Finite Group Databasebeta

← #4464291

Subgroups — #4464291

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17274C214yesmaximal
2#17269C214yesmaximal, complemented
3#17185C214yesmaximal, complemented
4#17133C214yesmaximal
5#15242C212yesmaximal, complemented
6#15228C212yesmaximal, complemented
7#15223C211yesmaximal, characteristic
8#15216C211yesmaximal, characteristic
9#15146C212yesmaximal, complemented
10#15137C212yesmaximal, complemented
11#15131C212yesmaximal
12#14340C212yesmaximal
13#13853C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#400C2214yes—
17C2×#398C2214yes—
18C2×#396C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384C2212yes—
23C2×#384C2212yes—
24C2×#382C2212yes—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1445C2212yes—
31#1445C2212yes—
32#1445C2214yes—
33#1445C2214yes—
34#1445C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2212yes—
53#1440C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4×#95C2212yes—
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C2×#125—42no—
69C2×#125—44no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117Q818yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2312yes—
96C22×C42C2312yes—
97#400—48no—
98#398—48no—
99#396—48no—
100#389—44no—
101#389—44no—
102#389—44no—
103#384—44no—
104#384—44no—
105#382—44no—
106#382—42no—
107#380—44no—
108#380—44no—
109C4×C22⋊C4—42no—
110C4×C22⋊C4—42no—
111C2×#95C2314yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C25C2411yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4C2×Q814yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2412yes—
150C23×C4C2412yes—
151C23×C4C2412yes—
152C23×C4C2412yes—
153C23×C4C2411yescharacteristic
154C23×C4C2412yes—
155C23×C4C2412yes—
156C23×C4C2411yescharacteristic
157#125—416no—
158#117—416no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
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×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C2×Q814yes—
200C2×C4⋊C4C2×Q814yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4C2×Q818yes—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42C2×Q818yes—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—42no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24C22×Q812yes—
264C24C2511yescharacteristic
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—28no—
294C22×C4—216no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4C22×Q814yes—
341C22×C4C22×Q814yes—
342C22×C4C22×Q812yes—
343C22×C4C22×Q812yes—
344C22×C4H(5,𝔽2)14yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)14yes—
348C22×C42−1+414yes—
349C22×C42−1+414yes—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—416no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—88no—
358C4⋊C4—88no—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C22⋊C4—48no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—48no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—216no—
371C42—44no—
372C42—48no—
373C42—44no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23C23×Q811yescharacteristic
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×H(5,𝔽2)11yescharacteristic
398C23C2×H(5,𝔽2)12yes—
399C23C2×H(5,𝔽2)12yes—
400C23C2×2−1+412yes—
401C23C2×2−1+412yes—
402C23#58512yes—
403C23#58512yes—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—416no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4#54314yes—
481C2×C4#54418yes—
482C2×C4#56014yes—
483C2×C4#56218yes—
484C2×C4#56314yes—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22C2×#54311yescharacteristic
495C22C2×#54412yes—
496C22#350312yes—
497C22#350411yescharacteristic
498C22#350411yescharacteristic
499C22C2×#56011yescharacteristic
500C22C2×#56212yes—
501C22C2×#56311yescharacteristic
502C22#353711yescharacteristic
503C22#353811yescharacteristic
504C22#353912yes—
505C22#354612yes—
506C22#354712yes—
507C22#354812yes—
508C22#355112yes—
509C22#355412yes—
510C22#355512yes—
511C22#355812yes—
512C22#355912yes—
513C22#356112yes—
514C22#356312yes—
515C4—44no—
516C4—44no—
517C4—42no—
518C4—44no—
519C4—44no—
520C4—82no—
521C4—84no—
522C4—84no—
523C4—82no—
524C4—84no—
525C4—84no—
526C4#284118yes—
527C2—82no—
528C2C2×#284111yescharacteristic
529C2#3730711yescharacteristic
530C2#3731312yes—
531C2#3731412yes—
532C2#3742511yescharacteristic
533C2#5297012yes—
534C2#5301012yes—
535C2#5306612yes—
536C2#5306712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17269C2semidirect
3. #17185C2semidirect
5. #15242C2semidirect
6. #15228C2semidirect
9. #15146C2semidirect
10. #15137C2semidirect

Maximal subgroups

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

subgroupindex
#172742
#172692
#171852
#171332
#152422
#152282
#152232
#152162
#151462
#151372
#151312
#143402
#138532