Finite Group Databasebeta

← C2×#60335 #10574544

Subgroups — C2×#60335 #10574544

Order 512.

6364 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3482C211yesmaximal, characteristic
2#60335C2132yesmaximal, direct product factor
3C2×#2994C213yesmaximal, complemented
4C2×#2993C213yesmaximal, complemented
5C2×#2990C213yesmaximal, complemented
6C2×#2988C211yesmaximal, characteristic, complemented
7C2×#2985C213yesmaximal, complemented
8C2×#2984C211yesmaximal, characteristic, complemented
9C2×#2982C211yesmaximal, characteristic, complemented
10C2×#2971C211yesmaximal, characteristic, complemented
11C2×#2969C213yesmaximal, complemented
12C2×#2966C213yesmaximal, complemented
13C2×#2965C213yesmaximal, complemented
14C2×Q8×M16C211yesmaximal, characteristic, complemented
15C2×#2962C213yesmaximal, complemented
16C22×C8×Q8C211yesmaximal, characteristic, complemented
17#3482C4116yes—
18#3482C22116yes—
19C2×#557C411yescharacteristic
20C2×#556C413yes—
21C2×#552C413yes—
22C2×#552C413yes—
23C2×D4×Q8C411yescharacteristic
24C2×#548C411yescharacteristic
25C2×#548C411yescharacteristic
26C2×#547C413yes—
27C2×#533C2213yes—
28C2×#529C2213yes—
29C2×#529C2213yes—
30C2×C4×#41C2213yes—
31C2×C4×#41C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33C22×C4×Q8C2211yescharacteristic
34#2994C22148yescomplemented
35#2993C22148yescomplemented
36#2990C22148yescomplemented
37#2988C22116yescomplemented
38#2985C22148yescomplemented
39#2984C22116yescomplemented
40#2982C22116yescomplemented
41#2971C22116yescomplemented
42#2969C22148yescomplemented
43#2966C22148yescomplemented
44#2965C22148yescomplemented
45Q8×M16C22116yescomplemented
46#2962C22148yescomplemented
47C2×#446C2213yes—
48C2×#446C2213yes—
49C2×#446C2213yes—
50C2×#446C2216yes—
51C2×#446C2213yes—
52C2×#446C2213yes—
53C2×#446C2213yes—
54C2×C8×Q8C2213yes—
55C2×C8×Q8C2211yescharacteristic
56C2×C8×Q8C2211yescharacteristic
57C2×C8×Q8C2213yes—
58C2×C8×Q8C2213yes—
59C2×C8×Q8C2211yescharacteristic
60C2×C8×Q8C2213yes—
61C2×C8×Q8C2211yescharacteristic
62C2×C8×Q8C22116yescomplemented
63C2×#436C2213yes—
64C2×#436C2213yes—
65C2×#436C2213yes—
66C2×#436C2213yes—
67C2×#435C2211yescharacteristic
68C2×#435C2213yes—
69C2×#435C2213yes—
70C2×#435C2213yes—
71C2×#435C2216yes—
72C2×C8×D4C2213yes—
73C2×C8×D4C2216yes—
74C2×C8×D4C2213yes—
75C2×#433C2216yes—
76C2×#433C2216yes—
77C2×#432C2213yes—
78C2×#432C2213yes—
79C2×#431C2213yes—
80C2×#431C2213yes—
81C2×#431C2213yes—
82C2×#431C2213yes—
83C2×#424C2216yes—
84C2×#423C2213yes—
85C22×C4⋊C8C2213yes—
86C22×C4⋊C8C2213yes—
87C2×#405C2216yes—
88C2×C4×M16C2213yes—
89C2×C4×M16C2213yes—
90C22×C4×C8C2213yes—
91C22×#41C2×C411yescharacteristic
92C22×#41C2×C413yes—
93C23×Q8C2×C411yescharacteristic
94C23×Q8C2×C411yescharacteristic
95C22×M16C2311yescharacteristic
96C22×M16C2313yes—
97C22×M16C2313yes—
98C22×M16C2313yes—
99C23×C8C2311yescharacteristic
100C23×C8C2313yes—
101C23×C8C2316yes—
102#557C2×C4116yes—
103#556C2×C4148yes—
104#552C2×C4148yes—
105#552C2×C4148yes—
106D4×Q8C2×C4116yes—
107#548C2×C4116yes—
108#548C2×C4116yes—
109#547C2×C4148yes—
110#533C2×C4124yes—
111#533C23124yes—
112C2×C4⋊Q8C2×C413yes—
113C2×C4⋊Q8C2×C413yes—
114#529C2×C4124yes—
115#529C2×C4124yes—
116#529C23124yes—
117#529C23124yes—
118C2×#126C2×C416yes—
119C2×#126C2×C416yes—
120C2×#125C2×C413yes—
121C2×#125C2×C413yes—
122C2×#125C2×C416yes—
123C2×#124C2×C413yes—
124C2×#124C2×C413yes—
125C2×#123C2×C416yes—
126C2×#123C2×C416yes—
127C2×C22⋊Q8C2×C413yes—
128C2×C22⋊Q8C2×C413yes—
129C2×C22⋊Q8C2×C416yes—
130C2×C22⋊Q8C2×C413yes—
131C2×C22⋊Q8C2×C413yes—
132C2×#121C2×C413yes—
133C2×#121C2×C413yes—
134C4×#41C2×C4124yes—
135C4×#41C2×C418yes—
136C4×#41C23124yes—
137C4×#41C2318yes—
138C2×C4×Q8C2×C411yescharacteristic
139C2×C4×Q8C2×C411yescharacteristic
140C2×C4×Q8C2×C413yes—
141C2×C4×Q8C2×C413yes—
142C2×C4×Q8C2×C418yes—
143C2×C4×Q8C2×C418yes—
144C2×C4×Q8C2×C411yescharacteristic
145C2×C4×Q8C2×C413yes—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2313yes—
148C2×C4×Q8C2313yes—
149C2×C4×Q8C2318yes—
150C2×C4×Q8C2318yes—
151C2×C4×Q8C2313yes—
152C2×C4×D4C2×C413yes—
153C2×C4×D4C2×C413yes—
154C2×C4×D4C2×C413yes—
155C2×C4×D4C2×C413yes—
156C2×C4×D4C2311yescharacteristic
157C2×C4×D4C2313yes—
158C2×C4×D4C2313yes—
159C2×C4×D4C2313yes—
160C2×#117C2×C413yes—
161C2×#117C2×C413yes—
162C2×#117C2×C413yes—
163C2×#117C2×C413yes—
164C2×#117C2316yes—
165C22×C4⋊C4C2×C413yes—
166C22×C4⋊C4C2×C413yes—
167C22×C42C2313yes—
168C22×C42C2313yes—
169C22×C42C2313yes—
170#446C23124yes—
171#446C23124yes—
172#446C23124yes—
173#446C23148yes—
174#446C23124yes—
175#446C23124yes—
176#446C23124yes—
177C8×Q8C23124yes—
178C8×Q8C2318yes—
179C8×Q8C2318yes—
180C8×Q8C23124yes—
181C8×Q8C23124yes—
182C8×Q8C2318yes—
183C8×Q8C23124yes—
184C8×Q8C2318yes—
185#436C23124yes—
186#436C23124yes—
187#436C23124yes—
188#436C23124yes—
189#435C2318yes—
190#435C23124yes—
191#435C23124yes—
192#435C23124yes—
193#435C23148yes—
194C8×D4C23124yes—
195C8×D4C23148yes—
196C8×D4C23124yes—
197#433C23148yes—
198#433C23148yes—
199#432C23124yes—
200#432C23124yes—
201#431C23124yes—
202#431C23124yes—
203#431C23124yes—
204#431C23124yes—
205#424C23148yes—
206#423C23124yes—
207C2×C4⋊C8C2313yes—
208C2×C4⋊C8C2313yes—
209C2×C4⋊C8C2313yes—
210C2×C4⋊C8C2313yes—
211C2×C4⋊C8C2316yes—
212C2×C4⋊C8C23124yes—
213C2×C4⋊C8C23124yes—
214C2×C4⋊C8C2316yes—
215C2×C4⋊C8C2313yes—
216C2×C4⋊C8C2313yes—
217C2×C4⋊C8C2311yescharacteristic
218C2×C4⋊C8C2313yes—
219C2×C4⋊C8C2313yes—
220C2×C4⋊C8C2313yes—
221C2×C22⋊C8C2311yescharacteristic
222C2×C22⋊C8C2313yes—
223C2×C22⋊C8C2311yescharacteristic
224C2×C22⋊C8C2313yes—
225C2×C22⋊C8C2313yes—
226C2×C22⋊C8C2313yes—
227C2×C22⋊C8C2316yes—
228#405C23148yes—
229C4×M16C23124yes—
230C4×M16C23124yes—
231C2×#97C2311yescharacteristic
232C2×#97C2313yes—
233C2×#97C2313yes—
234C2×#97C2313yes—
235C2×#97C2316yes—
236C2×C4×C8C2313yes—
237C2×C4×C8C2313yes—
238C2×C4×C8C2313yes—
239C2×C4×C8C2313yes—
240C2×C4×C8C23124yes—
241C2×C4×C8C2313yes—
242C2×C4×C8C2313yes—
243C2×C4×C8C2316yes—
244C2×#41—28no—
245C2×#41—412no—
246C2×#41C22×C418yes—
247C2×#41C22×C4124yes—
248C22×Q8—28no—
249C22×Q8—28no—
250C22×Q8C22×C413yes—
251C22×Q8C22×C413yes—
252C22×Q8C22×C418yes—
253C22×Q8C22×C418yes—
254C22×Q8C22×C413yes—
255C22×Q8C22×C411yescharacteristic
256C22×D4C22×C411yescharacteristic
257C22×D4C22×C413yes—
258C22×D4C22×C413yes—
259C22×D4C22×C413yes—
260C23×C4C22×C413yes—
261C23×C4C22×C413yes—
262C23×C4C22×C413yes—
263C23×C4C22×C413yes—
264C23×C4C22×C413yes—
265C23×C4C22×C413yes—
266C23×C4C2413yes—
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—212no—
271C2×M16—46no—
272C2×M16—46no—
273C2×M16C2414yes—
274C2×M16C24112yes—
275C2×M16C24112yes—
276C2×M16C24112yes—
277C22×C8—24no—
278C22×C8—412no—
279C22×C8—46no—
280C22×C8C2414yes—
281C22×C8C2413yes—
282C22×C8C2413yes—
283C22×C8C2413yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C22×C8C2413yes—
288C22×C8C24112yes—
289C22×C8C24124yes—
290C22×C8C2411yescharacteristic
291C4⋊Q8C22×C4124yes—
292C4⋊Q8C22×C4124yes—
293#126C22×C4148yes—
294#126C22×C4148yes—
295#125C22×C4124yes—
296#125C22×C4124yes—
297#125C22×C4148yes—
298#124C22×C4124yes—
299#124C22×C4124yes—
300#123C22×C4148yes—
301#123C22×C4148yes—
302C22⋊Q8C22×C4124yes—
303C22⋊Q8C22×C4124yes—
304C22⋊Q8C22×C4148yes—
305C22⋊Q8C22×C4124yes—
306C22⋊Q8C22×C4124yes—
307#121C22×C4124yes—
308#121C22×C4124yes—
309C4×Q8C22×C414yes—
310C4×Q8C22×C4112yes—
311C4×Q8C22×C4112yes—
312C4×Q8C22×C418yes—
313C4×Q8C22×C418yes—
314C4×Q8C22×C4124yes—
315C4×Q8C22×C4124yes—
316C4×Q8C22×C418yes—
317C4×Q8C22×C4124yes—
318C4×Q8C22×C4112yes—
319C4×Q8C2414yes—
320C4×Q8C24112yes—
321C4×Q8C24112yes—
322C4×Q8C24112yes—
323C4×D4C22×C4124yes—
324C4×D4C22×C4124yes—
325C4×D4C22×C4124yes—
326C4×D4C22×C4124yes—
327C4×D4C22×C414yes—
328C4×D4C22×C4112yes—
329C4×D4C22×C4112yes—
330C4×D4C22×C4112yes—
331C4×D4C2414yes—
332C4×D4C24112yes—
333C4×D4C24112yes—
334C4×D4C24112yes—
335#117C22×C4124yes—
336#117C22×C4124yes—
337#117C22×C4124yes—
338#117C22×C4124yes—
339#117C22×C4124yes—
340#117C24124yes—
341C2×C4⋊C4C22×C413yes—
342C2×C4⋊C4C22×C413yes—
343C2×C4⋊C4C22×C413yes—
344C2×C4⋊C4C22×C416yes—
345C2×C4⋊C4C22×C416yes—
346C2×C4⋊C4C22×C4124yes—
347C2×C4⋊C4C22×C4124yes—
348C2×C4⋊C4C22×C411yescharacteristic
349C2×C4⋊C4C22×C413yes—
350C2×C4⋊C4C22×C413yes—
351C2×C4⋊C4C22×C413yes—
352C2×C4⋊C4C22×C413yes—
353C2×C4⋊C4C22×C413yes—
354C2×C4⋊C4C22×C413yes—
355C2×C4⋊C4C22×C416yes—
356C2×C4⋊C4C22×C416yes—
357C2×C22⋊C4C22×C411yescharacteristic
358C2×C22⋊C4C22×C411yescharacteristic
359C2×C22⋊C4C22×C413yes—
360C2×C22⋊C4C22×C413yes—
361C2×C22⋊C4C22×C416yes—
362C2×C22⋊C4C22×C413yes—
363C2×C22⋊C4C22×C413yes—
364C2×C22⋊C4C22×C416yes—
365C2×C22⋊C4C22×C416yes—
366C2×C42C22×C413yes—
367C2×C42C22×C413yes—
368C2×C42C22×C413yes—
369C2×C42C22×C413yes—
370C2×C42C22×C413yes—
371C2×C42C22×C413yes—
372C2×C42C22×C4112yes—
373C2×C42C22×C4112yes—
374C2×C42C22×C4112yes—
375C2×C42C24112yes—
376C2×C42C24112yes—
377C2×C42C24112yes—
378C2×C42C2413yes—
379C2×C42C2413yes—
380C2×C42C2413yes—
381C2×C42C2411yescharacteristic
382C4⋊C8C24112yes—
383C4⋊C8C24112yes—
384C4⋊C8C24112yes—
385C4⋊C8C24112yes—
386C4⋊C8C24124yes—
387C4⋊C8C24124yes—
388C4⋊C8C24112yes—
389C4⋊C8C24112yes—
390C4⋊C8C2414yes—
391C4⋊C8C24112yes—
392C4⋊C8C24112yes—
393C4⋊C8C24112yes—
394C22⋊C8C2414yes—
395C22⋊C8C24112yes—
396C22⋊C8C2414yes—
397C22⋊C8C24112yes—
398C22⋊C8C24112yes—
399C22⋊C8C24112yes—
400C22⋊C8C24124yes—
401#97C2414yes—
402#97C24112yes—
403#97C24112yes—
404#97C24112yes—
405#97C24124yes—
406C4×C8C24112yes—
407C4×C8C24112yes—
408C4×C8C24112yes—
409C4×C8C24112yes—
410C4×C8C24112yes—
411C4×C8C24112yes—
412C4×C8C24124yes—
413C24C23×C413yes—
414C24C23×C411yescharacteristic
415C24C23×C411yescharacteristic
416#41—232no—
417#41—448no—
418C2×Q8—232no—
419C2×Q8—232no—
420C2×Q8—212no—
421C2×Q8—212no—
422C2×Q8—46no—
423C2×Q8#13118yes—
424C2×Q8C23×C4112yes—
425C2×Q8C23×C4112yes—
426C2×Q8C23×C4112yes—
427C2×Q8C23×C414yes—
428C2×D4—212no—
429C2×D4—24no—
430C2×D4—212no—
431C2×D4—46no—
432C2×D4C23×C414yes—
433C2×D4C23×C4112yes—
434C2×D4C23×C4112yes—
435C2×D4C23×C4112yes—
436C22×C4—212no—
437C22×C4—212no—
438C22×C4—212no—
439C22×C4—212no—
440C22×C4—22no—
441C22×C4—26no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—26no—
445C22×C4—22no—
446C22×C4—46no—
447C22×C4—46no—
448C22×C4C23×C411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C413yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C413yes—
456C22×C4C23×C416yes—
457C22×C4C23×C413yes—
458C22×C4C23×C413yes—
459C22×C4C23×C413yes—
460C22×C4C23×C413yes—
461C22×C4C23×C4112yes—
462C22×C4C23×C4112yes—
463C22×C4C23×C4112yes—
464C22×C4C23×C4112yes—
465C22×C4C23×C4112yes—
466C22×C4C23×C4112yes—
467C22×C4C23×C413yes—
468C22×C4C2512yes—
469C22×C4C2512yes—
470C22×C4C2516yes—
471C22×C4C2511yescharacteristic
472M16—28no—
473M16—224no—
474M16—412no—
475M16—412no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—26no—
479C2×C8—26no—
480C2×C8—26no—
481C2×C8—26no—
482C2×C8—22no—
483C2×C8—28no—
484C2×C8—424no—
485C2×C8—412no—
486C2×C82−1+414yes—
487C2×C8C2512yes—
488C2×C8C2516yes—
489C2×C8C2512yes—
490C2×C8C2516yes—
491C2×C8C2512yes—
492C2×C8C2516yes—
493C2×C8C2516yes—
494C2×C8C2512yes—
495C4⋊C4C23×C4112yes—
496C4⋊C4C23×C4112yes—
497C4⋊C4C23×C4112yes—
498C4⋊C4C23×C4124yes—
499C4⋊C4C23×C4124yes—
500C4⋊C4C23×C414yes—
501C4⋊C4C23×C4112yes—
502C4⋊C4C23×C4112yes—
503C4⋊C4C23×C4112yes—
504C4⋊C4C23×C4112yes—
505C4⋊C4C23×C4112yes—
506C4⋊C4C23×C4112yes—
507C4⋊C4C23×C4124yes—
508C4⋊C4C23×C4124yes—
509C22⋊C4C23×C414yes—
510C22⋊C4C23×C414yes—
511C22⋊C4C23×C4112yes—
512C22⋊C4C23×C4112yes—
513C22⋊C4C23×C4124yes—
514C22⋊C4C23×C4112yes—
515C22⋊C4C23×C4112yes—
516C22⋊C4C23×C4124yes—
517C22⋊C4C23×C4124yes—
518C42C23×C412yes—
519C42C23×C416yes—
520C42C23×C4112yes—
521C42C23×C416yes—
522C42C23×C416yes—
523C42C23×C4112yes—
524C42C23×C4112yes—
525C42C23×C4112yes—
526C42C23×C4112yes—
527C42C23×C4112yes—
528C42C2516yes—
529C42C2516yes—
530C42C2516yes—
531C42C2512yes—
532C23—22no—
533C23—22no—
534C23—26no—
535C23—22no—
536C23—22no—
537C23—26no—
538C23—22no—
539C23—26no—
540C23#56814yes—
541C23C24×C416yes—
542C23C24×C412yes—
543C23C24×C412yes—
544C23C24×C411yescharacteristic
545Q8—224no—
546Q8—224no—
547Q8—412no—
548Q8C2×#131116yes—
549D4—224no—
550D4—28no—
551D4—224no—
552D4—412no—
553C2×C4—22no—
554C2×C4—26no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—26no—
558C2×C4—26no—
559C2×C4—26no—
560C2×C4—26no—
561C2×C4—224no—
562C2×C4—22no—
563C2×C4—26no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—26no—
567C2×C4—22no—
568C2×C4—26no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—212no—
572C2×C4—24no—
573C2×C4—224no—
574C2×C4—224no—
575C2×C4—224no—
576C2×C4—412no—
577C2×C4—412no—
578C2×C4C2×#131112yes—
579C2×C4C24×C412yes—
580C2×C4C24×C412yes—
581C2×C4C24×C412yes—
582C2×C4C24×C416yes—
583C2×C4C24×C412yes—
584C2×C4C24×C416yes—
585C2×C4C24×C416yes—
586C2×C4C24×C416yes—
587C2×C4C24×C416yes—
588C2×C4C24×C416yes—
589C2×C4C24×C411yescharacteristic
590C2×C4C24×C416yes—
591C2×C4C2×2−1+412yes—
592C2×C4#58512yes—
593C2×C4C2611yescharacteristic
594C8—22no—
595C8—22no—
596C8—26no—
597C8—26no—
598C8—26no—
599C8—26no—
600C8—22no—
601C8C2×2−1+414yes—
602C22—22no—
603C22—28no—
604C22—22no—
605C22—22no—
606C22—26no—
607C22—22no—
608C22—26no—
609C22—26no—
610C22—22no—
611C22—44no—
612C22—412no—
613C22C4×2−1+412yes—
614C22C22×#13112yes—
615C22C2×#56814yes—
616C22C2×#56812yes—
617C22C25×C411yescharacteristic
618C4—22no—
619C4—26no—
620C4—26no—
621C4—26no—
622C4—26no—
623C4—22no—
624C4—26no—
625C4—22no—
626C4—26no—
627C4—22no—
628C4C4×2−1+412yes—
629C4C22×#131112yes—
630C4C22×2−1+411yescharacteristic
631C4C2×#58511yescharacteristic
632C2—24no—
633C2—42no—
634C2—46no—
635C2#6033514yesdirect product factor
636C2C2×C4×2−1+411yescharacteristic
637C2C23×#13111yescharacteristic
638C2C22×#56811yescharacteristic
subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, cosocleC2×C4C26
socleC23C24×C4
prime radicalC2×#3482C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60335C2direct (N × H)
3. C2×#2994C2semidirect
4. C2×#2993C2semidirect
5. C2×#2990C2semidirect
6. C2×#2988C2semidirect
7. C2×#2985C2semidirect
8. C2×#2984C2semidirect
9. C2×#2982C2semidirect
10. C2×#2971C2semidirect
11. C2×#2969C2semidirect
12. C2×#2966C2semidirect
13. C2×#2965C2semidirect
14. C2×Q8×M16C2semidirect
15. C2×#2962C2semidirect
16. C22×C8×Q8C2semidirect
34. #2994C22semidirect
35. #2993C22semidirect
36. #2990C22semidirect
37. #2988C22semidirect
38. #2985C22semidirect
39. #2984C22semidirect
40. #2982C22semidirect
41. #2971C22semidirect
42. #2969C22semidirect
43. #2966C22semidirect
44. #2965C22semidirect
45. Q8×M16C22semidirect
46. #2962C22semidirect
62. C2×C8×Q8C22semidirect
635. C2#60335direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#34822
#603352
C2×#29942
C2×#29932
C2×#29902
C2×#29882
C2×#29852
C2×#29842
C2×#29822
C2×#29712
C2×#29692
C2×#29662
C2×#29652
C2×Q8×M162
C2×#29622
C22×C8×Q82