Finite Group Databasebeta

← #410628

Subgroups — #410628

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21033C212yesmaximal
2#21020C211yesmaximal, characteristic
3C2×C4×#106C211yesmaximal, characteristic
4#11366C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11330C211yesmaximal, characteristic
7#11271C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11242C212yesmaximal
10#10864C211yesmaximal, characteristic
11#10802C212yesmaximal
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2296C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20#1921—22no—
21#1920—22no—
22#1919—22no—
23#1918—22no—
24#1916—24no—
25C4×#106—24no—
26C2×#343C2212yes—
27C2×#343C2212yes—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#341C2211yescharacteristic
31C2×#341C2211yescharacteristic
32#1736C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35C2×#340C2212yes—
36#1729C2211yescharacteristic
37#1729C2211yescharacteristic
38#1729C2211yescharacteristic
39#1728C2211yescharacteristic
40#1728C2212yes—
41C22×M16C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2311yescharacteristic
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C42—22no—
52C22×C42C2311yescharacteristic
53C2×#108—24no—
54C2×#108—24no—
55C2×#108—42no—
56#428—24no—
57#428—24no—
58C2×#107—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106Q814yes—
65#424—24no—
66C2×C4⋊C8—22no—
67C2×C4×C8—24no—
68C2×C4×C8D414yes—
69#389D412yes—
70#389D412yes—
71#385—21no—
72#385—21no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C2×#95—22no—
79#343—24no—
80#343—24no—
81#343—42no—
82#343—42no—
83#341—22no—
84#341—22no—
85#340—24no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4C2411yescharacteristic
103C2×M16—22no—
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—21no—
116C2×M16—21no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—21no—
120C2×M16—21no—
121C2×M16—22no—
122C2×M16#4114yes—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C22×C8C2×Q812yes—
135C22×C8#4112yes—
136C22×C8#4114yes—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—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⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×Q814yes—
157C2×C4⋊C4#4114yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—84no—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169#108—48no—
170#107—28no—
171#106—28no—
172#106—28no—
173#106—28no—
174C4⋊C8—44no—
175C4×C8—28no—
176#95—22no—
177#95—22no—
178#95—42no—
179#95—42no—
180C24C2×#4111yescharacteristic
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4#12112yes—
199C22×C4#12112yes—
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×Q811yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C22×C4C2×#4112yes—
205C22×C4C2×#4112yes—
206C22×C4C2×#4112yes—
207C22×C4H(5,𝔽2)11yescharacteristic
208C22×C42−1+412yes—
209C22×C42−1+411yescharacteristic
210M16—24no—
211M16—24no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—42no—
220M16—42no—
221M16—42no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8C22⋊Q814yes—
243C2×C8#12314yes—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—22no—
261C42—22no—
262C42—24no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—84no—
268C42—84no—
269C23—22no—
270C23—22no—
271C23C2×#12111yescharacteristic
272C23#52912yes—
273C23#55212yes—
274C23#55411yescharacteristic
275C23#55611yescharacteristic
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4#47114yes—
301C2×C4C2×C22⋊Q811yescharacteristic
302C2×C4C2×#12311yescharacteristic
303C2×C4#52511yescharacteristic
304C2×C4#52512yes—
305C2×C4#52912yes—
306C2×C4#53211yescharacteristic
307C2×C4#53312yes—
308C2×C4#53611yescharacteristic
309C2×C4#53711yescharacteristic
310C2×C4#54112yes—
311C2×C4#54212yes—
312C2×C4#54712yes—
313C2×C4D4×Q811yescharacteristic
314C2×C4#55011yescharacteristic
315C2×C4#55112yes—
316C2×C4#55212yes—
317C2×C4#55212yes—
318C2×C4#55712yes—
319C8—24no—
320C8—24no—
321C8—44no—
322C8—81no—
323C8—82no—
324C8—81no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22#239211yescharacteristic
332C22#248611yescharacteristic
333C22#252012yes—
334C22#253312yes—
335C22#253811yescharacteristic
336C22C2×#47112yes—
337C22#306712yes—
338C4—24no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—84no—
343C4—82no—
344C4—84no—
345C4#246212yes—
346C4#247912yes—
347C4#249811yescharacteristic
348C4#256511yescharacteristic
349C4#263711yescharacteristic
350C4#285511yescharacteristic
351C2—22no—
352C2—22no—
353C2#1508311yescharacteristic
354C2#2371812yes—
355C2#2767211yescharacteristic
356C2#2767411yescharacteristic
357C2#2829612yes—
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
#210202
C2×C4×#1062
#113662
#113302
#112712
#112492
#112422
#108642
#108022