Finite Group Databasebeta

← #409231

Subgroups — #409231

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21159C212yesmaximal
2C2×#1944C211yesmaximal, characteristic
3#21021C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11330C211yesmaximal, characteristic
6#11290C212yesmaximal
7#11271C211yesmaximal, characteristic
8#11270C211yesmaximal, characteristic
9#11216C212yesmaximal
10#10913C212yesmaximal
11#10705C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×C4×M16C2212yes—
18C22×#97C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×#389C2212yes—
21#1944—24no—
22#1944—22no—
23#1943—22no—
24#1942—22no—
25#1941—22no—
26#1917—24no—
27C2×#343C2212yes—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#341C2212yes—
32C2×#341C2211yescharacteristic
33C2×#341C2211yescharacteristic
34#1736C2212yes—
35#1736C2212yes—
36C2×#340C2212yes—
37#1729C2211yescharacteristic
38#1729C2211yescharacteristic
39#1729C2211yescharacteristic
40#1728C2211yescharacteristic
41C22×M16C2311yescharacteristic
42C22×M16C2311yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2311yescharacteristic
53#428—24no—
54#428—24no—
55#428—24no—
56#428—42no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—42no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—22no—
65#424—42no—
66C2×C4⋊C8—22no—
67C4×M16—24no—
68C4×M16—42no—
69C2×#97—22no—
70C2×#97—42no—
71C2×#97D414yes—
72C2×C4×C8—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389D412yes—
78#389D412yes—
79#385—21no—
80#385—21no—
81C4×C4⋊C4—22no—
82C2×#95—22no—
83#343—24no—
84#343—24no—
85#343—24no—
86#343—42no—
87#341—24no—
88#341Q814yes—
89#341Q814yes—
90#340—24no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4C2411yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16C2×Q814yes—
121C2×M16C2×Q814yes—
122C2×M16#4114yes—
123C2×M16#4114yes—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C22×C8C2×Q812yes—
137C22×C8C2×Q812yes—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4#4114yes—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170#107—28no—
171#107—44no—
172#107—44no—
173#106—28no—
174#106—28no—
175#106—44no—
176C4⋊C8—44no—
177#97—44no—
178#97—44no—
179C4×C8—44no—
180#95—22no—
181#95—22no—
182#95—42no—
183#95—42no—
184C24C2×#4111yescharacteristic
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4#12112yes—
203C22×C4#12112yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×Q811yescharacteristic
207C22×C4C22×Q811yescharacteristic
208C22×C4C2×#4112yes—
209C22×C4C2×#4112yes—
210C22×C4C2×#4112yes—
211C22×C4H(5,𝔽2)12yes—
212C22×C42−1+411yescharacteristic
213C22×C42−1+411yescharacteristic
214M16—24no—
215M16—24no—
216M16—24no—
217M16—24no—
218M16—24no—
219M16—42no—
220M16—42no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16#12518yes—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8C4⋊Q814yes—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—22no—
275C23—22no—
276C23C2×#12111yescharacteristic
277C23#52912yes—
278C23#55312yes—
279C23#55711yescharacteristic
280C23#55711yescharacteristic
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—162no—
305C2×C4—162no—
306C2×C4#52511yescharacteristic
307C2×C4#52512yes—
308C2×C4C2×#12512yes—
309C2×C4C2×C4⋊Q811yescharacteristic
310C2×C4#53211yescharacteristic
311C2×C4#53211yescharacteristic
312C2×C4#53312yes—
313C2×C4#53411yescharacteristic
314C2×C4#53611yescharacteristic
315C2×C4#53912yes—
316C2×C4#54212yes—
317C2×C4#54612yes—
318C2×C4D4×Q811yescharacteristic
319C2×C4D4×Q811yescharacteristic
320C2×C4#55212yes—
321C2×C4#55412yes—
322C2×C4#55412yes—
323C2×C4#55612yes—
324C8—24no—
325C8—28no—
326C8—42no—
327C8—44no—
328C8—81no—
329C8—81no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22#239211yescharacteristic
337C22#250912yes—
338C22#254311yescharacteristic
339C22#254611yescharacteristic
340C22#258512yes—
341C22#306812yes—
342C22#306912yes—
343C4—42no—
344C4—42no—
345C4—82no—
346C4—84no—
347C4—81no—
348C4—81no—
349C4—162no—
350C4#246512yes—
351C4#247312yes—
352C4#258311yescharacteristic
353C4#260411yescharacteristic
354C4#266311yescharacteristic
355C4#285911yescharacteristic
356C2—22no—
357C2—22no—
358C2#1501211yescharacteristic
359C2#2372112yes—
360C2#2859411yescharacteristic
361C2#2859511yescharacteristic
362C2#2870312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#211592
C2×#19442
#210212
#113312
#113302
#112902
#112712
#112702
#112162
#109132
#107052