Finite Group Databasebeta

← C2×#9184 #197314

Subgroups — C2×#9184 #197314

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2C2×#2638C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4C2×#1508C212yesmaximal, complemented
5#9184C218yesmaximal, direct product factor
6C2×#1284C212yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2975—24no—
10#2975—22no—
11#2975C2214yes—
12C2×#436—24no—
13C2×#431—21no—
14C22×#99—21no—
15C2×#407—21no—
16C2×#407C2212yes—
17#2638—22no—
18#2638—22no—
19#2638C414yescomplemented
20#2638C2214yes—
21C2×#398C411yescharacteristic, complemented
22C2×#397—21no—
23C2×#397C411yescharacteristic, complemented
24C2×#395C411yescharacteristic, complemented
25C2×#394—21no—
26C2×#393C411yescharacteristic, complemented
27C2×#387—21no—
28C2×#385—21no—
29C2×#381C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31C4×#99—22no—
32C4×#99C2214yes—
33C2×#342—21no—
34#1508—24no—
35#1508C2218yescomplemented
36C2×#328—24no—
37#1284C2218yescomplemented
38C23×Q8C2×C411yescharacteristic
39C23×D4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C22×M16—22no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×C4×D4C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42—22no—
73C22×C42D411yescharacteristic
74#436—216no—
75#436—216no—
76#431—24no—
77#431—24no—
78C2×C4⋊C8—22no—
79C2×#99—28no—
80C2×#99—24no—
81#407—24no—
82#407—24no—
83#407—22no—
84#407—22no—
85#407D414yes—
86#407C2314yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8D414yes—
90C2×C4×C8—22no—
91#398—22no—
92#398C2×C414yescomplemented
93#397—22no—
94#397—24no—
95#397—24no—
96#397C2×C414yescomplemented
97#395—22no—
98#395C2×C414yescomplemented
99#394—24no—
100#394—24no—
101#393—22no—
102#393C2×C414yescomplemented
103#387—24no—
104#387—24no—
105#385—24no—
106#385—24no—
107#381—21no—
108#381—21no—
109#381C2×C412yes—
110#381C2312yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2314yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123#342—24no—
124#328—216no—
125C25C22⋊C411yescharacteristic
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—82no—
131C22×Q8C22×C412yes—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—42no—
136C22×D4C22⋊C412yes—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—28no—
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—84no—
168C2×M16—84no—
169C22×C8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22×C412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—84no—
234C2×C42C22⋊C412yes—
235C2×C42C2×D412yes—
236C4⋊C8—28no—
237#99—216no—
238C22⋊C8—28no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8C2×D418yes—
243C4×C8—28no—
244#95—22no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—24no—
253#95—42no—
254#95C22×C412yes—
255#95C22×C412yes—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—22no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C412yes—
268C24C2×C22⋊C411yescharacteristic
269C24#12112yes—
270C24#12112yes—
271C24C22×D411yescharacteristic
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—84no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—84no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—48no—
285C2×D4—48no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—82no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4#10414yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12012yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12311yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4#12312yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4112yes—
359M16—84no—
360M16—84no—
361M16—88no—
362C2×C8—28no—
363C2×C8—48no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—84no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—48no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—84no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—84no—
392C42—84no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—48no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—81no—
420C23—81no—
421C23—84no—
422C23—84no—
423C23—84no—
424C23#38612yes—
425C23#38611yescharacteristic
426C23#38611yescharacteristic
427C23C2×#10412yes—
428C23#42112yes—
429C23C22×C22⋊C411yescharacteristic
430C23C2×#12112yes—
431Q8—84no—
432Q8—84no—
433D4—48no—
434D4—48no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—816no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#38611yescharacteristic
492C2×C4#38611yescharacteristic
493C2×C4#38612yes—
494C2×C4C2×#10414yes—
495C2×C4C2×C4×D412yes—
496C2×C4C2×#12011yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C8—48no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—48no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—84no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22#187112yes—
525C22#187112yes—
526C22#190912yes—
527C22#191312yes—
528C22C2×#38611yescharacteristic
529C22C22×#10411yescharacteristic
530C22C2×#42111yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—82no—
542C4—88no—
543C4—82no—
544C4—82no—
545C2—21no—
546C2—21no—
547C2—22no—
548C2—42no—
549C2—42no—
550C2—82no—
551C2#918414yesdirect product factor
552C2C2×#187111yescharacteristic
553C2C2×#190911yescharacteristic
554C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. C2×#1508C2semidirect
5. #9184C2direct (N × H)
6. C2×#1284C2semidirect
19. #2638C4semidirect
21. C2×#398C4semidirect
23. C2×#397C4semidirect
24. C2×#395C4semidirect
26. C2×#393C4semidirect
35. #1508C22semidirect
37. #1284C22semidirect
92. #398C2×C4semidirect
96. #397C2×C4semidirect
98. #395C2×C4semidirect
102. #393C2×C4semidirect
551. C2#9184direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29752
C2×#26382
C2×C4×#992
C2×#15082
#91842
C2×#12842