Finite Group Databasebeta

← #5482125

Subgroups — #5482125

Order 512.

2168 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17221C211yesmaximal, characteristic, complemented
2#17109C212yesmaximal, complemented
3#17107C212yesmaximal, complemented
4#16379C214yesmaximal, complemented
5#16317C214yesmaximal, complemented
6#15964C211yesmaximal, characteristic, complemented
7#15737C214yesmaximal, complemented
8#15340C212yesmaximal
9#15280C214yesmaximal, complemented
10#14969C212yesmaximal, complemented
11#14914C212yesmaximal, complemented
12#14650C212yesmaximal, complemented
13#14152C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×#121C2211yescharacteristic
16C2×#399C2212yes—
17C2×#398C2212yes—
18C2×#397C2214yes—
19C2×#396C2214yes—
20C2×#394C2214yes—
21C2×#393C2214yes—
22C2×#391C2214yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1446C2212yes—
31#1446C2212yes—
32#1445C2211yescharacteristic
33#1445C2214yes—
34#1445C2211yescharacteristic
35#1444C2212yes—
36#1444C2212yes—
37#1444C2212yes—
38#1444C2214yes—
39#1444C2212yes—
40#1443C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2214yes—
59#1438C2212yes—
60#1437C2214yes—
61#1437C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—48no—
74C2×#121—24no—
75C2×#121—42no—
76C2×#121—42no—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C42C2312yes—
97C22×C42C2312yes—
98#399—44no—
99#398—28no—
100#397—48no—
101#396—48no—
102#394—48no—
103#393—48no—
104#391—48no—
105#390—44no—
106#389—44no—
107#387—48no—
108#386—48no—
109#385—48no—
110C4×C22⋊C4—42no—
111C2×#95—42no—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C25C2412yes—
140C25C2411yescharacteristic
141C22×Q8—28no—
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—44no—
145C22×Q8—42no—
146C22×D4—24no—
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—42no—
153C22×D4—44no—
154C22×D4—44no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C22⋊Q8—432no—
176#121—416no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42—42no—
234C2×C42—44no—
235#95—24no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—42no—
245#95—41no—
246#95—41no—
247#95—42no—
248#95—41no—
249#95—41no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262C24—21no—
263C24—24no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—22no—
273C24—22no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24H(5,𝔽2)12yes—
278C242−1+414yes—
279C24C2511yescharacteristic
280C2×Q8—48no—
281C2×Q8—416no—
282C2×Q8—88no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×D4—48no—
286C2×D4—416no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—84no—
290C2×D4—88no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—84no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—84no—
353C22×C4—88no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—88no—
358C22×C4C2×#4114yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)14yes—
361C22×C4H(5,𝔽2)18yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+414yes—
364C4⋊C4—44no—
365C4⋊C4—416no—
366C4⋊C4—416no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—84no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—416no—
376C22⋊C4—416no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C42—84no—
384C42—84no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—28no—
391C23—22no—
392C23—28no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23#56314yes—
422C23C22×#4111yescharacteristic
423C23C2×H(5,𝔽2)11yescharacteristic
424C23C2×H(5,𝔽2)11yescharacteristic
425C23C2×H(5,𝔽2)14yes—
426C23C2×H(5,𝔽2)12yes—
427C23C2×2−1+412yes—
428C23C2×2−1+412yes—
429C23#58512yes—
430Q8—816no—
431D4—816no—
432D4—88no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4—816no—
496C22—24no—
497C22—28no—
498C22—44no—
499C22—48no—
500C22—42no—
501C22—44no—
502C22—42no—
503C22—44no—
504C22—84no—
505C22—82no—
506C22—84no—
507C22—84no—
508C22—82no—
509C22—84no—
510C22—84no—
511C22—82no—
512C22—84no—
513C22#349011yescharacteristic
514C22#350812yes—
515C22#351812yes—
516C22#352012yes—
517C22C2×#56311yescharacteristic
518C22#352911yescharacteristic
519C22#353212yes—
520C22#353414yes—
521C22#353512yes—
522C22#353614yes—
523C22#353612yes—
524C22#354114yes—
525C22#354311yescharacteristic
526C22#354312yes—
527C22#355011yescharacteristic
528C22#355214yes—
529C4—44no—
530C4—48no—
531C4—44no—
532C4—42no—
533C4—42no—
534C4—44no—
535C4—82no—
536C4—84no—
537C4—84no—
538C4—84no—
539C4—82no—
540C2—44no—
541C2—162no—
542C2#4619911yescharacteristic
543C2#4631611yescharacteristic
544C2#4703111yescharacteristic
545C2#4828614yes—
546C2#5241312yes—
547C2#5266312yes—
548C2#5349212yes—
549C2#5796212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17221C2semidirect
2. #17109C2semidirect
3. #17107C2semidirect
4. #16379C2semidirect
5. #16317C2semidirect
6. #15964C2semidirect
7. #15737C2semidirect
9. #15280C2semidirect
10. #14969C2semidirect
11. #14914C2semidirect
12. #14650C2semidirect

Maximal subgroups

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

subgroupindex
#172212
#171092
#171072
#163792
#163172
#159642
#157372
#153402
#152802
#149692
#149142
#146502
#141522