Finite Group Databasebeta

← #772062

Subgroups — #772062

Order 512.

2270 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#16244C211yesmaximal, characteristic, complemented
2#16167C212yesmaximal, complemented
3#15980C211yesmaximal, characteristic, complemented
4#15961C212yesmaximal
5#15786C214yesmaximal, complemented
6#15593C214yesmaximal, complemented
7#15429C212yesmaximal
8#15166C212yesmaximal, complemented
9#15121C212yesmaximal, complemented
10#14210C214yesmaximal, complemented
11#14004C214yesmaximal, complemented
12#13847C211yesmaximal, characteristic
13#13784C212yesmaximal
14C22×C4×D4C2211yescharacteristic
15C2×#398C2214yes—
16C2×#396C2214yes—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#387C2214yes—
20C2×#386C2214yes—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#385C2214yes—
24C2×#380C2214yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2214yes—
27C2×C4×C22⋊C4C2212yes—
28C2×#376C2211yescharacteristic
29C22×#95C2212yes—
30#1445C2212yes—
31#1445C2212yes—
32#1445C2211yescharacteristic
33#1445C2211yescharacteristic
34#1444C2212yes—
35#1444C2214yes—
36#1444C2212yes—
37#1442C2212yes—
38#1442C2214yes—
39#1442C2212yes—
40#1442C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1441C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2211yescharacteristic
52#1438C2211yescharacteristic
53#1437C2212yes—
54#1437C2212yes—
55#1436C2214yes—
56#1436C2212yes—
57#1436C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63#1434C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C23×D4C2311yescharacteristic
68C24×C4C2314yes—
69C24×C4C2312yes—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C2×C4×D4Q818yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2312yes—
93C22×C42C2312yes—
94C22×C42C2312yes—
95#398—48no—
96#396—216no—
97#389—44no—
98#389—44no—
99#387—48no—
100#386—216no—
101#385—216no—
102#385—48no—
103#385—48no—
104#380—48no—
105C4×C4⋊C4—42no—
106C4×C22⋊C4—44no—
107C4×C22⋊C4—48no—
108#376—42no—
109C2×#95—44no—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C25C2412yes—
139C22×D4—28no—
140C22×D4—41no—
141C22×D4—41no—
142C22×D4C2×Q814yes—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C2×Q818yes—
155C23×C4C2412yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2411yescharacteristic
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C4×D4—416no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
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—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C2×Q814yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4C2×Q818yes—
215C2×C22⋊C4#41116yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42C2×Q814yes—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—42no—
246#95—41no—
247#95—41no—
248#95—44no—
249#95—41no—
250#95—41no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24C22×Q814yes—
264C24C2×#4118yes—
265C24C2511yescharacteristic
266C2×D4—216no—
267C2×D4—44no—
268C2×D4—48no—
269C2×D4—48no—
270C22×C4—22no—
271C22×C4—22no—
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—24no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—216no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—416no—
330C22×C4—88no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4C22×Q812yes—
339C22×C4C22×Q812yes—
340C22×C4C22×Q814yes—
341C22×C4C22×Q812yes—
342C22×C4C2×#4118yes—
343C22×C4C2×#4118yes—
344C22×C4H(5,𝔽2)14yes—
345C22×C4H(5,𝔽2)14yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+414yes—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—416no—
353C4⋊C4—48no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C4⋊C4—84no—
359C4⋊C4—88no—
360C22⋊C4—232no—
361C22⋊C4—48no—
362C22⋊C4—416no—
363C22⋊C4—416no—
364C22⋊C4—416no—
365C22⋊C4—88no—
366C42—28no—
367C42—48no—
368C42—84no—
369C42—84no—
370C42—84no—
371C23—24no—
372C23—24no—
373C23—28no—
374C23—28no—
375C23—28no—
376C23—28no—
377C23—28no—
378C23—28no—
379C23—28no—
380C23—28no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23#53318yes—
392C23C23×Q811yescharacteristic
393C23C22×#4114yes—
394C23C2×H(5,𝔽2)11yescharacteristic
395C23C2×H(5,𝔽2)11yescharacteristic
396C23C2×H(5,𝔽2)12yes—
397C23C2×H(5,𝔽2)12yes—
398C23C2×2−1+412yes—
399C23#58512yes—
400D4—416no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
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—44no—
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—42no—
466C2×C4—44no—
467C2×C4—816no—
468C2×C4—832no—
469C2×C4#54314yes—
470C22—216no—
471C22—28no—
472C22—44no—
473C22—48no—
474C22—48no—
475C22—48no—
476C22—44no—
477C22—48no—
478C22—48no—
479C22C2×#53312yes—
480C22C2×#54311yescharacteristic
481C22#348112yes—
482C22#348112yes—
483C22#349214yes—
484C22#350112yes—
485C22#350312yes—
486C22#350514yes—
487C22#351514yes—
488C22#351914yes—
489C22#353412yes—
490C22#353612yes—
491C22#354311yescharacteristic
492C22#354511yescharacteristic
493C22#354911yescharacteristic
494C22#355011yescharacteristic
495C4—24no—
496C4—44no—
497C4—48no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—42no—
502C4—44no—
503C4—48no—
504C4—84no—
505C4—82no—
506C4—84no—
507C2—48no—
508C2#3495811yescharacteristic
509C2#3540014yes—
510C2#3713411yescharacteristic
511C2#3730611yescharacteristic
512C2#4340412yes—
513C2#4345212yes—
514C2#4347812yes—
515C2#4379012yes—
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
1. #16244C2semidirect
2. #16167C2semidirect
3. #15980C2semidirect
5. #15786C2semidirect
6. #15593C2semidirect
8. #15166C2semidirect
9. #15121C2semidirect
10. #14210C2semidirect
11. #14004C2semidirect

Maximal subgroups

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

subgroupindex
#162442
#161672
#159802
#159612
#157862
#155932
#154292
#151662
#151212
#142102
#140042
#138472
#137842