Finite Group Databasebeta

← #375495

Subgroups — #375495

Order 512.

2982 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C4×#379C211yesmaximal, characteristic
3C2×C22⋊#95C211yesmaximal, characteristic
4#11381C212yesmaximal, complemented
5#11353C212yesmaximal, complemented
6#10912C214yesmaximal, complemented
7#10735C214yesmaximal
8C25×C4C2211yescharacteristic
9C22×C22⋊Q8C412yescomplemented
10C22×C22⋊Q8C2211yescharacteristic
11C23×C22⋊C4C2212yes—
12#2848—22no—
13#2848—22no—
14#2848—22no—
15#2848—22no—
16C2×#385C414yescomplemented
17C2×#385C2212yes—
18C2×#380C412yescomplemented
19C2×#380C2211yescharacteristic
20C2×#379C2212yes—
21C2×#379C2212yes—
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2212yes—
24C22×#95—22no—
25C2×#342C2218yes—
26C2×#342C2218yes—
27C22⋊#95—22no—
28C22⋊#95—22no—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C22⋊#95C2212yes—
32C26C2311yescharacteristic
33C23×Q8C2×C411yescharacteristic, complemented
34C24×C4—22no—
35C24×C4—48no—
36C24×C4—44no—
37C24×C4C2×C412yes—
38C24×C4C2×C412yes—
39C24×C4D412yes—
40C24×C4D412yes—
41C24×C4D412yes—
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—44no—
52C2×C22⋊Q8—48no—
53C22×C4⋊C4C2×C412yescomplemented
54C22×C4⋊C4C2×C414yescomplemented
55C22×C4⋊C4C2×C411yescharacteristic, complemented
56C22×C4⋊C4C2×C412yescomplemented
57C22×C22⋊C4—22no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—28no—
60C22×C22⋊C4—28no—
61C22×C22⋊C4—44no—
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C42—22no—
69#385—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73#385—22no—
74#385—24no—
75#385—22no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380—22no—
82#379—22no—
83#379—24no—
84#379—22no—
85#379—24no—
86#379Q818yescomplemented
87C4×C22⋊C4—44no—
88C4×C22⋊C4—48no—
89C2×#95—22no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—48no—
94C2×#95C2×C414yescomplemented
95C2×#95C2×C412yescomplemented
96#342—216no—
97#342—48no—
98C25—24no—
99C25—24no—
100C25—44no—
101C25—44no—
102C25C22×C412yes—
103C25C2×D411yescharacteristic
104C25C2×D412yes—
105C25C2×D411yescharacteristic
106C25C2×Q814yes—
107C25#4114yes—
108C25C2411yescharacteristic
109C22×Q8—22no—
110C22×Q8—42no—
111C22×Q8—44no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4—28no—
116C23×C4—28no—
117C23×C4—28no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—464no—
138C23×C4—44no—
139C23×C4—48no—
140C23×C4—48no—
141C23×C4—44no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C414yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—48no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—432no—
196C2×C22⋊C4—48no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4C2×Q818yes—
203C2×C22⋊C4#4118yes—
204C2×C42—44no—
205C2×C42—48no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—48no—
210#95—48no—
211#95—44no—
212#95—48no—
213C24—22no—
214C24—22no—
215C24—24no—
216C24—24no—
217C24—24no—
218C24—24no—
219C24—22no—
220C24—24no—
221C24—22no—
222C24—24no—
223C24—216no—
224C24—216no—
225C24—24no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—416no—
237C24—416no—
238C24—42no—
239C24—416no—
240C24—432no—
241C24—416no—
242C24C2×C22⋊C412yes—
243C24C2×C22⋊C412yes—
244C24C4×Q818yes—
245C24C22⋊Q818yes—
246C24#12412yes—
247C24#12412yes—
248C24C4⋊Q812yes—
249C24C4⋊Q812yes—
250C24C23×C411yescharacteristic
251C24C22×D411yescharacteristic
252C24C22×D411yescharacteristic
253C24C22×Q812yes—
254C24C2×#4112yes—
255C24C2×#4114yes—
256C24H(5,𝔽2)14yes—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—88no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—432no—
313C22×C4—416no—
314C22×C4—432no—
315C22×C4—416no—
316C22×C4—432no—
317C22×C4—432no—
318C22×C4—432no—
319C22×C4—432no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—416no—
323C22×C4—48no—
324C22×C4—432no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—816no—
337C22×C4#9918yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C22⋊C4—216no—
349C22⋊C4—216no—
350C22⋊C4—432no—
351C22⋊C4—48no—
352C22⋊C4—416no—
353C22⋊C4—48no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C42—88no—
363C23—28no—
364C23—216no—
365C23—28no—
366C23—24no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—28no—
378C23—28no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—216no—
383C23—24no—
384C23—216no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—216no—
390C23—216no—
391C23—22no—
392C23—44no—
393C23—48no—
394C23—48no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—48no—
400C23—44no—
401C23—48no—
402C23—48no—
403C23—48no—
404C23—48no—
405C23—48no—
406C23—48no—
407C23—48no—
408C23—48no—
409C23—44no—
410C23—464no—
411C23—464no—
412C23—48no—
413C23—44no—
414C23#39114yes—
415C23#39114yes—
416C23C2×#9914yes—
417C23#41014yes—
418C23C22×C22⋊C411yescharacteristic
419C23C2×C4×Q812yes—
420C23C4×#4112yes—
421C23#51812yes—
422C23C2×C22⋊Q812yes—
423C23#52512yes—
424C23C2×#12411yescharacteristic
425C23C2×C4⋊Q811yescharacteristic
426C23#53212yes—
427C23#53412yes—
428C23#53412yes—
429C23#55118yes—
430C23#55418yes—
431Q8—88no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—416no—
459C2×C4—416no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—84no—
470C2×C4—88no—
471C2×C4—84no—
472C2×C4—832no—
473C2×C4—88no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4C2×#9914yes—
480C2×C4C2×#9914yes—
481C22—216no—
482C22—216no—
483C22—24no—
484C22—24no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—48no—
506C22—48no—
507C22—432no—
508C22—432no—
509C22—48no—
510C22—48no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22#199618yes—
516C22C2×#39111yescharacteristic
517C22#229711yescharacteristic
518C22#230811yescharacteristic
519C22#236814yes—
520C22#256112yes—
521C22#256414yes—
522C22#258312yes—
523C22C22×#9912yes—
524C22C2×#41012yes—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—88no—
530C4—88no—
531C4—84no—
532C4—82no—
533C4—82no—
534C2—22no—
535C2—24no—
536C2—22no—
537C2—28no—
538C2—24no—
539C2—44no—
540C2#1415911yescharacteristic
541C2C2×#199612yes—
542C2#2142012yes—
543C2#2174611yescharacteristic
544C2#2174811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11381C2semidirect
5. #11353C2semidirect
6. #10912C2semidirect
9. C22×C22⋊Q8C4semidirect
16. C2×#385C4semidirect
18. C2×#380C4semidirect
33. C23×Q8C2×C4semidirect
53. C22×C4⋊C4C2×C4semidirect
54. C22×C4⋊C4C2×C4semidirect
55. C22×C4⋊C4C2×C4semidirect
56. C22×C4⋊C4C2×C4semidirect
86. #379Q8semidirect
94. C2×#95C2×C4semidirect
95. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C4×#3792
C2×C22⋊#952
#113812
#113532
#109122
#107352