Finite Group Databasebeta

← C2×#9232 #198014

Subgroups — C2×#9232 #198014

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#9232C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic
8C2×#1276C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12#3104—22no—
13#3104—22no—
14#3104C414yescomplemented
15#3104C2214yes—
16C2×#484—21no—
17C2×#477C412yescomplemented
18C2×#462C412yescomplemented
19C2×#451—21no—
20C2×C4×Q16—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C42×Q8—22no—
25C42×Q8—22no—
26C42×Q8—22no—
27C42×Q8C2214yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—22no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C43—21no—
33C2×C43C2211yescharacteristic
34#1994—22no—
35#1994C2214yes—
36C2×#339—21no—
37C2×#339—21no—
38#1503—22no—
39#1503C2214yes—
40C2×#328—22no—
41#1487—22no—
42#1487C2214yes—
43C2×Q8⋊C8—22no—
44#1303C2214yes—
45#1276C2214yes—
46C23×Q8D411yescharacteristic
47C22×Q16—22no—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533C2×C412yes—
56#533C2312yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—216no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8D412yes—
77C2×C4×Q8D412yes—
78C2×C4×Q8C2312yes—
79C2×#117—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C42—22no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93#484—24no—
94#484—24no—
95#477—24no—
96#477C2×C418yescomplemented
97#462—24no—
98#462C2×C418yescomplemented
99#451—24no—
100#451—24no—
101C4×Q16—28no—
102C4×Q16—28no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2312yes—
110#431C2312yes—
111C2×#107—22no—
112#423—21no—
113#423—21no—
114#423—21no—
115#423C2312yes—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2×C412yes—
120C2×#103—22no—
121C2×#103—22no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4D412yes—
135C4×C4⋊C4C2312yes—
136C2×#95—22no—
137C2×#95D412yes—
138C43—24no—
139C43—24no—
140C43—21no—
141C43—21no—
142C43D412yes—
143C43C2312yes—
144#339—24no—
145#339—24no—
146#328—28no—
147Q8⋊C8—28no—
148C22×Q8—28no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C412yes—
156C22×Q8C2×D412yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×Q16—28no—
166C2×Q16—48no—
167C2×M16—22no—
168C2×M16—41no—
169C2×M16—42no—
170C2×M16—42no—
171C22×C8—22no—
172C22×C8—41no—
173C22×C8—41no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182#125—22no—
183#125—22no—
184#125—41no—
185#125—41no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C4×Q8—232no—
195C4×Q8—232no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8C22⋊C414yes—
217C4×Q8C22⋊C414yes—
218C4×Q8C22×C414yes—
219C4×Q8C2×D414yes—
220#117—22no—
221#117—22no—
222#117—41no—
223#117—41no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C22⋊C412yes—
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C22⋊C4—41no—
248C2×C42—28no—
249C2×C42—216no—
250C2×C42—28no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—44no—
270C2×C42—48no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42Q1614yescomplemented
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#107—28no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8—41no—
297C4⋊C8C22×C414yes—
298#103—28no—
299#103—28no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C4×C8—22no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—41no—
308#95—28no—
309#95—22no—
310#95C2×D414yes—
311C24#12011yescharacteristic
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8—28no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—44no—
326C2×Q8—82no—
327C2×Q8#10418yes—
328C2×Q8C2×C22⋊C414yes—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12311yescharacteristic
367C22×C4#12311yescharacteristic
368C22×C4C2×Q1612yes—
369C22×C4#13512yes—
370C22×C4#13712yes—
371C22×C4#13712yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375Q16—416no—
376M16—42no—
377M16—42no—
378M16—82no—
379C2×C8—24no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—82no—
384C2×C8—82no—
385C4⋊C4—232no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4C2×C22⋊C414yes—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—232no—
411C42—28no—
412C42—28no—
413C42—24no—
414C42—24no—
415C42—22no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—44no—
422C42—44no—
423C42—416no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—44no—
430C42C2×C22⋊C411yescharacteristic
431C42C2×C22⋊C411yescharacteristic
432C42C4×D411yescharacteristic
433C42C4×D411yescharacteristic
434C42C4×D411yescharacteristic
435C42C4×D411yescharacteristic
436C42#12111yescharacteristic
437C42#12111yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12311yescharacteristic
441C42#12311yescharacteristic
442C42C2×Q1614yescomplemented
443C42C23×C411yescharacteristic
444C42C22×D411yescharacteristic
445C42C2×#4111yescharacteristic
446C42C2×#4111yescharacteristic
447C23—22no—
448C23—22no—
449C23#38611yescharacteristic
450C23#38611yescharacteristic
451C23#45112yes—
452C23#45212yes—
453C23C2×#12011yescharacteristic
454Q8—28no—
455Q8—28no—
456Q8—24no—
457Q8—28no—
458Q8—44no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462Q8C2×#10418yes—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—216no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—416no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4C2×#10414yes—
505C2×C4C4×Q1612yes—
506C2×C4C4×Q1612yes—
507C2×C4#44112yes—
508C2×C4#44112yes—
509C2×C4#45112yes—
510C2×C4#45212yes—
511C2×C4#46212yes—
512C2×C4#46212yes—
513C2×C4#46412yes—
514C2×C4#46412yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4C22×Q1611yescharacteristic
522C2×C4C2×#13511yescharacteristic
523C2×C4C2×#13711yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C8—44no—
526C8—82no—
527C8—82no—
528C22—21no—
529C22—21no—
530C22—24no—
531C22—24no—
532C22—21no—
533C22—21no—
534C22#187812yes—
535C22#187812yes—
536C22#189612yes—
537C22#190612yes—
538C22C2×#38611yescharacteristic
539C22C2×#45111yescharacteristic
540C22C2×#45211yescharacteristic
541C4—28no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—28no—
547C4—21no—
548C4—21no—
549C4—44no—
550C4—48no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4#187812yes—
555C4#187812yes—
556C4#189612yes—
557C4C22×#10412yes—
558C4C2×C4×Q1611yescharacteristic
559C4C2×#44111yescharacteristic
560C4C2×#46211yescharacteristic
561C4C2×#46411yescharacteristic
562C2—21no—
563C2—21no—
564C2—22no—
565C2#923214yesdirect product factor
566C2C2×#187811yescharacteristic
567C2C2×#189611yescharacteristic
568C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9232C2direct (N × H)
14. #3104C4semidirect
17. C2×#477C4semidirect
18. C2×#462C4semidirect
96. #477C2×C4semidirect
98. #462C2×C4semidirect
274. C2×C42Q16semidirect
442. C42C2×Q16semidirect
565. C2#9232direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31042
C2×C42×Q82
C2×#19942
C2×#15032
C2×#14872
#92322
C2×#13032
C2×#12762