Finite Group Databasebeta

← #474818

Subgroups — #474818

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23747C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#21143C211yesmaximal, characteristic
4#11303C214yesmaximal
5#11239C212yesmaximal
6#11239C212yesmaximal
7#11204C212yesmaximal
8#10822C211yesmaximal, characteristic
9C22×#107C2211yescharacteristic
10C22×#107C2211yescharacteristic
11C2×#424C2212yes—
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15#2285C2211yescharacteristic
16#2101—24no—
17#2100—22no—
18#2099—22no—
19#1942—22no—
20#1942—22no—
21#1939—22no—
22#1939—22no—
23C2×#343C2214yes—
24C2×#343C2214yes—
25C2×#341C2212yes—
26C2×#341C2212yes—
27#1736C2214yes—
28#1736C2214yes—
29C2×#340C2212yes—
30#1729C2212yes—
31#1728C2212yes—
32C22×M16C2312yes—
33C22×M16C2314yes—
34C23×C8C2311yescharacteristic
35C23×C8C2311yescharacteristic
36C2×#117C2312yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2312yes—
39C22×C4⋊C4C2312yes—
40C22×C42C2311yescharacteristic
41C2×#107—22no—
42C2×#107—24no—
43C2×#107—22no—
44C2×#107—42no—
45#424—28no—
46#423—42no—
47C2×C4⋊C8—24no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4×C8—42no—
51C2×C4×C8D412yes—
52C2×C4×C8D412yes—
53#389—22no—
54#389—22no—
55#389—22no—
56#389—22no—
57#384D411yescharacteristic
58#384D411yescharacteristic
59#382D412yes—
60#380—21no—
61#380—21no—
62C4×C4⋊C4—21no—
63C4×C4⋊C4—21no—
64C2×#95—22no—
65#343—28no—
66#343—44no—
67#341—24no—
68#341—42no—
69#340—24no—
70#340—24no—
71#340—24no—
72#340—24no—
73#340—24no—
74#336—24no—
75#336—22no—
76#336—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4C2411yescharacteristic
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—22no—
98C22×C8C2×D412yes—
99C22×C8C2×D412yes—
100C22×C8#4114yes—
101#117—24no—
102#117—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—84no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4#4114yes—
122C2×C22⋊C4—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130#107—28no—
131#107—44no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4⋊C8—48no—
135C4×C8—42no—
136C4×C8—42no—
137#95—22no—
138#95—22no—
139#95—42no—
140#95—42no—
141C24C2×#4111yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4#12111yescharacteristic
160C22×C4#12111yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4C22×D411yescharacteristic
164C22×C4C22×D411yescharacteristic
165C22×C4C2×#4112yes—
166C22×C4C2×#4112yes—
167C22×C4H(5,𝔽2)12yes—
168C22×C42−1+412yes—
169C22×C42−1+412yes—
170C22×C42−1+412yes—
171M16—44no—
172M16—44no—
173M16—48no—
174M16—44no—
175M16—44no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8#12414yes—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C22⋊C4—44no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—22no—
211C42—22no—
212C42—42no—
213C42—42no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23C2×#12111yescharacteristic
219C23#53912yes—
220C23#55212yes—
221C23#55612yes—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—84no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—164no—
249C2×C4#47212yes—
250C2×C4#47312yes—
251C2×C4#52511yescharacteristic
252C2×C4C2×#12411yescharacteristic
253C2×C4#53411yescharacteristic
254C2×C4#53611yescharacteristic
255C2×C4#53611yescharacteristic
256C2×C4#53611yescharacteristic
257C2×C4#53712yes—
258C2×C4#54114yes—
259C2×C4#54114yes—
260C2×C4#54712yes—
261C2×C4#54812yes—
262C2×C4#55314yes—
263C2×C4#56314yes—
264C8—24no—
265C8—42no—
266C8—82no—
267C8—84no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22#253611yescharacteristic
275C22#254711yescharacteristic
276C22#261811yescharacteristic
277C22#277614yes—
278C22C2×#47211yescharacteristic
279C22C2×#47311yescharacteristic
280C22#306711yescharacteristic
281C22#306711yescharacteristic
282C4—22no—
283C4—22no—
284C4—41no—
285C4—41no—
286C4—84no—
287C4—82no—
288C4—82no—
289C4—162no—
290C4#259111yescharacteristic
291C4#263411yescharacteristic
292C4#264211yescharacteristic
293C4#272514yes—
294C4#281711yescharacteristic
295C2—24no—
296C2#1701011yescharacteristic
297C2#2821711yescharacteristic
298C2#2821911yescharacteristic
299C2#2873111yescharacteristic
300C2#2873411yescharacteristic
301C2#2978811yescharacteristic
302C2#2978911yescharacteristic
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
#237472
C2×#19422
#211432
#113032
#112392
#112042
#108222