Finite Group Databasebeta

← C2×#17630 #6343474

Subgroups — C2×#17630 #6343474

Order 512.

3472 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×#3506C211yesmaximal, characteristic
2C2×#2975C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic, complemented
4#17630C2116yesmaximal, direct product factor
5C2×#1518C211yesmaximal, characteristic, complemented
6C2×#1517C211yesmaximal, characteristic, complemented
7C2×#1513C214yesmaximal, complemented
8C2×#1504C214yesmaximal, complemented
9C2×#1503C211yesmaximal, characteristic, complemented
10C2×#1502C211yesmaximal, characteristic, complemented
11#3506—24no—
12#3506—24no—
13#3506C418yescomplemented
14#3506C2218yes—
15C2×#559—22no—
16C2×#556C412yescomplemented
17C2×#555C412yescomplemented
18C2×#553—22no—
19C2×#550C412yescomplemented
20C2×#547C412yescomplemented
21C2×#538—22no—
22C2×#535—21no—
23C2×#533C2211yescharacteristic
24C2×#532C2211yescharacteristic
25C2×#525C2214yes—
26C2×#520—21no—
27C22×C4×D4—21no—
28C22×C4×D4C2211yescharacteristic
29#2975—24no—
30#2975C2218yescomplemented
31C2×#436—22no—
32C2×#436—22no—
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#407C2214yes—
36#2349—24no—
37#2349C2218yescomplemented
38C2×#389—21no—
39C2×#385—22no—
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44#1518C2218yescomplemented
45#1517C2218yescomplemented
46#1513C22132yescomplemented
47#1504C22132yescomplemented
48#1503C2218yescomplemented
49#1502C2218yescomplemented
50C2×#328C2212yes—
51C2×#328C2212yes—
52C2×#328C2214yes—
53C2×#327C2212yes—
54C2×#327C2212yes—
55C2×#327C2214yes—
56C22×#41—21no—
57C22×#41C2×C414yescomplemented
58C23×D4D411yescharacteristic
59C24×C4—22no—
60C24×C4C2312yes—
61C22×M16—24no—
62#559—216no—
63#559—216no—
64#556—28no—
65#556C2×C4116yescomplemented
66#555—28no—
67#555C2×C4116yescomplemented
68#553—216no—
69#553—216no—
70#550—28no—
71#550C2×C4116yescomplemented
72#547—28no—
73#547C2×C4116yescomplemented
74#538—216no—
75#538—216no—
76#535—28no—
77#535—28no—
78#533—22no—
79#533—22no—
80#533C2×C414yescomplemented
81#533C2314yes—
82#532—22no—
83#532—22no—
84#532C2×C414yescomplemented
85#532C2314yes—
86C2×C4⋊Q8C2×C411yescharacteristic, complemented
87C2×C4⋊Q8C2×C411yescharacteristic, complemented
88C2×C4⋊Q8C2×C411yescharacteristic, complemented
89C2×#127—22no—
90C2×#127C2×C411yescharacteristic, complemented
91C2×#126—24no—
92C2×#125—21no—
93C2×#124—22no—
94C2×#124—24no—
95C2×#124—21no—
96#525—28no—
97#525—28no—
98#525C2×C4116yescomplemented
99#525C23116yes—
100C2×#123—24no—
101C2×#123—24no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2×C414yescomplemented
104C2×C22⋊Q8C2312yes—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121—21no—
108C2×#121—24no—
109C2×#121C2×C414yescomplemented
110C2×#121C2312yes—
111C2×#120—24no—
112#520—28no—
113#520—28no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2×C412yescomplemented
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—28no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—48no—
123C2×C4×D4C2×C412yes—
124C2×C4×D4C2×C412yes—
125C2×C4×D4C2×C414yes—
126C2×C4×D4C2×C412yescomplemented
127C2×C4×D4D412yes—
128C2×C4×D4D412yes—
129C2×C4×D4D414yes—
130C2×C4×D4D414yes—
131C2×C4×D4C2314yes—
132C2×#117—21no—
133C2×#117—22no—
134C2×#117C2×C414yescomplemented
135C22×C4⋊C4—22no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4D411yescharacteristic
140C22×C22⋊C4—22no—
141C22×C22⋊C4D412yes—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#436—216no—
145#436—216no—
146#431—22no—
147#431—22no—
148#431C2314yes—
149#431C2314yes—
150C2×C4⋊C8—21no—
151C2×C4⋊C8—21no—
152#407—28no—
153#407C23116yes—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C4×C8—21no—
158C2×C4×C8—21no—
159#389—28no—
160#385—216no—
161#382—22no—
162#382C2314yes—
163#380—24no—
164#380C2318yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C2×#95—24no—
168C2×#95—22no—
169C2×#95—21no—
170C2×#95C2318yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174#328C2318yes—
175#328C2318yes—
176#328C23116yes—
177#327C2318yes—
178#327C2318yes—
179#327C23116yes—
180C25C2×D412yes—
181C2×#41—24no—
182C2×#41—28no—
183C2×#41—24no—
184C2×#41—44no—
185C2×#41—44no—
186C2×#41—88no—
187C2×#41C22×C4116yescomplemented
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8C22×C412yes—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4C22⋊C412yes—
199C22×D4C22⋊C412yes—
200C22×D4C22⋊C414yes—
201C22×D4C22×C412yes—
202C22×D4C2×D414yes—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—48no—
210C23×C4—48no—
211C23×C4—84no—
212C23×C4—84no—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C2×M16—216no—
224C2×M16—88no—
225C22×C8—22no—
226C22×C8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8C22×C414yescomplemented
231C4⋊Q8C22×C414yescomplemented
232C4⋊Q8C22×C414yescomplemented
233#127—216no—
234#127—22no—
235#127C22×C414yescomplemented
236#126—232no—
237#125—24no—
238#125—24no—
239#124—216no—
240#124—232no—
241#124—24no—
242#124—24no—
243#123—216no—
244#123—216no—
245#123—232no—
246C22⋊Q8—24no—
247C22⋊Q8—28no—
248C22⋊Q8—24no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8C22×C414yes—
252C22⋊Q8C22×C4116yescomplemented
253C22⋊Q8C2414yes—
254#121—216no—
255#121—216no—
256#121—24no—
257#121—28no—
258#121—24no—
259#121—22no—
260#121—22no—
261#121—232no—
262#121C22×C414yes—
263#121C22×C4116yescomplemented
264#121C2414yes—
265#120—216no—
266#120—216no—
267C4×Q8—216no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8C22×C418yescomplemented
272C4×D4—216no—
273C4×D4—216no—
274C4×D4—216no—
275C4×D4—216no—
276C4×D4—216no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—432no—
281C4×D4C22⋊C418yes—
282C4×D4C22⋊C418yes—
283C4×D4C22⋊C418yes—
284C4×D4C22⋊C418yes—
285C4×D4C22×C418yes—
286C4×D4C22×C418yes—
287C4×D4C22×C418yescomplemented
288C4×D4C2×D418yes—
289C4×D4C2×D418yes—
290#117—24no—
291#117—28no—
292#117—24no—
293#117—216no—
294#117C22×C4116yescomplemented
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C22⋊C412yes—
310C2×C4⋊C4C22⋊C412yes—
311C2×C4⋊C4C22⋊C414yes—
312C2×C4⋊C4C22×C412yes—
313C2×C4⋊C4C22×C412yes—
314C2×C4⋊C4C2×D414yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4C22⋊C414yes—
324C2×C22⋊C4C22⋊C414yes—
325C2×C22⋊C4C22⋊C418yes—
326C2×C22⋊C4C2×D418yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C22×C412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C4⋊C8—24no—
340C4⋊C8—24no—
341C22⋊C8—216no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8C2414yes—
345C22⋊C8C2414yes—
346C4×C8—24no—
347C4×C8—24no—
348#95—21no—
349#95—22no—
350#95—28no—
351#95—24no—
352#95—21no—
353#95C2412yes—
354#95C2414yes—
355#95C2412yes—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—44no—
360C24—44no—
361C24C2×C22⋊C414yes—
362C24C2×C22⋊C414yes—
363C24C22×D411yescharacteristic
364C24C22×D414yes—
365#41—48no—
366#41—48no—
367#41—416no—
368#41—816no—
369#41—816no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—82no—
376C2×Q8C23×C414yes—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—82no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4C2×C22⋊C418yes—
397C2×D4C2×C22⋊C418yes—
398C2×D4C23×C414yes—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—416no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—832no—
431C22×C4C2×C22⋊C414yes—
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C414yes—
436C22×C4C2×C22⋊C418yes—
437C22×C4C2×C22⋊C418yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4#13714yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C23×C411yescharacteristic
445C22×C4C23×C418yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D418yes—
452C22×C4C22×D412yes—
453C22×C4C2511yescharacteristic
454M16—816no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—44no—
458C2×C8—44no—
459C4⋊C4—24no—
460C4⋊C4—216no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4C2×C22⋊C418yes—
473C4⋊C4C2×C22⋊C418yes—
474C4⋊C4C23×C414yes—
475C4⋊C4C23×C414yes—
476C22⋊C4—24no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—24no—
480C22⋊C4—216no—
481C22⋊C4—24no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—416no—
485C22⋊C4C2×C22⋊C4116yes—
486C22⋊C4C2×C22⋊C4116yes—
487C42—216no—
488C42—28no—
489C42—48no—
490C42C2×C22⋊C412yes—
491C42C2×C22⋊C412yes—
492C42C2×C22⋊C412yes—
493C42C2×C22⋊C412yes—
494C42C23×C412yes—
495C42C23×C412yes—
496C42C22×D412yes—
497C42C22×D412yes—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—416no—
512C23—416no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C22×C22⋊C418yes—
517C23C22×C22⋊C411yescharacteristic
518C23C22×C22⋊C411yescharacteristic
519C23C2×AGL(1,ℤ/8)12yes—
520C23C2×#13712yes—
521C23C23×D411yescharacteristic
522Q8—44no—
523Q8—44no—
524Q8—48no—
525Q8—84no—
526D4—216no—
527D4—216no—
528D4—44no—
529D4—44no—
530D4—48no—
531D4—44no—
532D4—44no—
533D4—48no—
534D4—84no—
535D4—816no—
536D4—816no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—816no—
572C2×C4—832no—
573C2×C4—84no—
574C2×C4—88no—
575C2×C4#41714yes—
576C2×C4#41714yes—
577C2×C4#41718yes—
578C2×C4C22×C22⋊C418yes—
579C2×C4C22×C22⋊C411yescharacteristic
580C2×C4C22×C22⋊C411yescharacteristic
581C2×C4C22×C22⋊C412yes—
582C2×C4C22×C22⋊C412yes—
583C2×C4C22×C22⋊C412yes—
584C2×C4C22×C22⋊C412yes—
585C2×C4C2×AGL(1,ℤ/8)14yes—
586C2×C4C2×#13714yes—
587C2×C4C24×C411yescharacteristic
588C2×C4C23×D411yescharacteristic
589C8—44no—
590C8—44no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—48no—
601C22—416no—
602C22—48no—
603C22—416no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—88no—
608C22#288312yes—
609C22C2×#41712yes—
610C22C2×#41712yes—
611C22#290112yes—
612C22C23×C22⋊C411yescharacteristic
613C22C22×AGL(1,ℤ/8)11yescharacteristic
614C22C22×#13711yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—28no—
620C4—22no—
621C4—22no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—84no—
628C4C2×#41714yes—
629C4C2×#41714yes—
630C2—21no—
631C2—22no—
632C2—21no—
633C2—48no—
634C2—84no—
635C2#1763014yesdirect product factor
636C2C2×#288311yescharacteristic
637C2C22×#41711yescharacteristic
638C2C2×#290111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3506C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2975C2semidirect
3. C2×#2349C2semidirect
4. #17630C2direct (N × H)
5. C2×#1518C2semidirect
6. C2×#1517C2semidirect
7. C2×#1513C2semidirect
8. C2×#1504C2semidirect
9. C2×#1503C2semidirect
10. C2×#1502C2semidirect
13. #3506C4semidirect
16. C2×#556C4semidirect
17. C2×#555C4semidirect
19. C2×#550C4semidirect
20. C2×#547C4semidirect
30. #2975C22semidirect
37. #2349C22semidirect
44. #1518C22semidirect
45. #1517C22semidirect
46. #1513C22semidirect
47. #1504C22semidirect
48. #1503C22semidirect
49. #1502C22semidirect
57. C22×#41C2×C4semidirect
65. #556C2×C4semidirect
67. #555C2×C4semidirect
71. #550C2×C4semidirect
73. #547C2×C4semidirect
80. #533C2×C4semidirect
84. #532C2×C4semidirect
86. C2×C4⋊Q8C2×C4semidirect
87. C2×C4⋊Q8C2×C4semidirect
88. C2×C4⋊Q8C2×C4semidirect
90. C2×#127C2×C4semidirect
98. #525C2×C4semidirect
103. C2×C22⋊Q8C2×C4semidirect
109. C2×#121C2×C4semidirect
116. C2×C4×Q8C2×C4semidirect
126. C2×C4×D4C2×C4semidirect
134. C2×#117C2×C4semidirect
187. C2×#41C22×C4semidirect
230. C4⋊Q8C22×C4semidirect
231. C4⋊Q8C22×C4semidirect
232. C4⋊Q8C22×C4semidirect
235. #127C22×C4semidirect
252. C22⋊Q8C22×C4semidirect
263. #121C22×C4semidirect
271. C4×Q8C22×C4semidirect
287. C4×D4C22×C4semidirect
294. #117C22×C4semidirect
635. C2#17630direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35062
C2×#29752
C2×#23492
#176302
C2×#15182
C2×#15172
C2×#15132
C2×#15042
C2×#15032
C2×#15022