Finite Group Databasebeta

← #410630

Subgroups — #410630

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21033C212yesmaximal
2#21021C211yesmaximal, characteristic
3C2×#1776C211yesmaximal, characteristic
4#11366C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11331C211yesmaximal, characteristic
7#11270C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11242C212yesmaximal
10#10865C211yesmaximal, characteristic
11#10802C212yesmaximal
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2296C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20#1921—22no—
21#1920—22no—
22#1919—22no—
23#1918—22no—
24#1917—22no—
25#1917—22no—
26#1776—22no—
27#1776—22no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#341C2211yescharacteristic
33C2×#341C2211yescharacteristic
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2212yes—
38#1729C2211yescharacteristic
39#1729C2211yescharacteristic
40#1729C2211yescharacteristic
41#1728C2211yescharacteristic
42#1728C2212yes—
43C22×M16C2311yescharacteristic
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42—22no—
54C22×C42C2311yescharacteristic
55C2×#108—24no—
56C2×#108—24no—
57C2×#108—42no—
58#428—24no—
59#428—24no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106Q814yes—
67#424—24no—
68C2×C4⋊C8—22no—
69C2×#97—24no—
70C2×#97D412yes—
71C2×#97D412yes—
72#389D412yes—
73#389D412yes—
74#385—22no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C2×#95—22no—
82#343—24no—
83#343—24no—
84#343—42no—
85#343—42no—
86#341—22no—
87#341—22no—
88#340—24no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—21no—
99#336—21no—
100#336—22no—
101#336—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4#4112yes—
105C23×C4C2411yescharacteristic
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—22no—
117C2×M16—22no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—22no—
125C2×M16#4114yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137C22×C8C2×Q812yes—
138C22×C8#4112yes—
139C22×C8#4114yes—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—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—22no—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×Q814yes—
160C2×C4⋊C4#4114yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—84no—
170C2×C42C2×D412yes—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173#108—48no—
174#107—28no—
175#107—28no—
176#106—28no—
177#106—28no—
178C4⋊C8—44no—
179#97—24no—
180#97—24no—
181#95—22no—
182#95—22no—
183#95—42no—
184#95—42no—
185C24C2×#4111yescharacteristic
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
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×C4C2×#4111yescharacteristic
208C22×C4C2×#4112yes—
209C22×C4C2×#4112yes—
210C22×C4C2×#4112yes—
211C22×C4H(5,𝔽2)11yescharacteristic
212C22×C42−1+412yes—
213C22×C42−1+411yescharacteristic
214M16—24no—
215M16—24no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—42no—
220M16—42no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—22no—
230C2×C8—24no—
231C2×C8—22no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8C22⋊Q814yes—
247C2×C8#12314yes—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—24no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—84no—
273C42—84no—
274C23—22no—
275C23—22no—
276C23C2×#12111yescharacteristic
277C23#52912yes—
278C23#55212yes—
279C23#55411yescharacteristic
280C23#55611yescharacteristic
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4C2×C22⋊Q811yescharacteristic
306C2×C4C2×#12311yescharacteristic
307C2×C4#52511yescharacteristic
308C2×C4#52512yes—
309C2×C4#52912yes—
310C2×C4#53211yescharacteristic
311C2×C4#53312yes—
312C2×C4#53611yescharacteristic
313C2×C4#53711yescharacteristic
314C2×C4#54112yes—
315C2×C4#54212yes—
316C2×C4#54712yes—
317C2×C4D4×Q811yescharacteristic
318C2×C4#55011yescharacteristic
319C2×C4#55112yes—
320C2×C4#55212yes—
321C2×C4#55212yes—
322C2×C4#55712yes—
323C8—24no—
324C8—24no—
325C8—44no—
326C8—81no—
327C8—82no—
328C8—81no—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22#239211yescharacteristic
335C22#248611yescharacteristic
336C22#252012yes—
337C22#253312yes—
338C22#253811yescharacteristic
339C22#306812yes—
340C22#306912yes—
341C4—42no—
342C4—42no—
343C4—82no—
344C4—84no—
345C4—81no—
346C4—84no—
347C4—81no—
348C4#246212yes—
349C4#247912yes—
350C4#249811yescharacteristic
351C4#256511yescharacteristic
352C4#263711yescharacteristic
353C4#285511yescharacteristic
354C2—22no—
355C2—22no—
356C2#1508311yescharacteristic
357C2#2371911yescharacteristic
358C2#2372011yescharacteristic
359C2#2767612yes—
360C2#2829812yes—
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
#210332
#210212
C2×#17762
#113662
#113312
#112702
#112492
#112422
#108652
#108022