Finite Group Databasebeta

← C2×#9199 #197494

Subgroups — C2×#9199 #197494

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3C2×#1520C212yesmaximal, complemented
4C22×#323C211yesmaximal, characteristic
5#9199C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C23×C22⋊C4—21no—
9C23×C22⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2626—22no—
12#2626—22no—
13#2626C414yescomplemented
14#2626C2214yes—
15C2×#399C412yescomplemented
16C2×#396—22no—
17C2×#393C412yescomplemented
18#2315—24no—
19#2315—22no—
20#2315C2214yes—
21C2×#385—24no—
22C2×#385—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×#379—21no—
27#1520—24no—
28#1520C2218yescomplemented
29C2×#328—24no—
30C2×#323—22no—
31C2×#323—22no—
32C2×#323C2214yes—
33#1289C2218yescomplemented
34C26D411yescharacteristic
35C23×Q8C2×C411yescharacteristic, complemented
36C24×C4—22no—
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8C2×C412yescomplemented
47C2×C22⋊Q8C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic, complemented
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—44no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4D412yes—
71C22×C22⋊C4D412yes—
72C22×C22⋊C4D412yes—
73C22×C22⋊C4C2312yes—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80#399—24no—
81#399C2×C418yescomplemented
82#396—28no—
83#396—28no—
84#393—24no—
85#393C2×C418yescomplemented
86#385—216no—
87#385—24no—
88#385—24no—
89#385—216no—
90#380—24no—
91#380—22no—
92#380—22no—
93#380—24no—
94#380—21no—
95#380—21no—
96#380C2×C412yescomplemented
97#380D414yes—
98#380C2312yes—
99#380C2314yes—
100#379—24no—
101#379—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2×C412yescomplemented
105C2×#95D414yes—
106#328—216no—
107#323—28no—
108C25—21no—
109C25—21no—
110C25—21no—
111C25—21no—
112C25—22no—
113C25—22no—
114C25—82no—
115C25—82no—
116C25C22⋊C412yes—
117C25C22⋊C412yes—
118C25SD1614yes—
119C25Q1614yes—
120C25C2×D412yes—
121C25C2×D411yescharacteristic
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—82no—
126C22×Q8C22×C412yescomplemented
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—24no—
135C23×C4—28no—
136C23×C4—48no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—84no—
140C23×C4—84no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—84no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4—84no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4C22×C412yescomplemented
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—416no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—84no—
218C2×C22⋊C4—84no—
219C2×C22⋊C4C22⋊C414yes—
220C2×C22⋊C4C22⋊C414yes—
221C2×C22⋊C4C22×C414yes—
222C2×C22⋊C4C2×D414yes—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226C22⋊C8—42no—
227#95—28no—
228#95—28no—
229#95—22no—
230#95—24no—
231#95C22×C414yescomplemented
232#95C2×D418yes—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—24no—
237C24—24no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—42no—
252C24—42no—
253C24—44no—
254C24—44no—
255C24—44no—
256C24—44no—
257C24—82no—
258C24—82no—
259C24—82no—
260C24—82no—
261C24—88no—
262C24—88no—
263C24—82no—
264C24—82no—
265C24—82no—
266C24—82no—
267C24—82no—
268C24—82no—
269C24—88no—
270C24—88no—
271C24#10318yes—
272C24#10318yes—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C414yes—
275C24C2×C22⋊C411yescharacteristic
276C24#12011yescharacteristic
277C24#12011yescharacteristic
278C24#12311yescharacteristic
279C24#12311yescharacteristic
280C24C2×SD1614yes—
281C24C2×SD1612yes—
282C24C2×Q1614yes—
283C24C2×Q1612yes—
284C24AGL(1,ℤ/8)12yes—
285C24#13712yes—
286C24C22×D411yescharacteristic
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—84no—
290C2×Q8—82no—
291C2×Q8—82no—
292C2×Q8—84no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—416no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—82no—
325C22×C4—81no—
326C22×C4—82no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4—832no—
336C22×C4C2×C22⋊C414yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12012yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4112yes—
353C2×C8—44no—
354C2×C8—84no—
355C2×C8—84no—
356C2×C8—84no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—416no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—28no—
391C23—28no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—28no—
399C23—28no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—416no—
415C23—44no—
416C23—44no—
417C23—416no—
418C23—44no—
419C23—44no—
420C23—416no—
421C23—44no—
422C23—44no—
423C23—82no—
424C23—82no—
425C23—88no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—88no—
431C23—88no—
432C23—88no—
433C23—84no—
434C23—84no—
435C23—84no—
436C23—84no—
437C23—88no—
438C23—88no—
439C23—88no—
440C23—88no—
441C23—88no—
442C23#38611yescharacteristic
443C23#38611yescharacteristic
444C23#38611yescharacteristic
445C23#38611yescharacteristic
446C23C2×#10312yes—
447C23C2×#10318yes—
448C23C2×#10312yes—
449C23#41712yes—
450C23#41712yes—
451C23#45012yes—
452C23#45012yes—
453C23#45112yes—
454C23#45112yes—
455C23#48312yes—
456C23#48312yes—
457C23#48412yes—
458C23#48412yes—
459C23C22×C22⋊C411yescharacteristic
460C23C2×#12011yescharacteristic
461C23C2×#12311yescharacteristic
462C23C22×SD1611yescharacteristic
463C23C22×Q1611yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466Q8—84no—
467Q8—84no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—88no—
503C2×C4—82no—
504C2×C4—816no—
505C2×C4—82no—
506C2×C4—832no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#38612yes—
510C2×C4#38612yes—
511C2×C4C2×C4×D412yes—
512C2×C4C2×#12112yes—
513C8—84no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—28no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—48no—
544C22—48no—
545C22—416no—
546C22—48no—
547C22—44no—
548C22—44no—
549C22—416no—
550C22—84no—
551C22—88no—
552C22—84no—
553C22—84no—
554C22—88no—
555C22#187212yes—
556C22#187212yes—
557C22#187212yes—
558C22#187212yes—
559C22#191112yes—
560C22#191312yes—
561C22C2×#38611yescharacteristic
562C22C22×#10311yescharacteristic
563C22C2×#41711yescharacteristic
564C22C2×#45011yescharacteristic
565C22C2×#45111yescharacteristic
566C22C2×#48311yescharacteristic
567C22C2×#48411yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—44no—
573C4—84no—
574C4—88no—
575C4—82no—
576C4—82no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—22no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2—22no—
585C2—22no—
586C2—48no—
587C2#919914yesdirect product factor
588C2C2×#187211yescharacteristic
589C2C2×#191111yescharacteristic
590C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
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
3. C2×#1520C2semidirect
5. #9199C2direct (N × H)
6. C2×#1289C2semidirect
13. #2626C4semidirect
15. C2×#399C4semidirect
17. C2×#393C4semidirect
28. #1520C22semidirect
33. #1289C22semidirect
35. C23×Q8C2×C4semidirect
46. C2×C22⋊Q8C2×C4semidirect
53. C22×C4⋊C4C2×C4semidirect
81. #399C2×C4semidirect
85. #393C2×C4semidirect
96. #380C2×C4semidirect
104. C2×#95C2×C4semidirect
126. C22×Q8C22×C4semidirect
183. C2×C4⋊C4C22×C4semidirect
231. #95C22×C4semidirect
587. C2#9199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23152
C2×#15202
C22×#3232
#91992
C2×#12892