Finite Group Databasebeta

← #382767

Subgroups — #382767

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11372C212yesmaximal, complemented
5#11357C212yesmaximal, complemented
6#11200C212yesmaximal
7#11199C212yesmaximal
8#10850C212yesmaximal, complemented
9#10849C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2314—24no—
15#2314C418yescomplemented
16C2×#388—22no—
17C2×#387—22no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25#1798—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117C2×C414yes—
44C2×#117D414yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D412yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—24no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—24no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67#388—216no—
68#387—216no—
69#384—22no—
70#384—24no—
71#384C2×C414yes—
72#382—22no—
73#382—24no—
74#382C2×C414yes—
75#380—24no—
76#380—42no—
77#380—42no—
78#380C2×C418yes—
79C4×C4⋊C4—24no—
80C4×C4⋊C4C2×C418yes—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—24no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130#117—216no—
131#117—216no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C22⋊C418yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—82no—
170C2×C22⋊C4—82no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4C22⋊C418yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—28no—
177C2×C42—82no—
178C2×C42—82no—
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42#4114yes—
184C2×C42#4114yes—
185C4⋊C8—48no—
186C22⋊C8—44no—
187C22⋊C8—44no—
188#95—28no—
189#95—28no—
190#95—28no—
191#95—44no—
192#95—44no—
193#95—82no—
194#95—82no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24C2×C22⋊C412yes—
206C24C22×D411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C414yes—
269C22×C4C2×C22⋊C414yes—
270C22×C4#12314yes—
271C22×C4#12314yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C4H(5,𝔽2)12yes—
279C22×C42−1+411yescharacteristic
280C22×C42−1+411yescharacteristic
281C22×C42−1+412yes—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—216no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C22⋊C4—216no—
316C22⋊C4—88no—
317C22⋊C4—88no—
318C42—24no—
319C42—24no—
320C42—88no—
321C42#11718yes—
322C42#12318yes—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—24no—
327C23—22no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C22×C22⋊C411yescharacteristic
347C23C2×#12311yescharacteristic
348C23C2×#12311yescharacteristic
349C23#53511yescharacteristic
350C23#53611yescharacteristic
351C23#53711yescharacteristic
352C23#53711yescharacteristic
353C23#57612yes—
354C23#57712yes—
355C23#57712yes—
356C23#57812yes—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#38118yes—
410C2×C4C2×#11712yes—
411C2×C4#52012yes—
412C2×C4#52011yescharacteristic
413C2×C4#52011yescharacteristic
414C2×C4C2×#12312yes—
415C2×C4#53511yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53712yes—
418C2×C4#54012yes—
419C2×C4#54012yes—
420C2×C4#54112yes—
421C2×C4#54112yes—
422C2×C4#54212yes—
423C2×C4#54212yes—
424C2×C4#54212yes—
425C2×C4#54212yes—
426C8—82no—
427C8—82no—
428C22—42no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—44no—
433C22—81no—
434C22—81no—
435C22—82no—
436C22—82no—
437C22—81no—
438C22—81no—
439C22C2×#38111yescharacteristic
440C22#231111yescharacteristic
441C22#231111yescharacteristic
442C22#262411yescharacteristic
443C22#262411yescharacteristic
444C22#262511yescharacteristic
445C22#262511yescharacteristic
446C22#289712yes—
447C22#289812yes—
448C22#309612yes—
449C22#309712yes—
450C22#309812yes—
451C22#309912yes—
452C22#322412yes—
453C22#322512yes—
454C22#324312yes—
455C22#324412yes—
456C22#325712yes—
457C22#325812yes—
458C22#325911yescharacteristic
459C22#326011yescharacteristic
460C22#326011yescharacteristic
461C22#326111yescharacteristic
462C4—28no—
463C4—42no—
464C4—44no—
465C4—84no—
466C4—84no—
467C4—162no—
468C4—162no—
469C4#231412yes—
470C4#231412yes—
471C4#262412yes—
472C4#262512yes—
473C2—82no—
474C2#1434311yescharacteristic
475C2#2070312yes—
476C2#2187112yes—
477C2#2189711yescharacteristic
478C2#2189811yescharacteristic
479C2#2992911yescharacteristic
480C2#2993011yescharacteristic
481C2#2993512yes—
482C2#2994512yes—
483C2#2994811yescharacteristic
484C2#2994911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11372C2semidirect
5. #11357C2semidirect
8. #10850C2semidirect
9. #10849C2semidirect
15. #2314C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#17982
#143402
#113722
#113572
#112002
#111992
#108502
#108492