Finite Group Databasebeta

← C2×#9236 #198019

Subgroups — C2×#9236 #198019

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#9236C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic
8C2×#1276C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3104—22no—
12#3104—22no—
13#3104C414yescomplemented
14#3104C2214yes—
15C2×#484—21no—
16C2×#477C412yescomplemented
17C2×#462C412yescomplemented
18C2×#451—21no—
19C2×C4×Q16—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23#2295—22no—
24#2295—22no—
25#2295—22no—
26#2295C2214yes—
27C2×#391—21no—
28C2×#389—21no—
29C2×#387—22no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376—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—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8D412yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8D412yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88#484—24no—
89#484—24no—
90#477—24no—
91#477C2×C418yescomplemented
92#462—24no—
93#462C2×C418yescomplemented
94#451—24no—
95#451—24no—
96C4×Q16—28no—
97C4×Q16—28no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431C2×C412yes—
104#431C2312yes—
105#431C2312yes—
106C2×#107—22no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423C2312yes—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2×C412yes—
115C2×#103—22no—
116C2×#103—22no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121#391—24no—
122#391—24no—
123#389—24no—
124#389—24no—
125#387—28no—
126#387—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133#376—24no—
134#376—24no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139C43—21no—
140C43—21no—
141C43D412yes—
142C43C2312yes—
143#339—24no—
144#339—24no—
145#328—28no—
146Q8⋊C8—28no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—44no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C412yes—
155C22×Q8C2×D412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×Q16—28no—
165C2×Q16—48no—
166C2×M16—22no—
167C2×M16—41no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—22no—
171C22×C8—41no—
172C22×C8—41no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C22×C412yes—
181#125—22no—
182#125—22no—
183#125—41no—
184#125—41no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
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—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8C22⋊C414yes—
214C4×Q8C22⋊C414yes—
215C4×Q8C22×C414yes—
216C4×Q8C2×D414yes—
217#117—22no—
218#117—22no—
219#117—41no—
220#117—41no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C22⋊C412yes—
242C2×C4⋊C4C22⋊C412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C22⋊C4—41no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
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—44no—
263C2×C42—44no—
264C2×C42—41no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42Q1614yescomplemented
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289C4⋊C8—41no—
290C4⋊C8C22×C414yes—
291#103—28no—
292#103—28no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—22no—
300C4×C8—41no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—22no—
305#95C2×D414yes—
306C24#12011yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×Q8C2×C22⋊C414yes—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4#12311yescharacteristic
363C22×C4C2×Q1612yes—
364C22×C4#13512yes—
365C22×C4#13712yes—
366C22×C4#13712yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370Q16—416no—
371M16—42no—
372M16—42no—
373M16—82no—
374C2×C8—24no—
375C2×C8—44no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—82no—
379C2×C8—82no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—416no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4C2×C22⋊C414yes—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—24no—
406C42—24no—
407C42—22no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—44no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—44no—
423C42C2×C22⋊C411yescharacteristic
424C42C2×C22⋊C411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42#12111yescharacteristic
430C42#12111yescharacteristic
431C42#12111yescharacteristic
432C42#12111yescharacteristic
433C42#12311yescharacteristic
434C42#12311yescharacteristic
435C42C2×Q1614yescomplemented
436C42C23×C411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—22no—
441C23—22no—
442C23#38611yescharacteristic
443C23#38611yescharacteristic
444C23#45112yes—
445C23#45212yes—
446C23C2×#12011yescharacteristic
447Q8—24no—
448Q8—24no—
449Q8—44no—
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—82no—
454Q8—82no—
455Q8—82no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—88no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#38611yescharacteristic
492C2×C4#38611yescharacteristic
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#42114yes—
498C2×C4C4×Q1612yes—
499C2×C4C4×Q1612yes—
500C2×C4#44112yes—
501C2×C4#44112yes—
502C2×C4#45112yes—
503C2×C4#45212yes—
504C2×C4#46212yes—
505C2×C4#46212yes—
506C2×C4#46412yes—
507C2×C4#46412yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4C22×Q1611yescharacteristic
515C2×C4C2×#13511yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C2×C4C2×#13711yescharacteristic
518C8—44no—
519C8—82no—
520C8—82no—
521C22—21no—
522C22—21no—
523C22—24no—
524C22—24no—
525C22—21no—
526C22—21no—
527C22#187812yes—
528C22#187812yes—
529C22#189712yes—
530C22#190612yes—
531C22C2×#38611yescharacteristic
532C22C2×#45111yescharacteristic
533C22C2×#45211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—44no—
541C4—44no—
542C4—44no—
543C4—44no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4#187812yes—
548C4#187812yes—
549C4#189712yes—
550C4C2×#42112yes—
551C4C2×C4×Q1611yescharacteristic
552C4C2×#44111yescharacteristic
553C4C2×#46211yescharacteristic
554C4C2×#46411yescharacteristic
555C2—21no—
556C2—21no—
557C2—22no—
558C2#923614yesdirect product factor
559C2C2×#187811yescharacteristic
560C2C2×#189711yescharacteristic
561C2C2×#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. #9236C2direct (N × H)
13. #3104C4semidirect
16. C2×#477C4semidirect
17. C2×#462C4semidirect
91. #477C2×C4semidirect
93. #462C2×C4semidirect
267. C2×C42Q16semidirect
435. C42C2×Q16semidirect
558. C2#9236direct (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×#22952
C2×#19942
C2×#15032
C2×#14872
#92362
C2×#13032
C2×#12762