Finite Group Databasebeta

← #411123

Subgroups — #411123

Order 512.

683 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21025C211yesmaximal, characteristic
2#20459C211yesmaximal, characteristic
3#11374C211yesmaximal, characteristic
4#11367C211yesmaximal, characteristic
5#11336C211yesmaximal, characteristic
6#11286C211yesmaximal, characteristic
7#11241C212yesmaximal
8#11221C212yesmaximal
9#10920C212yesmaximal
10#10893C212yesmaximal
11#10866C211yesmaximal, characteristic
12C2×#430C2211yescharacteristic
13C2×#430C2211yescharacteristic
14C2×#427C2212yes—
15C2×#424C2211yescharacteristic
16C2×#424C2212yes—
17#2873C2212yes—
18C2×#405C2211yescharacteristic
19#2285C2211yescharacteristic
20#1917—22no—
21#1916—22no—
22#1778—22no—
23C4×#108—22no—
24C2×#344C2212yes—
25C2×#344C2212yes—
26#1740C2212yes—
27C2×#341C2212yes—
28#1733C2212yes—
29#1733C2212yes—
30C2×#339C2212yes—
31C2×#339C2212yes—
32C2×H(3,ℤ/4)C2211yescharacteristic
33C2×H(3,ℤ/4)C2211yescharacteristic
34#1729C2211yescharacteristic
35#1729C2211yescharacteristic
36#1729C2212yes—
37#1728C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C22×M16C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42C22×M16C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×#117C2312yes—
46C2×#117C2312yes—
47C2×#117C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C42C2311yescharacteristic
50#430—24no—
51#430—22no—
52#430Q814yes—
53C2×#108—22no—
54C2×#108—22no—
55#427—24no—
56#427—42no—
57C2×#107—22no—
58C2×#106—22no—
59#424—24no—
60#424—22no—
61#424—22no—
62#424—42no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—42no—
66#405—22no—
67#405—22no—
68#405—42no—
69#405—42no—
70C4×M16—24no—
71C2×#97D412yes—
72C2×C4×C8D412yes—
73#384Q811yescharacteristic
74#384Q811yescharacteristic
75#382Q811yescharacteristic
76#382Q811yescharacteristic
77#380—22no—
78C4×C4⋊C4—22no—
79#344—24no—
80#344—42no—
81#342—24no—
82#341—22no—
83#341—22no—
84#339—24no—
85#339—24no—
86#339—42no—
87#339—42no—
88H(3,ℤ/4)—22no—
89H(3,ℤ/4)—22no—
90#336—21no—
91#336—22no—
92#336—22no—
93#336—21no—
94#336—22no—
95#336—42no—
96C23×C4—21no—
97C23×C4C2411yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
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—21no—
113C2×M16—21no—
114C2×M16C2×Q814yes—
115C2×M16C2×Q812yes—
116C2×M16#4114yes—
117C2×M16#4112yes—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8C2×D412yes—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8#4112yes—
133C22×C8#4112yes—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117#4114yes—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×Q811yescharacteristic
151C2×C4⋊C4C2×Q811yescharacteristic
152C2×C4⋊C4C2×Q811yescharacteristic
153C2×C4⋊C4C2×Q811yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4#4112yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C2×D412yes—
165C2×C42C2×Q811yescharacteristic
166C2×C42C2×Q811yescharacteristic
167#108—28no—
168#108—28no—
169#107—44no—
170#106—44no—
171C4⋊C8—48no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—42no—
175#97—24no—
176#97—44no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—44no—
180#95—22no—
181#95—22no—
182C24C2×#4111yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4C22⋊Q812yes—
193C22×C4C22⋊Q812yes—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×Q811yescharacteristic
196C22×C4C22×Q811yescharacteristic
197C22×C4C2×#4111yescharacteristic
198C22×C4C2×#4112yes—
199C22×C4C2×#4112yes—
200C22×C4C2×#4112yes—
201C22×C4H(5,𝔽2)12yes—
202C22×C4H(5,𝔽2)11yescharacteristic
203C22×C42−1+411yescharacteristic
204M16—24no—
205M16—24no—
206M16—24no—
207M16—24no—
208M16—24no—
209M16—24no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—24no—
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—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8C22⋊Q814yes—
241C2×C8#12311yescharacteristic
242C2×C8#12311yescharacteristic
243C2×C8#12312yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—82no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—24no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—42no—
270C42—42no—
271C42—42no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊Q811yescharacteristic
275C23#52912yes—
276C23#55312yes—
277C23#55711yescharacteristic
278C23#55911yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4C2×C22⋊Q811yescharacteristic
301C2×C4C2×#12311yescharacteristic
302C2×C4#52511yescharacteristic
303C2×C4#52912yes—
304C2×C4#53211yescharacteristic
305C2×C4#53312yes—
306C2×C4#53312yes—
307C2×C4#53511yescharacteristic
308C2×C4#53711yescharacteristic
309C2×C4#53812yes—
310C2×C4#53912yes—
311C2×C4#55112yes—
312C2×C4#55212yes—
313C2×C4#55312yes—
314C2×C4#55412yes—
315C2×C4#55412yes—
316C2×C4#55711yescharacteristic
317C2×C4Q8211yescharacteristic
318C8—24no—
319C8—21no—
320C8—22no—
321C8—21no—
322C8—44no—
323C8—44no—
324C8—81no—
325C8—81no—
326C22—21no—
327C22—22no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22#239611yescharacteristic
334C22#248711yescharacteristic
335C22#258211yescharacteristic
336C22#258512yes—
337C22#259712yes—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—81no—
342C4—84no—
343C4—81no—
344C4—81no—
345C4—162no—
346C4#249811yescharacteristic
347C4#256611yescharacteristic
348C4#258612yes—
349C4#259012yes—
350C4#263611yescharacteristic
351C4#285511yescharacteristic
352C2—21no—
353C2—22no—
354C2—21no—
355C2—22no—
356C2#1510511yescharacteristic
357C2#2769311yescharacteristic
358C2#2769411yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC23×C4C24
socleC22#2487
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ 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
#210252
#204592
#113742
#113672
#113362
#112862
#112412
#112212
#109202
#108932
#108662