Finite Group Databasebeta

← #413635

Subgroups — #413635

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal
2C2×#2086C212yesmaximal
3C2×#1847C211yesmaximal, characteristic
4C2×#1767C211yesmaximal, characteristic
5#14712C211yesmaximal, characteristic
6#14707C211yesmaximal, characteristic
7#11378C212yesmaximal
8#11377C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10843C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C2×#389—22no—
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2088—28no—
27#2086—28no—
28#1847—24no—
29#1767—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1436—22no—
38#1436—22no—
39C4⋊#95—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42D412yes—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8D418yes—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97D414yes—
76#389—24no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389—48no—
81#384—22no—
82#384—22no—
83#384—22no—
84#384Q814yes—
85#382—22no—
86#382—22no—
87#382—22no—
88#382Q814yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C43—22no—
98C43—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—24no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
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×C8C2×D414yes—
128C22×C8C2×D414yes—
129C22×C8C2×D414yes—
130C22×C8#4114yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4C2×Q812yes—
170C2×C4⋊C4C2×Q814yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42C2×D414yes—
185C2×C42C2×D412yes—
186C2×C42C2×Q812yes—
187C2×C42C2×Q812yes—
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191#107—48no—
192#106—48no—
193C4⋊C8—216no—
194#97—28no—
195#97—44no—
196#95—28no—
197#95—44no—
198#95—48no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—44no—
204C24C22×D411yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4#12114yes—
253C22×C4C22⋊Q814yes—
254C22×C4C22⋊Q814yes—
255C22×C4#12712yes—
256C22×C4#12712yes—
257C22×C4AGL(1,ℤ/8)14yes—
258C22×C4#13714yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C22×Q811yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C42−1+412yes—
268C22×C42−1+412yes—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8#12118yes—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—416no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—88no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—88no—
304C42—24no—
305C42—24no—
306C42—48no—
307C42—48no—
308C42—42no—
309C42—42no—
310C42#12314yes—
311C42#12514yes—
312C42#12514yes—
313C42C4⋊Q814yes—
314C23C2×#12111yescharacteristic
315C23C2×C22⋊Q811yescharacteristic
316C23C2×C22⋊Q811yescharacteristic
317C23C2×#12711yescharacteristic
318C23#53211yescharacteristic
319C23#53611yescharacteristic
320C23#53611yescharacteristic
321C23C2×AGL(1,ℤ/8)12yes—
322C23C2×#13712yes—
323C23#57512yes—
324C23#57512yes—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#39818yes—
378C2×C4#49614yes—
379C2×C4C2×#12112yes—
380C2×C4C2×#12311yescharacteristic
381C2×C4C2×#12511yescharacteristic
382C2×C4C2×#12511yescharacteristic
383C2×C4C2×C4⋊Q811yescharacteristic
384C2×C4#53212yes—
385C2×C4#53612yes—
386C2×C4#54112yes—
387C2×C4#54712yes—
388C2×C4#54712yes—
389C2×C4D4×Q812yes—
390C2×C4D4×Q812yes—
391C2×C4#55212yes—
392C2×C4#55212yes—
393C2×C4#55712yes—
394C2×C4#55712yes—
395C8—28no—
396C8—44no—
397C22C2×#39811yescharacteristic
398C22#254211yescharacteristic
399C22#254211yescharacteristic
400C22#256511yescharacteristic
401C22#256511yescharacteristic
402C22#262111yescharacteristic
403C22#285611yescharacteristic
404C22#305512yes—
405C22#305612yes—
406C22#307712yes—
407C22#307812yes—
408C22#307912yes—
409C22#308012yes—
410C22C2×#49611yescharacteristic
411C22#315011yescharacteristic
412C22#315211yescharacteristic
413C22#315311yescharacteristic
414C22#317212yes—
415C22#318012yes—
416C22#323512yes—
417C22#323612yes—
418C22#323912yes—
419C22#324012yes—
420C4—28no—
421C4—24no—
422C4—42no—
423C4—44no—
424C4—82no—
425C4—82no—
426C4—84no—
427C4—162no—
428C4—161no—
429C4—161no—
430C4#248612yes—
431C4#250612yes—
432C4#250612yes—
433C4#254612yes—
434C2#1515111yescharacteristic
435C2#2367012yes—
436C2#2851711yescharacteristic
437C2#2851811yescharacteristic
438C2#2852011yescharacteristic
439C2#2852111yescharacteristic
440C2#2918112yes—
441C2#2918412yes—
442C2#2988412yes—
443C2#3318311yescharacteristic
444C2#3318611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#20882
C2×#20862
C2×#18472
C2×#17672
#147122
#147072
#113782
#113772
#108432