Finite Group Databasebeta

← C2×#9108 #196309

Subgroups — C2×#9108 #196309

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2C2×#2322C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4C22×#324C212yesmaximal, complemented
5#9108C218yesmaximal, direct product factor
6C2×#1282C212yesmaximal, complemented
7C22×#123C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10#2972—24no—
11#2972—22no—
12#2972C2214yes—
13C2×C8×D4—24no—
14C2×#431—21no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2322—22no—
19#2322—22no—
20#2322C414yes—
21#2322C2214yes—
22C2×#389—21no—
23C2×#388—21no—
24C2×#387—21no—
25C2×#386C411yescharacteristic
26C2×#385—21no—
27C2×#385C411yescharacteristic
28C2×#381C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1459—22no—
31#1459C2214yes—
32C2×#324—24no—
33C2×#324—24no—
34C2×#324—21no—
35C2×#324C2218yescomplemented
36C2×#323—21no—
37#1282C2218yescomplemented
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—22no—
45C23×C8—22no—
46C2×#123—22no—
47C2×#123—24no—
48C2×#123C818yescomplemented
49C2×#123C2×C414yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42D411yescharacteristic
79C8×D4—216no—
80C8×D4—216no—
81#431—24no—
82#431—24no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yes—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99C2×C4×C8—22no—
100#389—24no—
101#389—24no—
102#388—24no—
103#388—24no—
104#387—24no—
105#387—24no—
106#386—22no—
107#386C2×C414yes—
108#385—24no—
109#385—24no—
110#385—22no—
111#385C2×C414yes—
112#381—22no—
113#381C2×C414yes—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4C2312yes—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123#324—24no—
124#324—216no—
125#323—24no—
126C25C22⋊C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4C22⋊C412yes—
132C22×D4C2×C814yes—
133C22×D4M1614yes—
134C22×D4C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C412yes—
158C23×C4C2×C814yescomplemented
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—28no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—84no—
180C22×C8—84no—
181C22×C8—84no—
182#123—216no—
183#123—28no—
184#123C2×C8116yescomplemented
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C2×C814yescomplemented
234C2×C22⋊C4C22×C412yes—
235C2×C22⋊C4C22×C412yes—
236C2×C22⋊C4C22×C412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—82no—
245C2×C42C22⋊C412yes—
246C2×C42C2×D412yes—
247C4⋊C8—28no—
248C22⋊C8—28no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251C22⋊C8—24no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256C22⋊C8C2×D418yes—
257C4×C8—28no—
258#95—22no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C22⋊C814yes—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C412yes—
274C24C2×C22⋊C411yescharacteristic
275C24C4×D412yes—
276C24C4×D412yes—
277C24C22×C812yes—
278C24C2×M1612yes—
279C24#13114yes—
280C24C23×C411yescharacteristic
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—28no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4C22⋊C818yes—
290C2×D4C22×C814yes—
291C2×D4C2×M1614yes—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—82no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4C22⋊C814yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12012yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12311yescharacteristic
371C22×C4#12311yescharacteristic
372C22×C4#12312yes—
373C22×C4C22×C814yescomplemented
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4112yes—
378C2×C8—28no—
379C2×C8—48no—
380C2×C8—44no—
381C2×C8—84no—
382C2×C8—84no—
383C2×C8—88no—
384C2×C8—84no—
385C2×C8—84no—
386C2×C8—84no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—216no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—84no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—216no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—42no—
406C22⋊C4—84no—
407C22⋊C4C22×C814yescomplemented
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—82no—
413C42—82no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—48no—
430C23—41no—
431C23—42no—
432C23—48no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—81no—
437C23—81no—
438C23—84no—
439C23#38612yes—
440C23#38611yescharacteristic
441C23#38611yescharacteristic
442C23C2×C22⋊C812yes—
443C23C2×C22⋊C814yes—
444C23C2×C22⋊C812yes—
445C23#40812yes—
446C23#40812yes—
447C23C8×D414yes—
448C23C8×D414yes—
449C23#43514yes—
450C23#43514yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×C4×D412yes—
453C23C23×C811yescharacteristic
454C23C22×M1611yescharacteristic
455C23C2×#13112yes—
456D4—216no—
457D4—28no—
458D4—28no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4#38611yescharacteristic
517C2×C4#38611yescharacteristic
518C2×C4#38612yes—
519C2×C4C2×C22⋊C814yes—
520C2×C4C2×#12011yescharacteristic
521C2×C4C2×#12112yes—
522C2×C4C2×#12311yescharacteristic
523C8—48no—
524C8—84no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—28no—
536C22—28no—
537C22—24no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—48no—
549C22—82no—
550C22—84no—
551C22#186912yes—
552C22#186914yes—
553C22#186912yes—
554C22#191312yes—
555C22#191512yes—
556C22C2×#38611yescharacteristic
557C22C22×C22⋊C811yescharacteristic
558C22C2×#40811yescharacteristic
559C22C2×C8×D412yes—
560C22C2×#43512yes—
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—22no—
567C4—24no—
568C4—44no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—82no—
573C4—82no—
574C2—21no—
575C2—21no—
576C2—22no—
577C2—24no—
578C2—22no—
579C2—22no—
580C2—44no—
581C2#910814yesdirect product factor
582C2C2×#186911yescharacteristic
583C2C2×#191311yescharacteristic
584C2C2×#191511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#324C2semidirect
5. #9108C2direct (N × H)
6. C2×#1282C2semidirect
35. C2×#324C22semidirect
37. #1282C22semidirect
48. C2×#123C8semidirect
158. C23×C4C2×C8semidirect
184. #123C2×C8semidirect
233. C2×C22⋊C4C2×C8semidirect
373. C22×C4C22×C8semidirect
407. C22⋊C4C22×C8semidirect
581. C2#9108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29722
C2×#23222
C2×#14592
C22×#3242
#91082
C2×#12822