Finite Group Databasebeta

← #196968

Subgroups — #196968

Order 512.

2340 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35305C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic
3#20581C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic, complemented
5C2×#1508C211yesmaximal, characteristic, complemented
6#9154C214yesmaximal, complemented
7#9143C214yesmaximal, complemented
8#8016C211yesmaximal, characteristic, complemented
9#8012C211yesmaximal, characteristic, complemented
10C23×#41C2211yescharacteristic
11C2×#536C412yescomplemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×D4—21no—
14C2×#408—22no—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2848C2214yes—
18#2616C414yescomplemented
19#2435—21no—
20#2434—21no—
21#2432—22no—
22#2406—22no—
23#2405—22no—
24#2339—24no—
25#2339—24no—
26#2311C412yescomplemented
27C2×#391—21no—
28C2×#386—22no—
29#2280C2211yescharacteristic
30C2×#381C2212yes—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2212yes—
34#1780—22no—
35#1779C2214yes—
36#1520—24no—
37#1508—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—48no—
49C22×#41C2×C412yes—
50C22×#41C2×C412yes—
51C22×#41D412yes—
52C22×#41D412yes—
53C23×Q8C2×C411yescharacteristic, complemented
54C23×Q8D411yescharacteristic
55C23×D4D411yescharacteristic
56C23×D4D412yes—
57C24×C4—22no—
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C22×M16—21no—
61C23×C8—21no—
62#536—48no—
63C2×C4⋊Q8—22no—
64C2×#124—22no—
65C2×#123—24no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8C2×C412yescomplemented
73C2×C22⋊Q8C2×C418yescomplemented
74C2×C22⋊Q8C2312yes—
75C2×C4×D4—28no—
76C2×C4×D4—48no—
77C2×#117C2×C412yescomplemented
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic, complemented
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—21no—
86#408—48no—
87#407—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—28no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96#395—24no—
97#394—28no—
98#394—24no—
99#391—28no—
100#391—22no—
101#390—22no—
102#386—216no—
103#386—24no—
104#385—24no—
105#385—28no—
106#381—21no—
107#381—24no—
108#381—24no—
109#380—22no—
110#380C2×C418yescomplemented
111#380C2×C412yescomplemented
112#380D414yes—
113#380C2312yes—
114#379—24no—
115#379—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4D414yes—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95C2311yescharacteristic
125C2×#95C2312yes—
126C2×#95C2311yescharacteristic
127#328—24no—
128#328—24no—
129#328—28no—
130C25C2×D412yes—
131C25C2×D411yescharacteristic
132C2×#41—28no—
133C2×#41—28no—
134C2×#41—24no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—24no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—44no—
145C2×#41—44no—
146C2×#41—416no—
147C2×#41—416no—
148C2×#41—416no—
149C2×#41—416no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8C22⋊C412yes—
158C22×Q8C22⋊C412yes—
159C22×Q8C22×C412yes—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—48no—
166C22×D4C22⋊C412yes—
167C22×D4C22⋊C414yes—
168C22×D4C22⋊C414yes—
169C22×D4C22⋊C412yes—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—28no—
179C23×C4—21no—
180C23×C4—44no—
181C23×C4—48no—
182C23×C4—48no—
183C23×C4—48no—
184C23×C4—84no—
185C23×C4C22×C414yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D414yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C2×M16—44no—
199C2×M16—42no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—44no—
203C22×C8—42no—
204C4⋊Q8—48no—
205#124—48no—
206#123—416no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C22⋊Q8—416no—
218C4×D4—432no—
219#117—44no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—82no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—416no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257#95—28no—
258#95—28no—
259#95—22no—
260#95—28no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95C2×D412yes—
266#95C2×D414yes—
267#95C2×D412yes—
268C24—22no—
269C24—22no—
270C24—28no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—44no—
276C24—42no—
277C24—42no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C414yes—
281C24#12012yes—
282C24#12312yes—
283C24C22×D411yescharacteristic
284#41—416no—
285#41—416no—
286#41—416no—
287#41—416no—
288#41—416no—
289#41—416no—
290#41—416no—
291#41—416no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—44no—
302C2×Q8—42no—
303C2×Q8—44no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×D4—216no—
307C2×D4—216no—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4#9918yes—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—416no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—82no—
367C22×C4—88no—
368C22×C4—832no—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12012yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4#12312yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388M16—84no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—84no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—416no—
412C22⋊C4—82no—
413C22⋊C4—88no—
414C22⋊C4—84no—
415C42—48no—
416C42—44no—
417C23—22no—
418C23—22no—
419C23—28no—
420C23—28no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—28no—
427C23—28no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—416no—
439C23—416no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—84no—
446C23—84no—
447C23#38612yes—
448C23#38612yes—
449C23C2×#9918yes—
450C23C22×C22⋊C411yescharacteristic
451C23C2×#12011yescharacteristic
452C23C2×#12311yescharacteristic
453C23#57712yes—
454C23#57812yes—
455Q8—48no—
456Q8—48no—
457Q8—82no—
458Q8—82no—
459Q8—84no—
460D4—216no—
461D4—216no—
462D4—416no—
463D4—416no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
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—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—832no—
508C2×C4—84no—
509C2×C4—88no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4#38612yes—
513C2×C4#38612yes—
514C2×C4C2×#9914yes—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C8—82no—
520C8—82no—
521C22—24no—
522C22—24no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—216no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—28no—
535C22—48no—
536C22—48no—
537C22—416no—
538C22—44no—
539C22—48no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22C2×#38611yescharacteristic
547C22C22×#9912yes—
548C22#289812yes—
549C22#290112yes—
550C22#300911yescharacteristic
551C22#301111yescharacteristic
552C22#309712yes—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4#189814yes—
566C2—21no—
567C2—21no—
568C2—22no—
569C2—48no—
570C2—42no—
571C2—42no—
572C2#1760814yes—
573C2#2091311yescharacteristic
574C2#2098011yescharacteristic
575C2C2×#189811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1520C2semidirect
5. C2×#1508C2semidirect
6. #9154C2semidirect
7. #9143C2semidirect
8. #8016C2semidirect
9. #8012C2semidirect
11. C2×#536C4semidirect
18. #2616C4semidirect
26. #2311C4semidirect
53. C23×Q8C2×C4semidirect
72. C2×C22⋊Q8C2×C4semidirect
73. C2×C22⋊Q8C2×C4semidirect
77. C2×#117C2×C4semidirect
80. C22×C4⋊C4C2×C4semidirect
110. #380C2×C4semidirect
111. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353052
C2×#23392
#205812
C2×#15202
C2×#15082
#91542
#91432
#80162
#80122