Finite Group Databasebeta

← #416659

Subgroups — #416659

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#14149C211yesmaximal, characteristic
3#13966C211yesmaximal, characteristic
4#13965C211yesmaximal, characteristic
5#11344C214yesmaximal
6#11222C214yesmaximal
7#10854C211yesmaximal, characteristic
8#10854C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10#2851—22no—
11#2646—22no—
12#2569—22no—
13#2569—22no—
14#2442—22no—
15#2442—22no—
16#2315C2212yes—
17#2315C2212yes—
18C2×#385C2214yes—
19C2×#385C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27C2×#342C2212yes—
28C2×#342C2214yes—
29C2×#342C2212yes—
30C2×#342C2214yes—
31#1444—22no—
32#1442—22no—
33#1441—22no—
34#1436—22no—
35C4⋊#95—22no—
36#1434—22no—
37C22⋊#95C2212yes—
38C22⋊#95C2214yes—
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×#125—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2311yescharacteristic
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2312yes—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42D412yes—
57#400—24no—
58#400—24no—
59#398—24no—
60#398—24no—
61#389—22no—
62#389—22no—
63#387—24no—
64#387—24no—
65#385—24no—
66#385—24no—
67#385—24no—
68#385—24no—
69#385—28no—
70#385—44no—
71#385—44no—
72#382—22no—
73#380—21no—
74#380—21no—
75#380—24no—
76#380—24no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—22no—
84#380Q814yes—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4Q814yes—
92C4×C22⋊C4Q814yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—24no—
107C2×#95—22no—
108C2×#95D412yes—
109C2×#95D412yes—
110C2×#95D412yes—
111#342—28no—
112#342—24no—
113#342—24no—
114#342—44no—
115C25C2411yescharacteristic
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C2×D414yes—
128C23×C4C2×D412yes—
129C23×C4C2×D414yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×Q812yes—
138C23×C4C2×Q812yes—
139C23×C4C2×Q812yes—
140C23×C4#4112yes—
141#125—84no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4—82no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C2×Q814yes—
188C2×C22⋊C4C2×Q814yes—
189C2×C22⋊C4C2×Q814yes—
190C2×C22⋊C4#4114yes—
191C2×C22⋊C4#4114yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—82no—
197C2×C42—82no—
198C2×C42—82no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—48no—
213#95—48no—
214#95—44no—
215#95—44no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—82no—
221#95—84no—
222C24—42no—
223C24—42no—
224C24C22×D411yescharacteristic
225C24C22×D411yescharacteristic
226C24C22×D411yescharacteristic
227C24C22×Q811yescharacteristic
228C24C22×Q811yescharacteristic
229C24C22×Q811yescharacteristic
230C24C2×#4111yescharacteristic
231C24C2×#4112yes—
232C24C2×#4112yes—
233C24H(5,𝔽2)12yes—
234C242−1+412yes—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—88no—
297C22×C4—88no—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q812yes—
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4C4⋊Q812yes—
309C22×C4C4⋊Q812yes—
310C22×C4C4⋊Q812yes—
311C22×C4C4⋊Q812yes—
312C22×C4C4⋊Q812yes—
313C22×C4C4⋊Q812yes—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—816no—
324C4⋊C4—88no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—28no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C22⋊C4#12518yescomplemented
345C42—44no—
346C42—82no—
347C42—84no—
348C42—84no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—44no—
360C23—44no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23C2×#12011yescharacteristic
368C23C2×C22⋊Q811yescharacteristic
369C23C2×C22⋊Q811yescharacteristic
370C23C2×C22⋊Q811yescharacteristic
371C23C2×#12512yes—
372C23C2×C4⋊Q811yescharacteristic
373C23C2×C4⋊Q811yescharacteristic
374C23C2×C4⋊Q811yescharacteristic
375C23#53211yescharacteristic
376C23#53211yescharacteristic
377C23#53312yes—
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380C23#54412yes—
381C23#54412yes—
382C23#54612yes—
383C23#54712yes—
384C23#54812yes—
385C23#55012yes—
386C23#55212yes—
387C23#55312yes—
388C23#55412yes—
389C23#55712yes—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—28no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—82no—
450C2×C4—88no—
451C2×C4—82no—
452C2×C4—84no—
453C2×C4#39512yes—
454C2×C4#39512yes—
455C2×C4#39512yes—
456C2×C4#39512yes—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—24no—
471C22—22no—
472C22—24no—
473C22—22no—
474C22—24no—
475C22—22no—
476C22—24no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22C2×#39511yescharacteristic
486C22#243111yescharacteristic
487C22#243211yescharacteristic
488C22#244112yes—
489C22#244212yes—
490C22#255012yes—
491C22#255212yes—
492C22#256511yescharacteristic
493C22#256611yescharacteristic
494C22#266011yescharacteristic
495C22#285911yescharacteristic
496C22#302512yes—
497C22#302612yes—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—48no—
503C4—84no—
504C4—84no—
505C4—82no—
506C4—84no—
507C4—84no—
508C4—82no—
509C4—84no—
510C2—22no—
511C2—21no—
512C2—22no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—22no—
517C2—22no—
518C2#1522811yescharacteristic
519C2#2356411yescharacteristic
520C2#2356511yescharacteristic
521C2#2558712yes—
522C2#2566512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
344. C22⋊C4#125semidirect

Maximal subgroups

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

subgroupindex
#350202
#141492
#139662
#139652
#113442
#112222
#108542
C4×#3422