Finite Group Databasebeta

← C2×#10174 #236917

Subgroups — C2×#10174 #236917

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2C22×#500C211yesmaximal, characteristic, complemented
3C2×#3152C211yesmaximal, characteristic
4C2×#1966C211yesmaximal, characteristic, complemented
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic
7#10174C218yesmaximal, direct product factor
8C2×#1278C211yesmaximal, characteristic, complemented
9C22×#137—22no—
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13#3226—22no—
14#3226—22no—
15#3226—22no—
16#3226C2214yes—
17C2×#500—22no—
18C2×#500—22no—
19C2×#500—21no—
20C2×#500—22no—
21C2×#500—22no—
22C2×#500—22no—
23C2×#500C2214yescomplemented
24#3152—22no—
25#3152—22no—
26#3152—22no—
27#3152C2214yes—
28C2×#497—22no—
29C2×#494—21no—
30C2×#492—21no—
31C2×#486—21no—
32C2×#480—22no—
33C2×#474—22no—
34C2×#431C2211yescharacteristic
35C22×#107—22no—
36C2×#423C2211yescharacteristic
37C22×#101—22no—
38C2×C4×M16C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#1966—22no—
41#1966C2214yescomplemented
42#1546—22no—
43#1546C2214yescomplemented
44C2×#332—22no—
45#1492—22no—
46#1492C2214yes—
47C2×D4⋊C8—22no—
48#1278C2214yescomplemented
49C22×#41—21no—
50C23×Q8—21no—
51C23×Q8—21no—
52C2×#137—28no—
53C2×#137—28no—
54C2×#137—48no—
55C2×#137—48no—
56C22×Q16—24no—
57C22×SD16—24no—
58C22×M16—21no—
59C22×M16D412yes—
60C23×C8—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532D412yes—
68#532C2312yes—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8D412yes—
90C2×C4⋊Q8D412yes—
91C2×C4⋊Q8D412yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#127D411yescharacteristic
95C2×#124—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D412yes—
104C22×C42C2311yescharacteristic
105#500—28no—
106#500—28no—
107#500—24no—
108#500—28no—
109#500—24no—
110#500—28no—
111#497—28no—
112#497—28no—
113#494—24no—
114#494—24no—
115#492—24no—
116#492—24no—
117#486—24no—
118#486—24no—
119#480—28no—
120#480—28no—
121#474—28no—
122#474—28no—
123#431—21no—
124#431—21no—
125#431D412yes—
126#431C2312yes—
127C2×#107—24no—
128C2×#107—28no—
129C2×#107—28no—
130C2×#107—28no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423C2312yes—
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8D412yes—
138C2×#102—24no—
139C2×#101—28no—
140C2×#101—48no—
141C2×C22⋊C8—21no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16D412yes—
145C4×M16C2312yes—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8D412yes—
153C2×C4×C8D412yes—
154C2×C4×C8C2312yes—
155#332—28no—
156D4⋊C8—28no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—41no—
160C2×#41—41no—
161C2×#41—44no—
162C2×#41—44no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—41no—
184C23×C4—22no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190#137—416no—
191#137—432no—
192#137—416no—
193C2×Q16—216no—
194C2×Q16—416no—
195C2×SD16—216no—
196C2×SD16—416no—
197C2×M16—22no—
198C2×M16—24no—
199C2×M16—22no—
200C2×M16—41no—
201C2×M16—44no—
202C2×M16D818yes—
203C2×M16C2×D414yes—
204C22×C8—28no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—41no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—21no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—21no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—21no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—41no—
232C4⋊Q8—41no—
233C4⋊Q8—41no—
234C4⋊Q8—41no—
235C4⋊Q8—41no—
236C4⋊Q8—41no—
237C4⋊Q8—41no—
238C4⋊Q8—41no—
239C4⋊Q8—41no—
240C4⋊Q8—41no—
241C4⋊Q8—41no—
242C4⋊Q8—41no—
243C4⋊Q8—41no—
244C4⋊Q8—41no—
245C4⋊Q8C2×D414yes—
246C4⋊Q8C2×D412yes—
247#127—21no—
248#127—21no—
249#127—21no—
250#127C2×D412yes—
251#124—22no—
252#124—22no—
253#124—41no—
254#124—41no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—41no—
260#121—41no—
261#121—41no—
262#121—41no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—41no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4C2×D414yes—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42C2411yescharacteristic
311#107—232no—
312#107—216no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8C2×D414yes—
319#102—216no—
320#101—416no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323#97—22no—
324#97—22no—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8C2×D414yes—
333C24—41no—
334C24#12111yescharacteristic
335#41—44no—
336#41—44no—
337#41—48no—
338#41—84no—
339#41—82no—
340#41—82no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—84no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—84no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4#12011yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12011yescharacteristic
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4#12011yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12711yescharacteristic
411C22×C4#12711yescharacteristic
412C22×C4C2×D814yes—
413C22×C4AGL(1,ℤ/8)14yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4111yescharacteristic
417Q16—432no—
418SD16—432no—
419M16—24no—
420M16—28no—
421M16—44no—
422M16—82no—
423M16C2×D818yes—
424C2×C8—28no—
425C2×C8—28no—
426C2×C8—28no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—28no—
430C2×C8—24no—
431C2×C8—44no—
432C2×C8—42no—
433C2×C8—82no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C42#12011yescharacteristic
459C42#12011yescharacteristic
460C42#12011yescharacteristic
461C42#12011yescharacteristic
462C42#12011yescharacteristic
463C42#12011yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42#12111yescharacteristic
467C42#12111yescharacteristic
468C42#12711yescharacteristic
469C42#12711yescharacteristic
470C42C22×D411yescharacteristic
471C42C22×D411yescharacteristic
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C23—22no—
475C23—22no—
476C23—42no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23#39211yescharacteristic
481C23#39211yescharacteristic
482C23#45914yes—
483C23C2×#12111yescharacteristic
484Q8—48no—
485Q8—44no—
486Q8—82no—
487Q8—82no—
488Q8—82no—
489Q8—82no—
490Q8—84no—
491Q8—84no—
492Q8—84no—
493Q8—84no—
494Q8—84no—
495Q8—82no—
496D4—44no—
497D4—44no—
498D4—44no—
499D4—84no—
500D4—84no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#39211yescharacteristic
528C2×C4#39211yescharacteristic
529C2×C4#39211yescharacteristic
530C2×C4#39211yescharacteristic
531C2×C4#39211yescharacteristic
532C2×C4#39211yescharacteristic
533C2×C4#44714yes—
534C2×C4#44714yes—
535C2×C4#45614yes—
536C2×C4#45914yes—
537C2×C4#47314yes—
538C2×C4#49312yes—
539C2×C4#49312yes—
540C2×C4#49612yes—
541C2×C4#49612yes—
542C2×C4C2×#12011yescharacteristic
543C2×C4C2×#12011yescharacteristic
544C2×C4C2×#12011yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12711yescharacteristic
548C2×C4C22×D812yes—
549C2×C4C2×AGL(1,ℤ/8)12yes—
550C8—28no—
551C8—28no—
552C8—44no—
553C8—82no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—28no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—164no—
563C22#200512yes—
564C22#201112yes—
565C22#201112yes—
566C22#202012yes—
567C22C2×#39211yescharacteristic
568C22C2×#45912yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4#200512yes—
577C4#201112yes—
578C4#201112yes—
579C4#202012yes—
580C4C2×#44712yes—
581C4C2×#45612yes—
582C4C2×#47312yes—
583C4C2×#49311yescharacteristic
584C4C2×#49611yescharacteristic
585C2—22no—
586C2—22no—
587C2—162no—
588C2#1017414yesdirect product factor
589C2C2×#200511yescharacteristic
590C2C2×#201111yescharacteristic
591C2C2×#202011yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
2. C22×#500C2semidirect
4. C2×#1966C2semidirect
5. C2×#1546C2semidirect
7. #10174C2direct (N × H)
8. C2×#1278C2semidirect
23. C2×#500C22semidirect
41. #1966C22semidirect
43. #1546C22semidirect
48. #1278C22semidirect
588. C2#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32262
C22×#5002
C2×#31522
C2×#19662
C2×#15462
C2×#14922
#101742
C2×#12782