Finite Group Databasebeta

← #469783

Subgroups — #469783

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1921C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#13936C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11358C212yesmaximal
7#11315C212yesmaximal
8#11268C212yesmaximal
9#11256C212yesmaximal
10#11255C211yesmaximal, characteristic
11#11228C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#382C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18#1988—24no—
19#1921—24no—
20C2×#340C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32#1446—21no—
33#1446—21no—
34#1443—22no—
35#1441—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—42no—
67#389—22no—
68#389—22no—
69#382—24no—
70#382—42no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4D414yes—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95D411yescharacteristic
86C2×#95D411yescharacteristic
87C2×#95D412yes—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—42no—
96#340—42no—
97#336—24no—
98#336—22no—
99#336—42no—
100C22×Q8—44no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—82no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—21no—
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—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8#4112yes—
133C4⋊Q8—28no—
134C4⋊Q8—48no—
135C4⋊Q8—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4#4114yes—
171C2×C4⋊C4#4112yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—82no—
182C2×C42C2×D412yes—
183C2×C42#4112yes—
184#107—28no—
185C4⋊C8—44no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—42no—
194#95—42no—
195#95—81no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201C24C22×D411yescharacteristic
202C2×Q8—48no—
203C2×Q8—88no—
204C2×Q8—88no—
205C2×Q8—84no—
206C2×Q8—84no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
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—41no—
248C22×C4—41no—
249C22×C4—41no—
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—41no—
260C22×C4—41no—
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4#13512yes—
264C22×C4#13712yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)12yes—
271C22×C42−1+412yes—
272C22×C42−1+411yescharacteristic
273C22×C42−1+411yescharacteristic
274C22×C42−1+412yes—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—48no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—84no—
316C42—22no—
317C42—22no—
318C42—24no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—84no—
324C23C2×#12111yescharacteristic
325C23#53612yes—
326C23#54612yes—
327C23#54711yescharacteristic
328C23#54811yescharacteristic
329C23C2×#13511yescharacteristic
330C23C2×#13711yescharacteristic
331C23#57512yes—
332C23#57612yes—
333C23#57711yescharacteristic
334C23#57811yescharacteristic
335Q8—816no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#46414yes—
394C2×C4#52511yescharacteristic
395C2×C4#52511yescharacteristic
396C2×C4#53511yescharacteristic
397C2×C4#53611yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#54112yes—
401C2×C4#54112yes—
402C2×C4#54112yes—
403C2×C4#54212yes—
404C2×C4#54212yes—
405C2×C4#55112yes—
406C2×C4#55212yes—
407C2×C4#55211yescharacteristic
408C2×C4#55612yes—
409C2×C4#55611yescharacteristic
410C2×C4#56312yes—
411C2×C4#56312yes—
412C8—42no—
413C8—81no—
414C8—82no—
415C22#241111yescharacteristic
416C22#241511yescharacteristic
417C22#262111yescharacteristic
418C22#269812yes—
419C22#271612yes—
420C22C2×#46411yescharacteristic
421C22#304011yescharacteristic
422C22#304111yescharacteristic
423C22#304611yescharacteristic
424C22#323312yes—
425C22#323712yes—
426C22#324112yes—
427C22#324312yes—
428C22#329712yes—
429C22#331412yes—
430C22#332112yes—
431C22#332712yes—
432C22#334211yescharacteristic
433C22#334712yes—
434C22#335611yescharacteristic
435C22#336511yescharacteristic
436C22#337012yes—
437C22#337411yescharacteristic
438C4—24no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—161no—
447C4—162no—
448C4#259911yescharacteristic
449C4#263211yescharacteristic
450C4#263411yescharacteristic
451C4#275812yes—
452C4#276212yes—
453C4#282111yescharacteristic
454C2#1666311yescharacteristic
455C2#2447911yescharacteristic
456C2#2449911yescharacteristic
457C2#2468311yescharacteristic
458C2#2469311yescharacteristic
459C2#2986711yescharacteristic
460C2#2987211yescharacteristic
461C2#3170112yes—
462C2#3171512yes—
463C2#3223612yes—
464C2#3225012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19212
#152302
#139362
#113842
#113582
#113152
#112682
#112562
#112552
#112282