Finite Group Databasebeta

← #475679

Subgroups — #475679

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1933C211yesmaximal, characteristic
4C2×#1844C211yesmaximal, characteristic
5#15240C211yesmaximal, characteristic
6#15197C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11325C212yesmaximal
9#11122C212yesmaximal
10#11122C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25#2099—24no—
26#2095—24no—
27#1933—24no—
28#1844—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1445—22no—
37#1445—22no—
38#1442—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—42no—
61C2×#107—42no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103Q814yes—
76C2×C4×C8—24no—
77C2×C4×C8D414yes—
78#391—22no—
79#391—22no—
80#391—42no—
81#391—42no—
82#389—22no—
83#389—22no—
84#384—22no—
85#384—22no—
86#384—22no—
87#384—22no—
88#382—42no—
89#382—42no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—22no—
102#340—22no—
103#336—24no—
104#336—24no—
105#336D414yes—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8C2×Q812yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C2×D414yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142C22×C8#4114yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×Q812yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189#107—48no—
190#106—48no—
191#106—48no—
192#103—28no—
193#103—48no—
194#103—44no—
195C4×C8—28no—
196#95—24no—
197#95—24no—
198#95—48no—
199#95—48no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—84no—
204#95—84no—
205C24C22×D411yescharacteristic
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
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—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4#12712yes—
264C22×C4#12712yes—
265C22×C4#13514yes—
266C22×C4#13714yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274C22×C42−1+411yescharacteristic
275C22×C42−1+411yescharacteristic
276C22×C42−1+412yes—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8C22⋊Q814yes—
293C2×C8#12314yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C42—22no—
318C42—22no—
319C42—24no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23C2×#12711yescharacteristic
325C23#53512yes—
326C23#54712yes—
327C23#54711yescharacteristic
328C23D4×Q811yescharacteristic
329C23C2×#13512yes—
330C23C2×#13712yes—
331C23#57712yes—
332C23#57712yes—
333Q8—42no—
334Q8—44no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338Q8—82no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—81no—
380C2×C4#49514yes—
381C2×C4#49714yes—
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×#12311yescharacteristic
384C2×C4#53511yescharacteristic
385C2×C4#53511yescharacteristic
386C2×C4#53512yes—
387C2×C4#53611yescharacteristic
388C2×C4#53611yescharacteristic
389C2×C4#53912yes—
390C2×C4#53912yes—
391C2×C4#54212yes—
392C2×C4#54412yes—
393C2×C4#54711yescharacteristic
394C2×C4#54712yes—
395C2×C4D4×Q811yescharacteristic
396C2×C4#55312yes—
397C2×C4#55412yes—
398C2×C4#56312yes—
399C2×C4#56312yes—
400C8—24no—
401C8—24no—
402C8—44no—
403C22#250111yescharacteristic
404C22#254211yescharacteristic
405C22#265711yescharacteristic
406C22#271812yes—
407C22#272012yes—
408C22C2×#49511yescharacteristic
409C22C2×#49711yescharacteristic
410C22#315411yescharacteristic
411C22#315411yescharacteristic
412C22#321014yes—
413C22#322114yes—
414C22#333714yes—
415C22#334012yes—
416C22#334414yes—
417C22#335612yes—
418C22#338112yes—
419C22#338712yes—
420C4—24no—
421C4—24no—
422C4—84no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—162no—
427C4—162no—
428C4—161no—
429C4#248711yescharacteristic
430C4#262311yescharacteristic
431C4#262811yescharacteristic
432C4#271812yes—
433C4#272012yes—
434C4#282111yescharacteristic
435C2#1706411yescharacteristic
436C2#2792611yescharacteristic
437C2#2793311yescharacteristic
438C2#2852911yescharacteristic
439C2#2854211yescharacteristic
440C2#3026111yescharacteristic
441C2#3027311yescharacteristic
442C2#3228514yes—
443C2#3234314yes—
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×#20992
C2×#20952
C2×#19332
C2×#18442
#152402
#151972
#113252
#111222
#107972