Finite Group Databasebeta

← #5486628

Subgroups — #5486628

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic
2#17241C214yesmaximal, complemented
3#16913C214yesmaximal, complemented
4#16553C214yesmaximal, complemented
5#16097C214yesmaximal, complemented
6#15949C211yesmaximal, characteristic
7#15441C214yesmaximal, complemented
8#15262C211yesmaximal, characteristic
9#15262C211yesmaximal, characteristic
10#14753C214yesmaximal, complemented
11#14662C211yesmaximal, characteristic
12#14658C211yesmaximal, characteristic
13#14167C211yesmaximal, characteristic
14C22×#124C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#123C2211yescharacteristic
17C2×#398C2214yes—
18C2×#395C2212yes—
19C2×#395C2212yes—
20C2×#389C2214yes—
21C2×#387C2214yes—
22C2×#385C2212yes—
23C2×#385C2212yes—
24C2×#385C2212yes—
25C2×#385C2212yes—
26C2×#382C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1446C2212yes—
30#1445C2214yes—
31#1445C2214yes—
32#1444C2214yes—
33#1444C2214yes—
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1440C2212yes—
45#1440C2212yes—
46#1440C2212yes—
47#1440C2212yes—
48#1440C2212yes—
49#1440C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2211yescharacteristic
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C2×#124—24no—
72C2×#124—44no—
73C2×#123—24no—
74C2×#123—24no—
75C2×#123—44no—
76C2×#123—44no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101#398—48no—
102#395—44no—
103#395—44no—
104#389—48no—
105#387—48no—
106#385—44no—
107#385—44no—
108#385—44no—
109#385—44no—
110#382—44no—
111#380—44no—
112C4×C22⋊C4—44no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C25C2411yescharacteristic
139C25C2411yescharacteristic
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4#4114yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2412yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162C23×C4C2411yescharacteristic
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167#124—416no—
168#123—416no—
169#123—416no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—44no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24C2×#4112yes—
267C24C2×#4112yes—
268C24C2511yescharacteristic
269C2×Q8—44no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—88no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—88no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)14yes—
344C22×C4H(5,𝔽2)14yes—
345C22×C4H(5,𝔽2)14yes—
346C22×C42−1+418yes—
347C4⋊C4—416no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—416no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—28no—
370C42—88no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—28no—
378C23—28no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23C22×#4111yescharacteristic
400C23C2×H(5,𝔽2)12yes—
401C23C2×H(5,𝔽2)12yes—
402C23C2×H(5,𝔽2)12yes—
403C23C2×2−1+414yes—
404C23#58514yes—
405Q8—88no—
406D4—88no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—88no—
470C2×C4—88no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22—84no—
483C22#349011yescharacteristic
484C22#349011yescharacteristic
485C22#349011yescharacteristic
486C22#351514yes—
487C22#352912yes—
488C22#353714yes—
489C22#353812yes—
490C22#353812yes—
491C22#354012yes—
492C22#354012yes—
493C22#354114yes—
494C22#355012yes—
495C22#355414yes—
496C22#355914yes—
497C4—48no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—84no—
506C4—84no—
507C4—84no—
508C2—82no—
509C2—82no—
510C2#4574212yes—
511C2#4638411yescharacteristic
512C2#5023712yes—
513C2#5164612yes—
514C2#5247714yes—
515C2#5505314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17241C2semidirect
3. #16913C2semidirect
4. #16553C2semidirect
5. #16097C2semidirect
7. #15441C2semidirect
10. #14753C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#172412
#169132
#165532
#160972
#159492
#154412
#152622
#147532
#146622
#146582
#141672