Finite Group Databasebeta

← #473981

Subgroups — #473981

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14283C211yesmaximal, characteristic
2#14282C211yesmaximal, characteristic
3#11385C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11373C211yesmaximal, characteristic
7#11371C211yesmaximal, characteristic
8#11316C212yesmaximal
9#11315C212yesmaximal
10#11268C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#376C2211yescharacteristic
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29#1446—22no—
30#1445—22no—
31#1441—22no—
32#1441—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8Q814yes—
46C2×#125—24no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—42no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#384—22no—
63#384—22no—
64#382—42no—
65#376—42no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95D412yes—
77C2×#95D412yes—
78#340—24no—
79#340—24no—
80#340—42no—
81#340—42no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#336—24no—
87#336—22no—
88#336—22no—
89#336—42no—
90C22×Q8—24no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8—82no—
94C22×Q8—82no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2411yescharacteristic
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C4⋊Q8—48no—
121C4⋊Q8—48no—
122#125—48no—
123#125—48no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
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—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C2×Q812yes—
159C2×C4⋊C4C2×Q812yes—
160C2×C4⋊C4#4114yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42C2×Q812yes—
170C2×C42#4112yes—
171C2×C42#4112yes—
172C2×C42#4112yes—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—82no—
178#95—82no—
179#95—82no—
180#95—82no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185C24C22×D411yescharacteristic
186C2×Q8—48no—
187C2×Q8—88no—
188C2×Q8—88no—
189C2×Q8—84no—
190C2×Q8—84no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
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×C4C22⋊Q812yes—
247C22×C4C22⋊Q812yes—
248C22×C4#12414yes—
249C22×C4C22×Q811yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4112yes—
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4H(5,𝔽2)12yes—
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+411yescharacteristic
257C22×C42−1+411yescharacteristic
258C22×C42−1+412yes—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—416no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—88no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—42no—
297C42—42no—
298C23C2×C22⋊Q811yescharacteristic
299C23#52511yescharacteristic
300C23C2×#12411yescharacteristic
301C23#53411yescharacteristic
302C23#53511yescharacteristic
303C23#53611yescharacteristic
304C23#53611yescharacteristic
305C23#57612yes—
306C23#57712yes—
307C23#57712yes—
308C23#57812yes—
309Q8—816no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#53812yes—
369C2×C4#54212yes—
370C2×C4#54212yes—
371C2×C4#54212yes—
372C2×C4#54212yes—
373C2×C4#54312yes—
374C2×C4#55112yes—
375C2×C4#55112yes—
376C2×C4#55611yescharacteristic
377C2×C4#55611yescharacteristic
378C2×C4#55612yes—
379C2×C4#55711yescharacteristic
380C2×C4#55711yescharacteristic
381C2×C4#55712yes—
382C2×C4#56312yes—
383C2×C4#56312yes—
384C8—82no—
385C8—82no—
386C22#256111yescharacteristic
387C22#256811yescharacteristic
388C22#258111yescharacteristic
389C22#262611yescharacteristic
390C22#263211yescharacteristic
391C22#264011yescharacteristic
392C22#281711yescharacteristic
393C22#308112yes—
394C22#308212yes—
395C22#312412yes—
396C22#312512yes—
397C22#313612yes—
398C22#313712yes—
399C22#319812yes—
400C22#320112yes—
401C22#322011yescharacteristic
402C22#322111yescharacteristic
403C22#322211yescharacteristic
404C22#322311yescharacteristic
405C22#324112yes—
406C22#324212yes—
407C22#324312yes—
408C22#324412yes—
409C4—42no—
410C4—42no—
411C4—44no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—84no—
416C4—162no—
417C4—161no—
418C4—161no—
419C4#275912yes—
420C4#275912yes—
421C4#279912yes—
422C4#279912yes—
423C2#1689111yescharacteristic
424C2#2912012yes—
425C2#2922912yes—
426C2#2928212yes—
427C2#2996211yescharacteristic
428C2#2996311yescharacteristic
429C2#3007211yescharacteristic
430C2#3007411yescharacteristic
431C2#3020911yescharacteristic
432C2#3021011yescharacteristic
433C2#3279312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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
#142832
#142822
#113852
#113842
#113732
#113712
#113162
#113152
#112682
#107962