Finite Group Databasebeta

← C2×#9201 #197500

Subgroups — C2×#9201 #197500

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3C2×#1520C212yesmaximal, complemented
4C22×#323C211yesmaximal, characteristic
5#9201C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C23×C22⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2626—22no—
11#2626—22no—
12#2626C414yescomplemented
13#2626C2214yes—
14C2×#399C412yescomplemented
15C2×#396—22no—
16C2×#393C412yescomplemented
17#2312—24no—
18#2312—22no—
19#2312C2214yes—
20C2×#388—24no—
21C2×#385—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×#379—21no—
25C2×#379—21no—
26C2×C4×C22⋊C4—21no—
27#1520—24no—
28#1520C2218yescomplemented
29C2×#328—24no—
30C2×#323—22no—
31C2×#323—22no—
32C2×#323C2214yes—
33#1289C2218yescomplemented
34C26D411yescharacteristic
35C23×Q8C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4D412yes—
66C22×C22⋊C4D412yes—
67C22×C22⋊C4D412yes—
68C22×C22⋊C4C2312yes—
69C22×C42—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2314yes—
76#399—24no—
77#399C2×C418yescomplemented
78#396—28no—
79#396—28no—
80#393—24no—
81#393C2×C418yescomplemented
82#388—216no—
83#388—216no—
84#385—24no—
85#385—24no—
86#380—22no—
87#380—22no—
88#380—21no—
89#380—21no—
90#380C2×C412yes—
91#380D414yes—
92#380C2312yes—
93#380C2314yes—
94#379—24no—
95#379—24no—
96#379—24no—
97#379—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2×C412yes—
103C2×#95D414yes—
104#328—216no—
105#323—28no—
106C25—21no—
107C25—21no—
108C25—21no—
109C25—21no—
110C25—24no—
111C25—24no—
112C25—82no—
113C25—82no—
114C25C22⋊C412yes—
115C25C22⋊C412yes—
116C25C2×D412yes—
117C25C2×D411yescharacteristic
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—82no—
122C22×Q8C22×C412yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—84no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—84no—
204C2×C22⋊C4—84no—
205C2×C22⋊C4—84no—
206C2×C22⋊C4—84no—
207C2×C22⋊C4C22⋊C414yes—
208C2×C22⋊C4C22⋊C414yes—
209C2×C22⋊C4C22×C414yes—
210C2×C22⋊C4C2×D414yes—
211C2×C42—28no—
212C2×C42—84no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216C22⋊C8—42no—
217#95—28no—
218#95—28no—
219#95—22no—
220#95—24no—
221#95C22×C414yes—
222#95C2×D418yes—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—24no—
227C24—22no—
228C24—24no—
229C24—24no—
230C24—22no—
231C24—22no—
232C24—24no—
233C24—24no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24—44no—
238C24—44no—
239C24—44no—
240C24—44no—
241C24—44no—
242C24—44no—
243C24—44no—
244C24—44no—
245C24—82no—
246C24—82no—
247C24—82no—
248C24—82no—
249C24—82no—
250C24—82no—
251C24—82no—
252C24—82no—
253C24—82no—
254C24—82no—
255C24—816no—
256C24—816no—
257C24C2×C22⋊C411yescharacteristic
258C24C2×C22⋊C414yes—
259C24C2×C22⋊C411yescharacteristic
260C24#12011yescharacteristic
261C24#12011yescharacteristic
262C24#12311yescharacteristic
263C24#12311yescharacteristic
264C24#13514yes—
265C24#13714yes—
266C24C22×D411yescharacteristic
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—84no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—84no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—81no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4C2×C22⋊C414yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4C4×D412yes—
319C22×C4#12012yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12312yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4112yes—
329C2×C8—44no—
330C2×C8—84no—
331C2×C8—84no—
332C2×C8—84no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—82no—
341C4⋊C4—88no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—88no—
359C42—88no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—48no—
378C23—48no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—416no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—88no—
401C23—816no—
402C23—816no—
403C23—88no—
404C23—88no—
405C23—88no—
406C23—88no—
407C23—816no—
408C23—816no—
409C23#38611yescharacteristic
410C23#38611yescharacteristic
411C23#38611yescharacteristic
412C23#38611yescharacteristic
413C23#41612yes—
414C23#41612yes—
415C23#41912yes—
416C23#41912yes—
417C23#45214yes—
418C23#45214yes—
419C23#48514yes—
420C23#48514yes—
421C23C22×C22⋊C411yescharacteristic
422C23C2×#12011yescharacteristic
423C23C2×#12311yescharacteristic
424C23C2×#13512yes—
425C23C2×#13712yes—
426Q8—84no—
427Q8—84no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—88no—
463C2×C4—816no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4C2×C4×D412yes—
471C2×C4C2×#12112yes—
472C8—84no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—28no—
487C22—24no—
488C22—28no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—44no—
494C22—44no—
495C22—48no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—84no—
500C22—88no—
501C22—88no—
502C22—88no—
503C22—88no—
504C22—88no—
505C22—88no—
506C22#187312yes—
507C22#187312yes—
508C22#187312yes—
509C22#187312yes—
510C22#191112yes—
511C22#191312yes—
512C22C2×#38611yescharacteristic
513C22C2×#41611yescharacteristic
514C22C2×#41911yescharacteristic
515C22C2×#45212yes—
516C22C2×#48512yes—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—84no—
523C4—88no—
524C4—82no—
525C4—82no—
526C2—21no—
527C2—22no—
528C2—21no—
529C2—22no—
530C2—22no—
531C2—22no—
532C2—22no—
533C2—84no—
534C2#920114yesdirect product factor
535C2C2×#187311yescharacteristic
536C2C2×#191111yescharacteristic
537C2C2×#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. #9201C2direct (N × H)
6. C2×#1289C2semidirect
12. #2626C4semidirect
14. C2×#399C4semidirect
16. C2×#393C4semidirect
28. #1520C22semidirect
33. #1289C22semidirect
77. #399C2×C4semidirect
81. #393C2×C4semidirect
534. C2#9201direct (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×#23122
C2×#15202
C22×#3232
#92012
C2×#12892