Finite Group Databasebeta

← C2×#9175 #197216

Subgroups — C2×#9175 #197216

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#9175C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12#3077—22no—
13#3077—22no—
14#3077C414yescomplemented
15#3077C2214yes—
16C2×#478—22no—
17C2×#476C412yescomplemented
18C2×#474C412yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#428—21no—
22C2×#423C2211yescharacteristic
23C2×#418—21no—
24#2349—22no—
25#2349—22no—
26#2349—22no—
27#2349C2214yes—
28C2×#389—21no—
29C2×#385—22no—
30C2×#382C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C22×#95—21no—
34#1518—22no—
35#1518C2214yescomplemented
36#1503—22no—
37#1503C2214yescomplemented
38C2×#328—21no—
39C2×#328—21no—
40C2×#328—22no—
41#1478—22no—
42#1478C2214yes—
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1286C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C22×M16—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yescomplemented
57#533C2312yes—
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—44no—
74C2×C4×D4—44no—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4D411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#478—28no—
91#478—28no—
92#476—24no—
93#476C2×C418yescomplemented
94#474—24no—
95#474C2×C418yescomplemented
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2312yes—
103#431C2312yes—
104#428—24no—
105#428—24no—
106C2×#107—22no—
107C2×#106—22no—
108#423—21no—
109#423—21no—
110#423C2×C412yescomplemented
111#423C2312yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8C2×C412yescomplemented
116#418—24no—
117#418—24no—
118C2×#102—22no—
119C2×#102—22no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124#389—24no—
125#389—24no—
126#385—28no—
127#385—28no—
128#382—21no—
129#382—21no—
130#382D412yes—
131#382C2312yes—
132#380—24no—
133#380—24no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D412yes—
137C4×C4⋊C4C2312yes—
138C2×#95—24no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—24no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144C2×#95D412yes—
145#328—24no—
146#328—24no—
147#328—28no—
148D4⋊C8—28no—
149C25—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C22⋊C412yes—
158C22×D4C22⋊C412yes—
159C22×D4SD1614yes—
160C22×D4Q1614yes—
161C22×D4C2×D412yes—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—28no—
173C23×C4—48no—
174C23×C4—48no—
175C23×C4—84no—
176C23×C4—84no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—42no—
183C2×M16—42no—
184C22×C8—22no—
185C22×C8—41no—
186C22×C8—41no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yescomplemented
194C4⋊Q8C22×C412yescomplemented
195#125—22no—
196#125—22no—
197#125—41no—
198#125—41no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—48no—
228C4×D4—416no—
229C4×D4—48no—
230C4×D4C22⋊C414yes—
231C4×D4C22⋊C414yes—
232C4×D4C22×C414yes—
233C4×D4C2×D414yes—
234#117—22no—
235#117—22no—
236#117—41no—
237#117—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—84no—
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—41no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—41no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42C2411yescharacteristic
285#107—28no—
286#106—28no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294C4⋊C8C22×C414yescomplemented
295#102—28no—
296#102—28no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C4×C8—22no—
302C4×C8—22no—
303C4×C8—41no—
304C4×C8—41no—
305#95—28no—
306#95—21no—
307#95—22no—
308#95—21no—
309#95—22no—
310#95—22no—
311#95C2×D412yes—
312#95C2×D412yes—
313#95C2×D414yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—44no—
320C24—44no—
321C24#12011yescharacteristic
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4#10318yes—
342C2×D4#10318yes—
343C2×D4C2×C22⋊C414yes—
344C2×D4C2×SD1614yes—
345C2×D4C2×Q1614yes—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—416no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—416no—
367C22×C4—44no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—88no—
379C22×C4—832no—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4#12011yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12311yescharacteristic
394C22×C4#12311yescharacteristic
395C22×C4C2×SD1612yes—
396C22×C4C2×Q1612yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402M16—42no—
403M16—42no—
404M16—44no—
405C2×C8—24no—
406C2×C8—44no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—82no—
410C2×C8—82no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—88no—
432C4⋊C4C2×C22⋊C414yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—416no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42C2×C22⋊C411yescharacteristic
446C42C2×C22⋊C411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42#12111yescharacteristic
452C42#12111yescharacteristic
453C42#12111yescharacteristic
454C42#12111yescharacteristic
455C42#12311yescharacteristic
456C42#12311yescharacteristic
457C42C23×C411yescharacteristic
458C42C22×D411yescharacteristic
459C42C2×#4111yescharacteristic
460C42C2×#4111yescharacteristic
461C23—22no—
462C23—22no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—416no—
472C23—44no—
473C23—416no—
474C23—44no—
475C23—44no—
476C23#38611yescharacteristic
477C23#38611yescharacteristic
478C23#45012yes—
479C23#45112yes—
480C23C2×#12011yescharacteristic
481Q8—82no—
482Q8—82no—
483Q8—82no—
484Q8—82no—
485D4—24no—
486D4—24no—
487D4—24no—
488D4—28no—
489D4—28no—
490D4C2×#10318yes—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—832no—
524C2×C4—84no—
525C2×C4—88no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4#38611yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4C2×#10312yes—
535C2×C4C2×#10312yes—
536C2×C4#41712yes—
537C2×C4#41712yes—
538C2×C4#45012yes—
539C2×C4#45112yes—
540C2×C4#48312yes—
541C2×C4#48312yes—
542C2×C4#48412yes—
543C2×C4#48412yes—
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C22×SD1611yescharacteristic
551C2×C4C22×Q1611yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C8—44no—
555C8—82no—
556C8—82no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—24no—
562C22—24no—
563C22—28no—
564C22—48no—
565C22—416no—
566C22—48no—
567C22—44no—
568C22—44no—
569C22—416no—
570C22#187212yes—
571C22#187212yes—
572C22#190412yes—
573C22#190612yes—
574C22C2×#38611yescharacteristic
575C22C2×#45011yescharacteristic
576C22C2×#45111yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—84no—
589C4—82no—
590C4—82no—
591C4#187212yes—
592C4#187212yes—
593C4C22×#10311yescharacteristic
594C4C2×#41711yescharacteristic
595C4C2×#48311yescharacteristic
596C4C2×#48411yescharacteristic
597C2—21no—
598C2—22no—
599C2—21no—
600C2—48no—
601C2#917514yesdirect product factor
602C2C2×#187211yescharacteristic
603C2C2×#190411yescharacteristic
604C2C2×#190611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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×#1518C2semidirect
4. C2×#1503C2semidirect
6. #9175C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1286C2semidirect
14. #3077C4semidirect
17. C2×#476C4semidirect
18. C2×#474C4semidirect
35. #1518C22semidirect
37. #1503C22semidirect
44. #1303C22semidirect
45. #1286C22semidirect
56. #533C2×C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
93. #476C2×C4semidirect
95. #474C2×C4semidirect
110. #423C2×C4semidirect
115. C2×C4⋊C8C2×C4semidirect
193. C4⋊Q8C22×C4semidirect
194. C4⋊Q8C22×C4semidirect
294. C4⋊C8C22×C4semidirect
601. C2#9175direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#23492
C2×#15182
C2×#15032
C2×#14782
#91752
C2×#13032
C2×#12862