Finite Group Databasebeta

← #472431

Subgroups — #472431

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1943C211yesmaximal, characteristic
3#21021C211yesmaximal, characteristic
4#11365C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11330C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11271C211yesmaximal, characteristic
9#11216C212yesmaximal
10#11204C212yesmaximal
11#10845C211yesmaximal, characteristic
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#423C2212yes—
17C22×#97C2211yescharacteristic
18#2296C2211yescharacteristic
19C2×#389C2212yes—
20#2088—22no—
21#2088—22no—
22#2087—22no—
23#2086—22no—
24#1943—24no—
25#1917—24no—
26C2×#343C2212yes—
27C2×#343C2212yes—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#341C2211yescharacteristic
31C2×#341C2211yescharacteristic
32C2×#341C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35#1736C2212yes—
36C2×#340C2212yes—
37#1729C2211yescharacteristic
38#1729C2211yescharacteristic
39#1729C2211yescharacteristic
40#1728C2211yescharacteristic
41C22×M16C2312yes—
42C22×M16C2311yescharacteristic
43C22×M16C2311yescharacteristic
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—42no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107Q814yes—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—42no—
62#424—42no—
63#423—24no—
64#423—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×#97—42no—
69C2×#97D414yes—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389D412yes—
75#389D412yes—
76#385—22no—
77C4×C4⋊C4—22no—
78C2×#95—22no—
79#343—24no—
80#343—42no—
81#343—42no—
82#343—42no—
83#341—22no—
84#341—22no—
85#341—42no—
86#340—24no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—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—21no—
111C2×M16—21no—
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#4114yes—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8C2×Q812yes—
134C22×C8#4112yes—
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⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—84no—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×Q814yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167#107—28no—
168#107—44no—
169#106—44no—
170#106—44no—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C4⋊C8—44no—
174#97—44no—
175#95—22no—
176#95—22no—
177#95—42no—
178#95—42no—
179C24C2×#4111yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×Q811yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C22×C4C2×#4112yes—
203C22×C4H(5,𝔽2)12yes—
204C22×C4H(5,𝔽2)11yescharacteristic
205C22×C42−1+412yes—
206C22×C42−1+412yes—
207C22×C42−1+411yescharacteristic
208M16—24no—
209M16—24no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—42no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8C22⋊Q814yes—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C23—22no—
268C23—22no—
269C23C2×#12111yescharacteristic
270C23#54112yes—
271C23#55312yes—
272C23#55411yescharacteristic
273C23#55611yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—162no—
297C2×C4—162no—
298C2×C4C2×C22⋊Q811yescharacteristic
299C2×C4#53211yescharacteristic
300C2×C4#53411yescharacteristic
301C2×C4#53611yescharacteristic
302C2×C4#53611yescharacteristic
303C2×C4#53612yes—
304C2×C4#53711yescharacteristic
305C2×C4#53912yes—
306C2×C4#53912yes—
307C2×C4#54112yes—
308C2×C4#54412yes—
309C2×C4#54612yes—
310C2×C4D4×Q811yescharacteristic
311C2×C4#55011yescharacteristic
312C2×C4#55212yes—
313C2×C4#55712yes—
314C2×C4#56212yes—
315C2×C4#56312yes—
316C8—24no—
317C8—44no—
318C8—42no—
319C8—81no—
320C8—82no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—24no—
329C22#253811yescharacteristic
330C22#254311yescharacteristic
331C22#262111yescharacteristic
332C22#277612yes—
333C22#280212yes—
334C22#306812yes—
335C22#306912yes—
336C4—42no—
337C4—42no—
338C4—84no—
339C4—82no—
340C4—82no—
341C4—162no—
342C4#256511yescharacteristic
343C4#262711yescharacteristic
344C4#266411yescharacteristic
345C4#269412yes—
346C4#271312yes—
347C4#281411yescharacteristic
348C2—22no—
349C2—22no—
350C2#1682011yescharacteristic
351C2#2829911yescharacteristic
352C2#2830011yescharacteristic
353C2#2859612yes—
354C2#2988912yes—
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
#236702
C2×#19432
#210212
#113652
#113302
#113032
#112712
#112162
#112042
#108452