Finite Group Databasebeta

← #410631

Subgroups — #410631

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 353 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#11365C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11330C211yesmaximal, characteristic
7#11271C211yesmaximal, 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—24no—
25#1776—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×#107—22no—
60C2×#107—22no—
61C2×#107Q814yes—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—24no—
65#424—24no—
66C2×C4⋊C8—22no—
67C2×#97—24no—
68C2×#97D414yes—
69#389D412yes—
70#389D412yes—
71#385—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C2×#95—22no—
78#343—24no—
79#343—24no—
80#343—42no—
81#343—42no—
82#341—22no—
83#341—22no—
84#340—24no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
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—21no—
114C2×M16—21no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—22no—
120C2×M16#4114yes—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C22×C8C2×Q812yes—
133C22×C8#4112yes—
134C22×C8#4114yes—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—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⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×Q814yes—
154C2×C4⋊C4#4114yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—84no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D412yes—
167#108—48no—
168#107—28no—
169#107—28no—
170#106—28no—
171#106—28no—
172C4⋊C8—44no—
173#97—28no—
174#95—22no—
175#95—22no—
176#95—42no—
177#95—42no—
178C24C2×#4111yescharacteristic
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C22×Q811yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C22×C4C2×#4112yes—
201C22×C4C2×#4112yes—
202C22×C4C2×#4112yes—
203C22×C4H(5,𝔽2)11yescharacteristic
204C22×C42−1+412yes—
205C22×C42−1+411yescharacteristic
206M16—24no—
207M16—24no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216M16—42no—
217M16—42no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8C22⋊Q814yes—
239C2×C8#12314yes—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—84no—
264C42—84no—
265C23—22no—
266C23—22no—
267C23C2×#12111yescharacteristic
268C23#52912yes—
269C23#55212yes—
270C23#55411yescharacteristic
271C23#55611yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4C2×C22⋊Q811yescharacteristic
296C2×C4C2×#12311yescharacteristic
297C2×C4#52511yescharacteristic
298C2×C4#52512yes—
299C2×C4#52912yes—
300C2×C4#53211yescharacteristic
301C2×C4#53312yes—
302C2×C4#53611yescharacteristic
303C2×C4#53711yescharacteristic
304C2×C4#54112yes—
305C2×C4#54212yes—
306C2×C4#54712yes—
307C2×C4D4×Q811yescharacteristic
308C2×C4#55011yescharacteristic
309C2×C4#55112yes—
310C2×C4#55212yes—
311C2×C4#55212yes—
312C2×C4#55712yes—
313C8—24no—
314C8—24no—
315C8—44no—
316C8—81no—
317C8—82no—
318C8—81no—
319C22—24no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22#239211yescharacteristic
327C22#248611yescharacteristic
328C22#252012yes—
329C22#253312yes—
330C22#253811yescharacteristic
331C22#306812yes—
332C22#306912yes—
333C4—42no—
334C4—42no—
335C4—82no—
336C4—84no—
337C4—82no—
338C4—84no—
339C4#246212yes—
340C4#247912yes—
341C4#249811yescharacteristic
342C4#256511yescharacteristic
343C4#263711yescharacteristic
344C4#285511yescharacteristic
345C2—22no—
346C2—22no—
347C2#1508311yescharacteristic
348C2#2372112yes—
349C2#2767612yes—
350C2#2829911yescharacteristic
351C2#2830011yescharacteristic
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
#113652
#113302
#112712
#112492
#112422
#108652
#108022