Finite Group Databasebeta

← C2×#17617 #6343453

Subgroups — C2×#17617 #6343453

Order 512.

3288 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3506C211yesmaximal, characteristic
2C2×#2972C211yesmaximal, characteristic, complemented
3C2×#2357C211yesmaximal, characteristic, complemented
4#17617C2116yesmaximal, direct product factor
5C2×#1518C211yesmaximal, characteristic, complemented
6C2×#1517C211yesmaximal, characteristic, complemented
7C2×#1516C214yesmaximal, complemented
8C2×#1503C211yesmaximal, characteristic, complemented
9C2×#1502C211yesmaximal, characteristic, complemented
10C2×#1501C214yesmaximal, 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×D4C2211yescharacteristic
28#2972—24no—
29#2972C2218yescomplemented
30C2×C8×D4—22no—
31C2×C8×D4—22no—
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C22×C22⋊C8C2214yes—
35#2357—24no—
36#2357C2218yescomplemented
37C2×#390—21no—
38C2×#388—22no—
39C2×#382C2211yescharacteristic
40C2×#381C2212yes—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1518C2218yescomplemented
45#1517C2218yescomplemented
46#1516C22132yescomplemented
47#1503C2218yescomplemented
48#1502C2218yescomplemented
49#1501C22132yescomplemented
50C2×#328C2212yes—
51C2×#328C2214yes—
52C2×#328C2212yes—
53C2×#327C2212yes—
54C2×#327C2214yes—
55C2×#327C2212yes—
56C22×#41—21no—
57C22×#41C2×C414yes—
58C23×D4D411yescharacteristic
59C24×C4C2312yes—
60C23×C8—24no—
61#559—216no—
62#559—216no—
63#556—28no—
64#556C2×C4116yescomplemented
65#555—28no—
66#555C2×C4116yescomplemented
67#553—216no—
68#553—216no—
69#550—28no—
70#550C2×C4116yescomplemented
71#547—28no—
72#547C2×C4116yescomplemented
73#538—216no—
74#538—216no—
75#535—28no—
76#535—28no—
77#533—22no—
78#533—22no—
79#533C2×C414yes—
80#533C2314yes—
81#532—22no—
82#532—22no—
83#532C2×C414yes—
84#532C2314yes—
85C2×C4⋊Q8C2×C411yescharacteristic
86C2×C4⋊Q8C2×C411yescharacteristic
87C2×C4⋊Q8C2×C411yescharacteristic
88C2×#127—22no—
89C2×#127C2×C411yescharacteristic
90C2×#126—24no—
91C2×#125—21no—
92C2×#124—22no—
93C2×#124—24no—
94C2×#124—21no—
95#525—28no—
96#525—28no—
97#525C2×C4116yes—
98#525C23116yes—
99C2×#123—24no—
100C2×#123—24no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2×C414yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—22no—
105C2×#121—22no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121C2×C414yes—
109C2×#121C2312yes—
110C2×#120—24no—
111#520—28no—
112#520—28no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2×C412yes—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4C2×C412yes—
121C2×C4×D4C2×C412yes—
122C2×C4×D4C2×C412yes—
123C2×C4×D4C2×C414yes—
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D414yes—
127C2×C4×D4D414yes—
128C2×C4×D4C2314yes—
129C2×#117—21no—
130C2×#117—22no—
131C2×#117C2×C414yes—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4D411yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C22⋊C4D412yes—
139C22×C42—21no—
140C22×C42—22no—
141C22×C42C2311yescharacteristic
142C8×D4—216no—
143C8×D4—216no—
144#431—22no—
145#431—22no—
146#431C2314yes—
147#431C2314yes—
148C2×C4⋊C8—21no—
149C2×C4⋊C8—21no—
150C2×C22⋊C8—24no—
151C2×C22⋊C8—28no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C23116yes—
155C2×C4×C8—21no—
156C2×C4×C8—21no—
157#390—28no—
158#388—216no—
159#382—22no—
160#382C2314yes—
161#381—24no—
162#381C2318yes—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4C2314yes—
166C4×C22⋊C4—24no—
167C4×C22⋊C4C2318yes—
168C2×#95—22no—
169C2×#95—21no—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173#328C2318yes—
174#328C23116yes—
175#328C2318yes—
176#327C2318yes—
177#327C23116yes—
178#327C2318yes—
179C25C2×D412yes—
180C2×#41—24no—
181C2×#41—28no—
182C2×#41—24no—
183C2×#41—44no—
184C2×#41—44no—
185C2×#41—88no—
186C2×#41C22×C4116yes—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8C22×C412yes—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4C22⋊C412yes—
198C22×D4C22⋊C412yes—
199C22×D4C22⋊C414yes—
200C22×D4C22×C412yes—
201C22×D4C2×D414yes—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—84no—
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—216no—
221C22×C8—88no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8C22×C414yes—
226C4⋊Q8C22×C414yes—
227C4⋊Q8C22×C414yes—
228#127—216no—
229#127—22no—
230#127C22×C414yes—
231#126—232no—
232#125—24no—
233#125—24no—
234#124—216no—
235#124—232no—
236#124—24no—
237#124—24no—
238#123—216no—
239#123—216no—
240#123—232no—
241C22⋊Q8—24no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8C22×C4116yes—
247C22⋊Q8C22×C414yes—
248C22⋊Q8C2414yes—
249#121—216no—
250#121—216no—
251#121—24no—
252#121—28no—
253#121—24no—
254#121—22no—
255#121—22no—
256#121—232no—
257#121C22×C4116yes—
258#121C22×C414yes—
259#121C2414yes—
260#120—216no—
261#120—216no—
262C4×Q8—216no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8C22×C418yes—
267C4×D4—216no—
268C4×D4—216no—
269C4×D4—216no—
270C4×D4—216no—
271C4×D4—216no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4C22⋊C418yes—
276C4×D4C22⋊C418yes—
277C4×D4C22⋊C418yes—
278C4×D4C22⋊C418yes—
279C4×D4C22×C418yes—
280C4×D4C22×C418yes—
281C4×D4C22×C418yes—
282C4×D4C2×D418yes—
283C4×D4C2×D418yes—
284#117—24no—
285#117—28no—
286#117—24no—
287#117—216no—
288#117C22×C4116yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—216no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C22⋊C412yes—
302C2×C4⋊C4C22⋊C412yes—
303C2×C4⋊C4C22⋊C414yes—
304C2×C4⋊C4C22×C412yes—
305C2×C4⋊C4C22×C412yes—
306C2×C4⋊C4C2×D414yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4C22⋊C414yes—
318C2×C22⋊C4C22⋊C414yes—
319C2×C22⋊C4C22⋊C418yes—
320C2×C22⋊C4C2×D418yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—44no—
326C2×C42—84no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—216no—
340C22⋊C8C2414yes—
341C22⋊C8C2414yes—
342C4×C8—24no—
343C4×C8—24no—
344#95—28no—
345#95—24no—
346#95—21no—
347#95—22no—
348#95—21no—
349#95C2412yes—
350#95C2414yes—
351#95C2412yes—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—44no—
356C24—44no—
357C24C2×C22⋊C414yes—
358C24C2×C22⋊C414yes—
359C24C22×D411yescharacteristic
360C24C22×D414yes—
361#41—48no—
362#41—48no—
363#41—416no—
364#41—816no—
365#41—816no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—82no—
372C2×Q8C23×C414yes—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—82no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4C2×C22⋊C418yes—
393C2×D4C2×C22⋊C418yes—
394C2×D4C23×C414yes—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4C2×C22⋊C414yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C414yes—
428C22×C4C2×C22⋊C418yes—
429C22×C4C2×C22⋊C418yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4#13514yes—
434C22×C4#13514yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C23×C411yescharacteristic
437C22×C4C23×C418yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D418yes—
444C22×C4C22×D412yes—
445C22×C4C2511yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8—816no—
451C4⋊C4—24no—
452C4⋊C4—216no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4C2×C22⋊C418yes—
464C4⋊C4C2×C22⋊C418yes—
465C4⋊C4C23×C414yes—
466C4⋊C4C23×C414yes—
467C22⋊C4—24no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—24no—
471C22⋊C4—216no—
472C22⋊C4—24no—
473C22⋊C4—28no—
474C22⋊C4—24no—
475C22⋊C4—416no—
476C22⋊C4—88no—
477C22⋊C4C2×C22⋊C4116yes—
478C22⋊C4C2×C22⋊C4116yes—
479C42—216no—
480C42—28no—
481C42—48no—
482C42—88no—
483C42C2×C22⋊C412yes—
484C42C2×C22⋊C412yes—
485C42C2×C22⋊C412yes—
486C42C2×C22⋊C412yes—
487C42C23×C412yes—
488C42C23×C412yes—
489C42C22×D412yes—
490C42C22×D412yes—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—88no—
508C23—88no—
509C23C22×C22⋊C418yes—
510C23C22×C22⋊C411yescharacteristic
511C23C22×C22⋊C411yescharacteristic
512C23C2×#13512yes—
513C23C2×#13512yes—
514C23C23×D411yescharacteristic
515Q8—44no—
516Q8—44no—
517Q8—48no—
518Q8—84no—
519D4—216no—
520D4—216no—
521D4—44no—
522D4—44no—
523D4—48no—
524D4—44no—
525D4—44no—
526D4—48no—
527D4—84no—
528D4—816no—
529D4—816no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—416no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—84no—
566C2×C4—88no—
567C2×C4#41614yes—
568C2×C4#41614yes—
569C2×C4#41618yes—
570C2×C4C22×C22⋊C418yes—
571C2×C4C22×C22⋊C411yescharacteristic
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C22×C22⋊C412yes—
574C2×C4C22×C22⋊C412yes—
575C2×C4C22×C22⋊C412yes—
576C2×C4C22×C22⋊C412yes—
577C2×C4C2×#13514yes—
578C2×C4C2×#13514yes—
579C2×C4C24×C411yescharacteristic
580C2×C4C23×D411yescharacteristic
581C8—44no—
582C8—44no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—24no—
588C22—24no—
589C22—28no—
590C22—28no—
591C22—28no—
592C22—416no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—84no—
597C22—84no—
598C22—88no—
599C22—88no—
600C22#288312yes—
601C22C2×#41612yes—
602C22C2×#41612yes—
603C22#290112yes—
604C22C23×C22⋊C411yescharacteristic
605C22C22×#13511yescharacteristic
606C22C22×#13511yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—28no—
612C4—22no—
613C4—22no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—84no—
620C4C2×#41614yes—
621C4C2×#41614yes—
622C2—21no—
623C2—21no—
624C2—22no—
625C2—48no—
626C2—84no—
627C2#1761714yesdirect product factor
628C2C2×#288311yescharacteristic
629C2C22×#41611yescharacteristic
630C2C2×#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×#2972C2semidirect
3. C2×#2357C2semidirect
4. #17617C2direct (N × H)
5. C2×#1518C2semidirect
6. C2×#1517C2semidirect
7. C2×#1516C2semidirect
8. C2×#1503C2semidirect
9. C2×#1502C2semidirect
10. C2×#1501C2semidirect
13. #3506C4semidirect
16. C2×#556C4semidirect
17. C2×#555C4semidirect
19. C2×#550C4semidirect
20. C2×#547C4semidirect
29. #2972C22semidirect
36. #2357C22semidirect
44. #1518C22semidirect
45. #1517C22semidirect
46. #1516C22semidirect
47. #1503C22semidirect
48. #1502C22semidirect
49. #1501C22semidirect
64. #556C2×C4semidirect
66. #555C2×C4semidirect
70. #550C2×C4semidirect
72. #547C2×C4semidirect
627. C2#17617direct (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×#29722
C2×#23572
#176172
C2×#15182
C2×#15172
C2×#15162
C2×#15032
C2×#15022
C2×#15012