SD16C2313yes—
98C22×SD16C2316yes—
99C22×SD16C2313yes—
100C22×M16C2311yescharacteristic
101C22×M16C2313yes—
102#559—212no—
103#559C23124yes—
104Q82—24no—
105Q82—212no—
106Q82—24no—
107Q82C2318yes—
108Q82C2318yes—
109Q82C23124yes—
110#557—216no—
111#557—248no—
112#557—24no—
113#557—212no—
114#557C2318yes—
115#557C23124yes—
116#554—212no—
117#554C23124yes—
118#550—24no—
119#550C2318yes—
120#548—216no—
121#547—248no—
122#544—248no—
123#542—248no—
124#533—212no—
125#533—212no—
126#533C23124yes—
127#533C23124yes—
128#532—212no—
129#532C23124yes—
130C2×C4⋊Q8—23no—
131C2×C4⋊Q8—23no—
132C2×C4⋊Q8—23no—
133C2×C4⋊Q8—26no—
134C2×C4⋊Q8C2313yes—
135C2×C4⋊Q8C2313yes—
136C2×C4⋊Q8C2313yes—
137C2×C4⋊Q8C2313yes—
138C2×C4⋊Q8C2313yes—
139C2×#127C2313yes—
140#529—212no—
141#529C23124yes—
142C2×#126—23no—
143C2×#125—23no—
144C2×#125—23no—
145C2×#125—26no—
146C2×#125—23no—
147C2×#125—23no—
148C2×#125C2313yes—
149C2×#124—23no—
150C2×#124—23no—
151C2×#124C2313yes—
152C2×#123—23no—
153C2×#123—26no—
154C2×C22⋊Q8—23no—
155C2×C22⋊Q8—23no—
156C2×C22⋊Q8—23no—
157C2×C22⋊Q8—26no—
158C2×C22⋊Q8C2311yescharacteristic
159C2×C22⋊Q8C2311yescharacteristic
160C2×C22⋊Q8C2313yes—
161C2×#121—23no—
162C2×#121—23no—
163C2×#121C2313yes—
164#520—216no—
165#519—216no—
166C2×C4×Q8—23no—
167C2×C4×Q8—21no—
168C2×C4×Q8—23no—
169C2×C4×Q8—21no—
170C2×C4×Q8—21no—
171C2×C4×Q8—23no—
172C2×C4×Q8—23no—
173C2×C4×Q8D411yescharacteristic
174C2×C4×Q8D413yes—
175C2×C4×Q8D418yes—
176C2×C4×Q8C2313yes—
177C2×C4×Q8C2313yes—
178C2×C4×Q8C2313yes—
179C2×C4×Q8C2316yes—
180C2×C4×Q8C2311yescharacteristic
181C2×C4×Q8C2313yes—
182C2×C4×Q8C2318yes—
183C2×C4×D4—23no—
184C2×C4×D4—23no—
185C2×C4×D4—23no—
186C2×C4×D4C2313yes—
187C2×C4×D4C2316yes—
188C2×#117—23no—
189C2×#117—23no—
190C2×#117—23no—
191C22×C4⋊C4—23no—
192C22×C4⋊C4D413yes—
193C22×C42C2313yes—
194#501C23124yes—
195#500C23124yes—
196#497C23124yes—
197#492C23124yes—
198#491C23124yes—
199#489C23124yes—
200#487C23124yes—
201#486C23124yes—
202#479C23148yes—
203#477C23124yes—
204#477C23124yes—
205#477C23124yes—
206#476C23124yes—
207#474C23148yes—
208#470C2318yes—
209#470C23124yes—
210#468C23124yes—
211#468C23124yes—
212#468C23148yes—
213#464C23148yes—
214#462C23124yes—
215#461C23124yes—
216#461C23148yes—
217#460C23124yes—
218#460C23124yes—
219#446C23124yes—
220#446C23124yes—
221C8×Q8C2318yes—
222C8×Q8C2318yes—
223#440C23148yes—
224#440C23148yes—
225C4×Q16C23124yes—
226C4×Q16C23124yes—
227C4×SD16C23124yes—
228C4×SD16C23124yes—
229C2×#107C2316yes—
230C2×#107C2313yes—
231C2×#107C2313yes—
232C2×#106C2313yes—
233C2×#106C2311yescharacteristic
234#423C23124yes—
235C2×C4⋊C8C2313yes—
236C2×C4⋊C8C2313yes—
237C2×C4⋊C8C2316yes—
238C2×#103C2313yes—
239C2×#103C2313yes—
240C2×#103C2316yes—
241C2×#103C2313yes—
242C2×#103C2313yes—
243C2×#103C2311yescharacteristic
244C2×#103C2311yescharacteristic
245C2×#102C2313yes—
246C2×#102C2316yes—
247C2×#102C2313yes—
248C4×M16C23124yes—
249C2×#97C2313yes—
250C2×#97C2313yes—
251C2×C4×C8C2313yes—
252C2×C4×C8C2313yes—
253C2×#41—28no—
254C2×#41—224no—
255C2×#41—48no—
256C2×#41—812no—
257C22×Q8—28no—
258C22×Q8—23no—
259C22×Q8—23no—
260C22×Q8—23no—
261C22×Q8C2×D413yes—
262C22×Q8C2×D418yes—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×Q8C2413yes—
266C22×Q8C2411yescharacteristic
267C22×D4—23no—
268C22×D4—23no—
269C22×D4—23no—
270C22×D4C2413yes—
271C23×C4—23no—
272C23×C4—26no—
273C23×C4C2×D413yes—
274C23×C4C2×D413yes—
275C23×C4C2411yescharacteristic
276C2×Q16—24no—
277C2×Q16—46no—
278C2×Q16C2414yes—
279C2×Q16C24112yes—
280C2×SD16—212no—
281C2×SD16—412no—
282C2×SD16—46no—
283C2×SD16C24112yes—
284C2×SD16C24112yes—
285C2×SD16C24124yes—
286C2×M16—24no—
287C2×M16—46no—
288C2×M16C2414yes—
289C2×M16C24112yes—
290C22×C8C2413yes—
291C22×C8C2413yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C4⋊Q8—224no—
295C4⋊Q8—224no—
296C4⋊Q8—224no—
297C4⋊Q8—248no—
298C4⋊Q8—26no—
299C4⋊Q8—26no—
300C4⋊Q8—26no—
301C4⋊Q8—26no—
302C4⋊Q8—26no—
303C4⋊Q8C24112yes—
304C4⋊Q8C24112yes—
305C4⋊Q8C24112yes—
306C4⋊Q8C24112yes—
307C4⋊Q8C24112yes—
308#127—26no—
309#127C24112yes—
310#126—224no—
311#125—224no—
312#125—224no—
313#125—248no—
314#125—224no—
315#125—224no—
316#125—26no—
317#125C24112yes—
318#124—224no—
319#124—224no—
320#124—26no—
321#124C24112yes—
322#123—224no—
323#123—248no—
324C22⋊Q8—224no—
325C22⋊Q8—224no—
326C22⋊Q8—224no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—26no—
330C22⋊Q8—248no—
331C22⋊Q8C2414yes—
332C22⋊Q8C2414yes—
333C22⋊Q8C24112yes—
334#121—224no—
335#121—26no—
336#121—224no—
337#121C24112yes—
338C4×Q8—224no—
339C4×Q8—28no—
340C4×Q8—224no—
341C4×Q8—28no—
342C4×Q8—28no—
343C4×Q8—224no—
344C4×Q8—224no—
345C4×Q8—26no—
346C4×Q8—212no—
347C4×Q8—26no—
348C4×Q8—26no—
349C4×Q8C2×D418yes—
350C4×Q8C2×D4124yes—
351C4×Q8C2×D414yes—
352C4×Q8C2×D4112yes—
353C4×Q8C24112yes—
354C4×Q8C24112yes—
355C4×Q8C24112yes—
356C4×Q8C24124yes—
357C4×Q8C2414yes—
358C4×Q8C24112yes—
359C4×D4—224no—
360C4×D4—224no—
361C4×D4—224no—
362C4×D4—26no—
363C4×D4—212no—
364C4×D4C24112yes—
365C4×D4C24124yes—
366#117—224no—
367#117—224no—
368#117—224no—
369C2×C4⋊C4—23no—
370C2×C4⋊C4—23no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—23no—
373C2×C4⋊C4—23no—
374C2×C4⋊C4—23no—
375C2×C4⋊C4—23no—
376C2×C4⋊C4—23no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—23no—
379C2×C4⋊C4—23no—
380C2×C4⋊C4—23no—
381C2×C4⋊C4—26no—
382C2×C4⋊C4—23no—
383C2×C4⋊C4—23no—
384C2×C4⋊C4—23no—
385C2×C4⋊C4—224no—
386C2×C4⋊C4C2×D413yes—
387C2×C4⋊C4C2×D413yes—
388C2×C4⋊C4C2×D416yes—
389C2×C4⋊C4C2×D4124yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2413yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2413yes—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—23no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—23no—
398C2×C22⋊C4—23no—
399C2×C22⋊C4—23no—
400C2×C22⋊C4—23no—
401C2×C22⋊C4—26no—
402C2×C42—23no—
403C2×C42—26no—
404C2×C42—23no—
405C2×C42—23no—
406C2×C42C2×D413yes—
407C2×C42C2×D413yes—
408C2×C42C2×D4112yes—
409C2×C42C2413yes—
410C2×C42C2413yes—
411C2×C42C24112yes—
412#107C24112yes—
413#107C24112yes—
414#107C24124yes—
415#106C2414yes—
416#106C24112yes—
417C4⋊C8C24112yes—
418C4⋊C8C24112yes—
419C4⋊C8C24124yes—
420#103C2414yes—
421#103C24112yes—
422#103C2414yes—
423#103C24112yes—
424#103C24112yes—
425#103C24112yes—
426#103C24124yes—
427#102C24112yes—
428#102C24112yes—
429#102C24124yes—
430#97C24112yes—
431#97C24112yes—
432C4×C8C24112yes—
433C4×C8C24112yes—
434C24—23no—
435C24C22×D411yescharacteristic
436#41—432no—
437#41—848no—
438C2×Q8—232no—
439C2×Q8—212no—
440C2×Q8—21no—
441C2×Q8—23no—
442C2×Q8—21no—
443C2×Q8—212no—
444C2×Q8—212no—
445C2×Q8—212no—
446C2×Q8—26no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—26no—
451C2×Q8—22no—
452C2×Q8—86no—
453C2×Q8—86no—
454C2×Q8—86no—
455C2×Q8#13718yes—
456C2×Q8C22×D4112yes—
457C2×Q8C22×D414yes—
458C2×Q8C2512yes—
459C2×Q8C2516yes—
460C2×Q8C2512yes—
461C2×D4—212no—
462C2×D4—23no—
463C2×D4—212no—
464C2×D4—212no—
465C2×D4—26no—
466C2×D4—26no—
467C2×D4—412no—
468C2×D4—412no—
469C2×D4—86no—
470C2×D4C2516yes—
471C22×C4—212no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—23no—
476C22×C4—21no—
477C22×C4—23no—
478C22×C4—21no—
479C22×C4—23no—
480C22×C4—21no—
481C22×C4—212no—
482C22×C4—224no—
483C22×C4—46no—
484C22×C4—412no—
485C22×C4—812no—
486C22×C4C22×D412yes—
487C22×C4C22×D4112yes—
488C22×C4C22×D413yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D413yes—
491C22×C4C22×D413yes—
492C22×C4C22×D4112yes—
493C22×C4C22×D413yes—
494C22×C4C2512yes—
495C22×C4C2511yescharacteristic
496Q16—28no—
497Q16—412no—
498SD16—224no—
499SD16—424no—
500SD16—412no—
501M16—28no—
502M16—412no—
503C2×C8—22no—
504C2×C8—26no—
505C2×C8—26no—
506C2×C82−1+414yes—
507C2×C8C2512yes—
508C2×C8C2516yes—
509C2×C8C2516yes—
510C2×C8C2512yes—
511C4⋊C4—212no—
512C4⋊C4—212no—
513C4⋊C4—24no—
514C4⋊C4—212no—
515C4⋊C4—212no—
516C4⋊C4—212no—
517C4⋊C4—212no—
518C4⋊C4—212no—
519C4⋊C4—24no—
520C4⋊C4—212no—
521C4⋊C4—212no—
522C4⋊C4—212no—
523C4⋊C4—224no—
524C4⋊C4—212no—
525C4⋊C4—212no—
526C4⋊C4—212no—
527C4⋊C4—21no—
528C4⋊C4—23no—
529C4⋊C4—21no—
530C4⋊C4—23no—
531C4⋊C4C22×D4112yes—
532C4⋊C4C22×D4112yes—
533C4⋊C4C22×D4124yes—
534C4⋊C4C2516yes—
535C4⋊C4C2512yes—
536C4⋊C4C2512yes—
537C4⋊C4C2516yes—
538C22⋊C4—24no—
539C22⋊C4—212no—
540C22⋊C4—24no—
541C22⋊C4—212no—
542C22⋊C4—212no—
543C22⋊C4—212no—
544C22⋊C4—212no—
545C22⋊C4—224no—
546C42—212no—
547C42—224no—
548C42—212no—
549C42—212no—
550C42C22×D4112yes—
551C42C22×D416yes—
552C42C22×D416yes—
553C42C22×D4112yes—
554C42C2516yes—
555C42C2516yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—26no—
560C23—46no—
561C23—46no—
562C23—46no—
563C23C23×D412yes—
564C23C23×D411yescharacteristic
565Q8—224no—
566Q8—26no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—26no—
571Q8—22no—
572Q8—42no—
573Q8—46no—
574Q8—42no—
575Q8—812no—
576Q8—812no—
577Q8—812no—
578Q8C2×#137116yes—
579D4—26no—
580D4—26no—
581D4—424no—
582D4—46no—
583D4—424no—
584D4—812no—
585C2×C4—26no—
586C2×C4—26no—
587C2×C4—22no—
588C2×C4—26no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—224no—
593C2×C4—22no—
594C2×C4—26no—
595C2×C4—22no—
596C2×C4—26no—
597C2×C4—22no—
598C2×C4—26no—
599C2×C4—22no—
600C2×C4—412no—
601C2×C4—46no—
602C2×C4—42no—
603C2×C4—46no—
604C2×C4—42no—
605C2×C4—46no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—424no—
609C2×C4—824no—
610C2×C4C2×#137112yes—
611C2×C4C23×D416yes—
612C2×C4C23×D412yes—
613C2×C4C23×D416yes—
614C2×C4C23×D416yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C23×D416yes—
617C2×C4C2×2−1+412yes—
618C2×C4#58512yes—
619C2×C4C2611yescharacteristic
620C8—22no—
621C8—26no—
622C8—26no—
623C8C2×2−1+414yes—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—44no—
628C22—46no—
629C22—46no—
630C22—46no—
631C22—812no—
632C22#349712yes—
633C22C22×#13712yes—
634C22#358512yes—
635C22C24×D411yescharacteristic
636C4—22no—
637C4—26no—
638C4—26no—
639C4—26no—
640C4—42no—
641C4—46no—
642C4—42no—
643C4—42no—
644C4—46no—
645C4—42no—
646C4—46no—
647C4#349712yes—
648C4C22×#137112yes—
649C4C22×2−1+411yescharacteristic
650C4C2×#58511yescharacteristic
651C2—42no—
652C2—86no—
653C2#6239414yesdirect product factor
654C2C2×#349711yescharacteristic
655C2C23×#13711yescharacteristic
656C2C2×#358511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC2×#3522C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3502C2semidirect
3. #62394C2direct (N × H)
4. C2×#3400C2semidirect
5. C2×#3392C2semidirect
6. C2×#3385C2semidirect
7. C2×Q8×Q16C2semidirect
8. C2×#3375C2semidirect
9. C2×#3372C2semidirect
10. C2×#3363C2semidirect
11. C2×#3362C2semidirect
12. C2×#3254C2semidirect
13. C2×#3252C2semidirect
14. C2×#3182C2semidirect
15. C2×#3172C2semidirect
16. C2×Q8×M16C2semidirect
20. #3502C22semidirect
42. #3400C22semidirect
43. #3392C22semidirect
44. #3385C22semidirect
45. Q8×Q16C22semidirect
46. #3375C22semidirect
47. #3372C22semidirect
48. #3363C22semidirect
49. #3362C22semidirect
50. #3254C22semidirect
51. #3252C22semidirect
52. #3182C22semidirect
53. #3172C22semidirect
79. Q8×M16C22semidirect
653. C2#62394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35222
C2×#35022
#623942
C2×#34002
C2×#33922
C2×#33852
C2×Q8×Q162
C2×#33752
C2×#33722
C2×#33632
C2×#33622
C2×#32542
C2×#32522
C2×#31822
C2×#31722
C2×Q8×M162