Finite Group Databasebeta

← #4453478

Subgroups — #4453478

Order 512.

3730 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2487C212yesmaximal, complemented
3C22×#400C211yesmaximal, characteristic, complemented
4#16674C214yesmaximal, complemented
5#16634C214yesmaximal, complemented
6#16366C214yesmaximal, complemented
7#16360C214yesmaximal, complemented
8#14914C212yesmaximal, complemented
9#14913C212yesmaximal
10#14912C214yesmaximal, complemented
11#14339C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13C2×#539—216no—
14C2×#539—28no—
15C22×#126C2212yes—
16C22×#125C2212yes—
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×#121C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2487—216no—
26C2×#400—28no—
27C2×#400C2214yes—
28C2×#400C2214yes—
29C2×#400C2214yes—
30C2×#399C2214yes—
31C2×#389C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#381C2212yes—
36C2×#380C2212yes—
37C2×#380C2214yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41#1446C2214yes—
42#1444C2214yes—
43#1442C2218yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2218yes—
48#1441C2218yes—
49#1441C2214yes—
50#1440C2218yes—
51#1440C2218yes—
52#1439C2218yes—
53#1439C2214yes—
54#1438C2214yes—
55#1434C2214yes—
56C22⋊#95C2212yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C23×Q8C2311yescharacteristic
60C23×D4C2312yes—
61C23×D4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68#539—2128no—
69C2×#126—28no—
70C2×#126—28no—
71C2×#126—28no—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#123—28no—
76C2×#123—28no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—28no—
80C2×#123—28no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×#121—28no—
87C2×#121—24no—
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4D4116yes—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4D418yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C42C2312yes—
115#400—216no—
116#400—216no—
117#400—216no—
118#399—216no—
119#389—216no—
120#386—216no—
121#386—48no—
122#385—216no—
123#381—44no—
124#380—216no—
125#380—28no—
126C2×#95—28no—
127C2×#95—28no—
128C2×#95—24no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C25C2412yes—
143C25C2411yescharacteristic
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—44no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—44no—
156C22×D4C2×D418yes—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—48no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4C2×D418yes—
179C23×C4C2×D418yes—
180C23×C4C2×D414yes—
181C23×C4C2418yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191#126—264no—
192#125—264no—
193#123—264no—
194#123—264no—
195C22⋊Q8—264no—
196C22⋊Q8—232no—
197#121—232no—
198C4×D4—264no—
199C2×C4⋊C4—232no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4C2×D418yes—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×D418yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4C2×D418yes—
254C2×C22⋊C4C2×D418yes—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42C2×D418yes—
258#95—28no—
259#95—216no—
260#95—28no—
261#95—28no—
262#95—216no—
263#95—28no—
264#95—216no—
265#95—48no—
266#95—44no—
267#95—48no—
268#95—44no—
269#95—48no—
270#95—48no—
271C24—24no—
272C24—24no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—28no—
280C24—28no—
281C24C22×D414yes—
282C24C22×D414yes—
283C24C22×D412yes—
284C24H(5,𝔽2)12yes—
285C242−1+414yes—
286C242−1+412yes—
287C24C2511yescharacteristic
288C2×Q8—28no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×D4—216no—
292C2×D4—216no—
293C2×D4—216no—
294C2×D4—28no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—84no—
300C2×D4—88no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—416no—
347C22×C4—48no—
348C22×C4—416no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—816no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4#120116yes—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D412yes—
368C22×C4C22×D414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C4⋊C4—28no—
375C4⋊C4—216no—
376C4⋊C4—216no—
377C4⋊C4—28no—
378C4⋊C4—216no—
379C4⋊C4—216no—
380C4⋊C4—216no—
381C4⋊C4—216no—
382C4⋊C4—416no—
383C4⋊C4—416no—
384C4⋊C4—816no—
385C22⋊C4—216no—
386C22⋊C4—216no—
387C22⋊C4—216no—
388C22⋊C4—216no—
389C22⋊C4—216no—
390C22⋊C4—416no—
391C22⋊C4—816no—
392C22⋊C4—816no—
393C42—216no—
394C23—24no—
395C23—28no—
396C23—24no—
397C23—28no—
398C23—216no—
399C23—28no—
400C23—28no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—28no—
405C23—28no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—48no—
415C23—48no—
416C23—48no—
417C23—48no—
418C23C2×#12014yes—
419C23#53614yes—
420C23#53618yes—
421C23#53718yes—
422C23#53714yes—
423C23C23×D411yescharacteristic
424C23C23×D412yes—
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×2−1+412yes—
427C23C2×2−1+411yescharacteristic
428C23#58514yes—
429C23#58514yes—
430Q8—88no—
431D4—432no—
432D4—88no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—416no—
451C2×C4—416no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
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—816no—
484C2×C4—88no—
485C2×C4—816no—
486C2×C4—816no—
487C2×C4C2×#12018yes—
488C22—28no—
489C22—24no—
490C22—28no—
491C22—24no—
492C22—216no—
493C22—216no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—48no—
498C22—48no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22#261618yes—
504C22C22×#12011yescharacteristic
505C22C2×#53611yescharacteristic
506C22C2×#53612yes—
507C22C2×#53711yescharacteristic
508C22C2×#53712yes—
509C22#348512yes—
510C22#348514yes—
511C22#348712yes—
512C22#350014yes—
513C22#355318yes—
514C22#356014yes—
515C22#356414yes—
516C4—28no—
517C4—44no—
518C4—44no—
519C4—44no—
520C4—42no—
521C4—44no—
522C4—42no—
523C4—84no—
524C4—84no—
525C4—88no—
526C2—28no—
527C2—84no—
528C2C2×#261611yescharacteristic
529C2#3544611yescharacteristic
530C2#3567611yescharacteristic
531C2#3697112yes—
532C2#3712012yes—
533C2#4137214yes—
534C2#4137414yes—
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. C2×#2487C2semidirect
3. C22×#400C2semidirect
4. #16674C2semidirect
5. #16634C2semidirect
6. #16366C2semidirect
7. #16360C2semidirect
8. #14914C2semidirect
10. #14912C2semidirect
11. #14339C2semidirect
12. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24872
C22×#4002
#166742
#166342
#163662
#163602
#149142
#149132
#149122
#143392
#139602