Finite Group Databasebeta

← #5486288

Subgroups — #5486288

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17263C212yesmaximal, complemented
2#17202C212yesmaximal, complemented
3#17135C211yesmaximal, characteristic, complemented
4#16730C214yesmaximal, complemented
5#16352C214yesmaximal, complemented
6#15984C212yesmaximal
7#15966C211yesmaximal, characteristic, complemented
8#15385C214yesmaximal, complemented
9#15324C214yesmaximal, complemented
10#15017C212yesmaximal, complemented
11#15014C212yesmaximal, complemented
12#14666C212yesmaximal, complemented
13#14152C211yesmaximal, characteristic
14C22×#121C2211yescharacteristic
15C2×#400C2214yes—
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#386C2212yes—
19C2×#386C2214yes—
20C2×#386C2212yes—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#382C2212yes—
24C2×#381C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1445C2212yes—
29#1445C2211yescharacteristic
30#1445C2214yes—
31#1445C2211yescharacteristic
32#1445C2214yes—
33#1445C2212yes—
34#1444C2212yes—
35#1444C2212yes—
36#1444C2212yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2212yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2212yes—
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2212yes—
59#1438C2214yes—
60#1438C2212yes—
61#1437C2214yes—
62#1436C2214yes—
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×#121—24no—
73C2×#121—42no—
74C2×#121—42no—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C42C2312yes—
95#400—48no—
96#396—48no—
97#396—48no—
98#386—28no—
99#386—48no—
100#386—44no—
101#385—48no—
102#385—48no—
103#382—44no—
104#381—44no—
105#380—44no—
106C4×C22⋊C4—42no—
107C2×#95—42no—
108C2×#95C2312yes—
109C2×#95C2312yes—
110C2×#95C2312yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C25C2412yes—
137C25C2411yescharacteristic
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—28no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4—42no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4#4114yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172#121—416no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
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—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C42—24no—
225C2×C42—44no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—42no—
234#95—41no—
235#95—41no—
236#95—44no—
237#95—42no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—41no—
243#95—41no—
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—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—22no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24C2×#4114yes—
269C24H(5,𝔽2)12yes—
270C242−1+414yes—
271C24C2511yescharacteristic
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—416no—
276C2×D4—48no—
277C2×D4—88no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—84no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
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—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4—88no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)14yes—
355C22×C4H(5,𝔽2)14yes—
356C4⋊C4—44no—
357C4⋊C4—88no—
358C4⋊C4—88no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C22⋊C4—416no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—416no—
368C22⋊C4—84no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C22⋊C4—88no—
375C42—48no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—28no—
382C23—22no—
383C23—28no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—28no—
388C23—28no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—44no—
394C23—41no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23#56314yes—
413C23C22×#4111yescharacteristic
414C23C2×H(5,𝔽2)11yescharacteristic
415C23C2×H(5,𝔽2)11yescharacteristic
416C23C2×H(5,𝔽2)12yes—
417C23C2×H(5,𝔽2)12yes—
418C23C2×2−1+412yes—
419C23#58514yes—
420C23#58512yes—
421D4—88no—
422D4—816no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—88no—
485C2×C4—816no—
486C2×C4—816no—
487C22—24no—
488C22—28no—
489C22—48no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—42no—
495C22—48no—
496C22—48no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22#349011yescharacteristic
505C22#349112yes—
506C22#351412yes—
507C22#352012yes—
508C22C2×#56311yescharacteristic
509C22#352911yescharacteristic
510C22#353312yes—
511C22#353312yes—
512C22#353414yes—
513C22#353412yes—
514C22#353612yes—
515C22#354114yes—
516C22#354311yescharacteristic
517C22#354911yescharacteristic
518C22#355114yes—
519C22#355314yes—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—84no—
525C4—82no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4—84no—
530C4—84no—
531C2—44no—
532C2—84no—
533C2#4619911yescharacteristic
534C2#4628711yescharacteristic
535C2#4699911yescharacteristic
536C2#4891814yes—
537C2#5245312yes—
538C2#5276112yes—
539C2#5341412yes—
540C2#5570212yes—
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. #17263C2semidirect
2. #17202C2semidirect
3. #17135C2semidirect
4. #16730C2semidirect
5. #16352C2semidirect
7. #15966C2semidirect
8. #15385C2semidirect
9. #15324C2semidirect
10. #15017C2semidirect
11. #15014C2semidirect
12. #14666C2semidirect

Maximal subgroups

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

subgroupindex
#172632
#172022
#171352
#167302
#163522
#159842
#159662
#153852
#153242
#150172
#150142
#146662
#141522