Finite Group Databasebeta

← #409416

Subgroups — #409416

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1933C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#15242C211yesmaximal, characteristic
6#15240C211yesmaximal, characteristic
7#11372C212yesmaximal
8#11052C212yesmaximal
9#10926C212yesmaximal
10#10842C212yesmaximal
11C8×#95C211yesmaximal, characteristic
12C22×#103C2211yescharacteristic
13C22×#103C2211yescharacteristic
14C22×#103C2212yes—
15C22×C4×C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#391C2212yes—
18C2×#391C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×#382C2212yes—
24#2099—24no—
25#2072—24no—
26#1933—24no—
27#1834—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1445—22no—
39#1445—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—22no—
65C2×#103—24no—
66C2×#103—42no—
67C2×C4×C8—24no—
68C2×C4×C8—24no—
69C2×C4×C8—24no—
70C2×C4×C8—42no—
71C2×C4×C8D414yes—
72#391—22no—
73#391—22no—
74#391—24no—
75#391—24no—
76#389—22no—
77#389—22no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—22no—
82#382—24no—
83#382—24no—
84#382—24no—
85#382—42no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#336—24no—
103#336D414yes—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8#4112yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D414yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8#4114yes—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
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—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4#4112yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42#4114yes—
186C2×C42#4114yes—
187#103—28no—
188#103—48no—
189#103—44no—
190C4×C8—28no—
191C4×C8—48no—
192#95—24no—
193#95—24no—
194#95—48no—
195#95—48no—
196#95—48no—
197#95—42no—
198#95—42no—
199#95—42no—
200#95—84no—
201C24C22×D411yescharacteristic
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4#12418yes—
259C22×C4#12712yes—
260C22×C4#12712yes—
261C22×C4#13514yes—
262C22×C4#13514yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4112yes—
269C22×C4C2×#4112yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+412yes—
272C22×C42−1+411yescharacteristic
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8#12314yes—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C42—22no—
314C42—24no—
315C42—22no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C23C2×#12412yes—
321C23C2×#12711yescharacteristic
322C23#54611yescharacteristic
323C23#54712yes—
324C23#54811yescharacteristic
325C23C2×#13512yes—
326C23C2×#13512yes—
327C23#57712yes—
328C23#57812yes—
329Q8—42no—
330Q8—42no—
331Q8—44no—
332Q8—81no—
333Q8—81no—
334Q8—82no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4#49514yes—
377C2×C4#49514yes—
378C2×C4C2×#12311yescharacteristic
379C2×C4#52511yescharacteristic
380C2×C4#52912yes—
381C2×C4#52912yes—
382C2×C4#53211yescharacteristic
383C2×C4#53211yescharacteristic
384C2×C4#53212yes—
385C2×C4#53611yescharacteristic
386C2×C4#53711yescharacteristic
387C2×C4#53912yes—
388C2×C4#54112yes—
389C2×C4#54611yescharacteristic
390C2×C4#54712yes—
391C2×C4#54811yescharacteristic
392C2×C4#55112yes—
393C2×C4#55212yes—
394C2×C4#55212yes—
395C2×C4#55212yes—
396C8—24no—
397C8—42no—
398C8—44no—
399C22#248812yes—
400C22#248912yes—
401C22#250111yescharacteristic
402C22#254011yescharacteristic
403C22#285811yescharacteristic
404C22#313114yes—
405C22#313714yes—
406C22C2×#49511yescharacteristic
407C22C2×#49511yescharacteristic
408C22#315411yescharacteristic
409C22#315611yescharacteristic
410C22#330212yes—
411C22#330412yes—
412C22#333614yes—
413C22#333714yes—
414C22#336412yes—
415C22#336512yes—
416C4—24no—
417C4—24no—
418C4—48no—
419C4—82no—
420C4—82no—
421C4—161no—
422C4—162no—
423C4—161no—
424C4—162no—
425C4#239511yescharacteristic
426C4#249811yescharacteristic
427C4#251912yes—
428C4#255312yes—
429C4#261111yescharacteristic
430C4#266411yescharacteristic
431C2#1502011yescharacteristic
432C2#2770014yes—
433C2#2773214yes—
434C2#2792611yescharacteristic
435C2#2793111yescharacteristic
436C2#2840211yescharacteristic
437C2#2840311yescharacteristic
438C2#3322711yescharacteristic
439C2#3322911yescharacteristic
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×#20722
C2×#19332
C2×#18342
#152422
#152402
#113722
#110522
#109262
#108422
C8×#952