Finite Group Databasebeta

← C2×#9169 #197117

Subgroups — C2×#9169 #197117

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#2906C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic, complemented
4C22×#333C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#9169C218yesmaximal, direct product factor
7C2×#1301C211yesmaximal, characteristic, complemented
8C2×#1292C211yesmaximal, characteristic, complemented
9C22×#131—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—22no—
13#3079—22no—
14#3079C414yes—
15#3079C2214yes—
16C2×#477—21no—
17C2×#475C412yes—
18C2×#476—21no—
19C2×#474C412yes—
20C2×#431C2211yescharacteristic
21C2×#428—21no—
22#2906—22no—
23#2906—22no—
24#2906—22no—
25#2906C2214yes—
26C2×#424—22no—
27C2×#424—21no—
28C2×#423—21no—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8—21no—
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#417—21no—
35#1549—22no—
36#1549C2214yescomplemented
37C2×#333—21no—
38C2×#333—21no—
39C2×#333—22no—
40C2×#333C2214yescomplemented
41C2×#332—22no—
42#1477—22no—
43#1477C2214yes—
44C2×Q8⋊C8—21no—
45C2×D4⋊C8—21no—
46#1301C2214yescomplemented
47#1292C2214yescomplemented
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C2×#131—24no—
51C2×#131—24no—
52C2×#131—44no—
53C2×#131—44no—
54C22×M16—22no—
55C22×M16—21no—
56C22×M16—21no—
57C23×C8—22no—
58C23×C8—21no—
59C23×C8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊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⋊Q8C2×C412yes—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2312yes—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41C2×C412yes—
77C4×#41D412yes—
78C4×#41D412yes—
79C4×#41C2312yes—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#477—24no—
93#477—24no—
94#476—24no—
95#476—24no—
96#475—24no—
97#475C2×C418yes—
98#474—24no—
99#474C2×C418yes—
100#431—21no—
101#431—21no—
102#431—21no—
103#431C2312yes—
104#428—24no—
105#428—24no—
106C2×#107—22no—
107C2×#106—22no—
108#424—24no—
109#424—28no—
110#424—24no—
111#424—28no—
112#423—24no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423—24no—
118#423C2×C412yes—
119#423D412yes—
120#423C2312yes—
121#423C2312yes—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—24no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—24no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8C2×C412yes—
138C2×C4⋊C8D412yes—
139C2×C4⋊C8D411yescharacteristic
140C2×C4⋊C8D411yescharacteristic
141C2×C4⋊C8D412yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144#417—24no—
145#417—24no—
146C2×#103—22no—
147C2×#102—22no—
148C2×C22⋊C8—21no—
149C2×C4×C8—21no—
150#333—24no—
151#333—24no—
152#332—28no—
153Q8⋊C8—24no—
154D4⋊C8—24no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41C22⋊C412yes—
160C2×#41SD1614yes—
161C2×#41Q1614yes—
162C2×#41C2×D412yes—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—84no—
170C22×Q8C22⋊C411yescharacteristic
171C22×D4—21no—
172C22×D4C22⋊C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180#131—48no—
181#131—416no—
182#131—48no—
183C2×M16—28no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—48no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—28no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—48no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8—41no—
207C22×C8—82no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—41no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8C22×C414yes—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C22⋊C412yes—
229C4×Q8C22⋊C412yes—
230C4×Q8C22×C412yes—
231C4×Q8C2×D412yes—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4C22⋊C412yes—
245C4×D4C22⋊C412yes—
246C4×D4C22×C412yes—
247C4×D4C2×D412yes—
248#117—21no—
249#117—21no—
250#117—22no—
251#117—22no—
252#117—22no—
253#117—22no—
254#117C22⋊C412yes—
255#117C2×D412yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42C2411yescharacteristic
294#107—28no—
295#106—28no—
296C4⋊C8—28no—
297C4⋊C8—28no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—21no—
301C4⋊C8—22no—
302C4⋊C8—21no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8C22×C414yes—
315C4⋊C8C2×D414yes—
316C4⋊C8C2×D412yes—
317C4⋊C8C2×D412yes—
318#103—28no—
319#102—28no—
320C22⋊C8—22no—
321C22⋊C8—41no—
322C4×C8—22no—
323C4×C8—41no—
324C24—21no—
325C24#12311yescharacteristic
326#41—22no—
327#41—22no—
328#41—24no—
329#41—24no—
330#41—24no—
331#41—24no—
332#41—24no—
333#41#10318yes—
334#41C2×SD1614yes—
335#41C2×Q1614yes—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8#10314yes—
349C2×Q8C2×C22⋊C412yes—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4#10314yes—
357C2×D4C2×C22⋊C412yes—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—82no—
378C22×C4—84no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4#12011yescharacteristic
386C22×C4#12011yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12311yescharacteristic
392C22×C4#12311yescharacteristic
393C22×C4#12311yescharacteristic
394C22×C4C2×SD1612yes—
395C22×C4C2×Q1612yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401M16—416no—
402M16—42no—
403M16—42no—
404M16—42no—
405M16—42no—
406M16—44no—
407M16—44no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—416no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—44no—
416C2×C8—42no—
417C2×C8—44no—
418C2×C8—42no—
419C2×C8—44no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—82no—
423C2×C8—82no—
424C2×C8—82no—
425C2×C8—82no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C2×C22⋊C412yes—
441C4⋊C4C2×C22⋊C412yes—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42C2×C22⋊C411yescharacteristic
455C42C2×C22⋊C411yescharacteristic
456C42C4×D411yescharacteristic
457C42C4×D411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42#12011yescharacteristic
461C42#12011yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42C23×C411yescharacteristic
467C42C22×D411yescharacteristic
468C42C22×D411yescharacteristic
469C42C22×D411yescharacteristic
470C23—22no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23#38611yescharacteristic
477C23#38611yescharacteristic
478C23#48312yes—
479C23#48412yes—
480C23C2×#12311yescharacteristic
481Q8—22no—
482Q8—24no—
483Q8—84no—
484Q8—84no—
485Q8—84no—
486Q8—84no—
487Q8C2×#10314yes—
488D4—24no—
489D4—24no—
490D4—28no—
491D4—22no—
492D4—24no—
493D4C2×#10314yes—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#38611yescharacteristic
526C2×C4#38611yescharacteristic
527C2×C4#38611yescharacteristic
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4C2×#10312yes—
532C2×C4C2×#10312yes—
533C2×C4#41712yes—
534C2×C4#41712yes—
535C2×C4#45012yes—
536C2×C4#45012yes—
537C2×C4#45112yes—
538C2×C4#45112yes—
539C2×C4#47112yes—
540C2×C4#47312yes—
541C2×C4#48312yes—
542C2×C4#48412yes—
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C22×SD1611yescharacteristic
550C2×C4C22×Q1611yescharacteristic
551C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
552C2×C4C2×#13711yescharacteristic
553C8—44no—
554C8—42no—
555C8—44no—
556C8—42no—
557C8—82no—
558C8—82no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—48no—
569C22#187212yes—
570C22#187212yes—
571C22#190312yes—
572C22#190812yes—
573C22C2×#38611yescharacteristic
574C22C2×#48311yescharacteristic
575C22C2×#48411yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—84no—
586C4#187212yes—
587C4#187212yes—
588C4#190812yes—
589C4C22×#10311yescharacteristic
590C4C2×#41711yescharacteristic
591C4C2×#45011yescharacteristic
592C4C2×#45111yescharacteristic
593C4C2×#47111yescharacteristic
594C4C2×#47311yescharacteristic
595C2—21no—
596C2—21no—
597C2—22no—
598C2—44no—
599C2#916914yesdirect product factor
600C2C2×#187211yescharacteristic
601C2C2×#190311yescharacteristic
602C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1549C2semidirect
4. C22×#333C2semidirect
6. #9169C2direct (N × H)
7. C2×#1301C2semidirect
8. C2×#1292C2semidirect
36. #1549C22semidirect
40. C2×#333C22semidirect
46. #1301C22semidirect
47. #1292C22semidirect
599. C2#9169direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#29062
C2×#15492
C22×#3332
C2×#14772
#91692
C2×#13012
C2×#12922