Finite Group Databasebeta

← #196970

Subgroups — #196970

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35325C211yesmaximal, characteristic
2C2×#2343C211yesmaximal, characteristic
3#20717C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic, complemented
5C2×#1506C211yesmaximal, characteristic, complemented
6#9154C212yesmaximal
7#9153C212yesmaximal, complemented
8#9144C212yesmaximal, complemented
9#9143C212yesmaximal
10#8017C211yesmaximal, characteristic, complemented
11#8013C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2855C2212yes—
17#2848C2212yes—
18#2628C412yescomplemented
19#2621C412yescomplemented
20#2611—22no—
21#2599—22no—
22#2583—21no—
23#2561—21no—
24#2484—21no—
25#2482—21no—
26#2343—24no—
27#2343—24no—
28#2323C414yescomplemented
29C2×#387—21no—
30C2×#387—21no—
31C2×#385—21no—
32#2283C2211yescharacteristic
33C2×#383C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38C2×#376C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1817C2212yes—
41#1798—22no—
42#1779C2212yes—
43#1750—22no—
44#1518—24no—
45#1506—24no—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60C2×C4⋊Q8C2×C411yescharacteristic, complemented
61C2×C4⋊Q8C2×C411yescharacteristic, complemented
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2×C414yes—
65C2×C22⋊Q8C2312yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yescomplemented
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C412yes—
74C2×#117C2×C412yes—
75C2×#117D412yes—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—22no—
89#432—22no—
90#431—22no—
91#431—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#400—22no—
100#399—22no—
101#398—22no—
102#398—22no—
103#397—22no—
104#397—22no—
105#395—22no—
106#393—22no—
107#391C2×C414yescomplemented
108#389—24no—
109#389C2×C414yescomplemented
110#388—24no—
111#388—24no—
112#387—28no—
113#387—28no—
114#387—24no—
115#387—24no—
116#385—28no—
117#385—22no—
118#385—22no—
119#385—24no—
120#383—21no—
121#383—22no—
122#383—22no—
123#382—22no—
124#382C2×C412yescomplemented
125#382D414yes—
126#381—22no—
127#381—21no—
128#381—22no—
129#381—22no—
130#380C2×C414yes—
131#380C2312yes—
132#379—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C412yescomplemented
136C4×C22⋊C4—28no—
137C4×C22⋊C4—21no—
138#376—22no—
139#376—21no—
140#376D414yes—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153#336—24no—
154#328—24no—
155#328—24no—
156#328—24no—
157#328—24no—
158C25C2×D411yescharacteristic
159C22×Q8—21no—
160C22×Q8C22×C412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—82no—
174C23×C4C22×C412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188#125—42no—
189#125—42no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—48no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×Q8—42no—
201#117—24no—
202#117—24no—
203#117—24no—
204#117—24no—
205#117—24no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—44no—
255C2×C42—82no—
256C2×C42C22×C412yes—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—48no—
273#97—42no—
274C4×C8—42no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95C2×D412yes—
294#95C2×D412yes—
295#95C2×D412yes—
296#95C2×D412yes—
297C24—21no—
298C24—21no—
299C24—24no—
300C24—22no—
301C24—22no—
302C24—42no—
303C24C2×C22⋊C412yes—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—82no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—84no—
358C22×C4—816no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—48no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—84no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—48no—
413C42—44no—
414C42—82no—
415C42—84no—
416C42C4×D412yes—
417C42C4×D412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12112yes—
421C42#12112yes—
422C42#12112yes—
423C42#12112yes—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—48no—
433C23—48no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23#41014yes—
440C23C22×C22⋊C411yescharacteristic
441C23C2×#12011yescharacteristic
442C23C2×#12311yescharacteristic
443C23#57712yes—
444C23#57812yes—
445Q8—82no—
446Q8—82no—
447Q8—84no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—816no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4#38612yes—
506C2×C4#38612yes—
507C2×C4#38612yes—
508C2×C4#38612yes—
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C8—82no—
514C8—82no—
515C22—24no—
516C22—24no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—28no—
522C22—24no—
523C22—44no—
524C22—44no—
525C22—48no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22C2×#38611yescharacteristic
530C22C2×#41012yes—
531C22#289812yes—
532C22#290112yes—
533C22#300911yescharacteristic
534C22#301111yescharacteristic
535C22#309712yes—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—84no—
552C4—82no—
553C4—82no—
554C2—21no—
555C2—21no—
556C2—22no—
557C2—44no—
558C2#1760914yes—
559C2#2091311yescharacteristic
560C2#2098011yescharacteristic
561C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1518C2semidirect
5. C2×#1506C2semidirect
7. #9153C2semidirect
8. #9144C2semidirect
10. #8017C2semidirect
11. #8013C2semidirect
18. #2628C4semidirect
19. #2621C4semidirect
28. #2323C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
67. C2×C4×Q8C2×C4semidirect
107. #391C2×C4semidirect
109. #389C2×C4semidirect
124. #382C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353252
C2×#23432
#207172
C2×#15182
C2×#15062
#91542
#91532
#91442
#91432
#80172
#80132