Finite Group Databasebeta

← #4415053

Subgroups — #4415053

Order 512.

2852 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3#17011C212yesmaximal, complemented
4#17008C212yesmaximal, complemented
5#16992C212yesmaximal, complemented
6#16978C214yesmaximal, complemented
7#16811C212yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#14912C214yesmaximal, complemented
10#14821C212yesmaximal, complemented
11#14777C212yesmaximal, complemented
12#14719C212yesmaximal
13#14684C211yesmaximal, characteristic, complemented
14#14628C214yesmaximal, complemented
15C22×#123C2212yes—
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2854—28no—
22C2×#399C2214yes—
23C2×#398C2214yes—
24C2×#394C2214yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389—28no—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C22×#95C2212yes—
43#1445C2214yes—
44#1444C2214yes—
45#1442C2214yes—
46#1442C2218yes—
47#1442C2214yes—
48#1441C2218yes—
49#1441C2218yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2218yes—
53#1439C2218yes—
54#1439C2214yes—
55#1439C2218yes—
56#1439C2214yes—
57#1439C2214yes—
58#1434C2212yes—
59#1434C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C23×D4C2312yes—
66C23×D4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#123—28no—
73C2×#123—48no—
74C2×#121—24no—
75C2×#121—44no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—48no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2318yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2318yes—
103C22×C42—44no—
104C22×C42D418yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#399—216no—
110#398—48no—
111#394—48no—
112#390—24no—
113#390—24no—
114#389—24no—
115#389—24no—
116#389—28no—
117#389—28no—
118#389—28no—
119#389—28no—
120#388—28no—
121#388—28no—
122#386—28no—
123#386—48no—
124#386—48no—
125#385—48no—
126#381—44no—
127#380—24no—
128C2×#95—28no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C25C2412yes—
146C25C2411yescharacteristic
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C2×D414yes—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2418yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188#123—432no—
189#121—416no—
190C2×C4⋊C4—232no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—216no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4—432no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4#4118yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—216no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C42—416no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D414yes—
248C2×C42C2×D414yes—
249C2×C42C2×D414yes—
250C2×C42C2×D414yes—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—44no—
260#95—48no—
261#95—44no—
262#95—44no—
263#95—48no—
264#95—48no—
265#95—48no—
266#95—48no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24C22×D412yes—
280C24H(5,𝔽2)14yes—
281C24H(5,𝔽2)14yes—
282C242−1+414yes—
283C24C2511yescharacteristic
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—432no—
290C2×D4—44no—
291C2×D4—88no—
292C22×C4—232no—
293C22×C4—28no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—24no—
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—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—44no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—416no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—416no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—816no—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C2×#4118yes—
365C22×C4C2×#4114yes—
366C22×C4H(5,𝔽2)14yes—
367C22×C42−1+414yes—
368C4⋊C4—216no—
369C4⋊C4—216no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—416no—
374C4⋊C4—44no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—416no—
379C4⋊C4—816no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—416no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—816no—
388C22⋊C4—816no—
389C42—44no—
390C42—44no—
391C42—44no—
392C42—44no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—28no—
398C23—28no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—48no—
411C23—48no—
412C23—48no—
413C23—48no—
414C23—48no—
415C23—48no—
416C23—48no—
417C23#53414yes—
418C23#53514yes—
419C23#53614yes—
420C23#562116yes—
421C23C23×D411yescharacteristic
422C23C22×#4112yes—
423C23C2×H(5,𝔽2)12yes—
424C23C2×H(5,𝔽2)12yes—
425C23C2×H(5,𝔽2)12yes—
426C23C2×2−1+412yes—
427C23C2×2−1+412yes—
428C23#58512yes—
429D4—88no—
430D4—816no—
431C2×C4—232no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—416no—
454C2×C4—432no—
455C2×C4—416no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—816no—
488C2×C4#52514yes—
489C2×C4#53514yes—
490C2×C4#53614yes—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—232no—
496C22—24no—
497C22—24no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—816no—
504C22—82no—
505C22—82no—
506C22#281618yes—
507C22C2×#52511yescharacteristic
508C22C2×#53411yescharacteristic
509C22C2×#53511yescharacteristic
510C22C2×#53511yescharacteristic
511C22C2×#53611yescharacteristic
512C22C2×#53611yescharacteristic
513C22#344911yescharacteristic
514C22#349214yes—
515C22#351614yes—
516C22#351914yes—
517C22C2×#56214yes—
518C22#353414yes—
519C22#355214yes—
520C22#355914yes—
521C4—24no—
522C4—24no—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—48no—
528C4—82no—
529C4—84no—
530C4—82no—
531C4—82no—
532C4—88no—
533C2—28no—
534C2—162no—
535C2#3505711yescharacteristic
536C2#3523711yescharacteristic
537C2#3524511yescharacteristic
538C2C2×#281611yescharacteristic
539C2#3744011yescharacteristic
540C2#3753911yescharacteristic
541C2#3761711yescharacteristic
542C2#4574918yes—
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
2. C22×#389C2semidirect
3. #17011C2semidirect
4. #17008C2semidirect
5. #16992C2semidirect
6. #16978C2semidirect
7. #16811C2semidirect
8. #15091C2semidirect
9. #14912C2semidirect
10. #14821C2semidirect
11. #14777C2semidirect
13. #14684C2semidirect
14. #14628C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C22×#3892
#170112
#170082
#169922
#169782
#168112
#150912
#149122
#148212
#147772
#147192
#146842
#146282