Finite Group Databasebeta

← #330400

Subgroups — #330400

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#21339C211yesmaximal, characteristic
5#10589C214yesmaximal
6#10482C214yesmaximal
7#9896C211yesmaximal, characteristic
8#9691C211yesmaximal, characteristic
9#8805C211yesmaximal, characteristic
10C2×#477—21no—
11C2×#477C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C42×Q8C2211yescharacteristic
15#1990—22no—
16#1990C2214yes—
17#1987—21no—
18#1987C2212yes—
19#1984C2212yes—
20#1947—21no—
21#1947—21no—
22#1947—21no—
23#1947C2212yes—
24#1939C2212yes—
25#1916—22no—
26#1842C2211yescharacteristic
27#1842C2211yescharacteristic
28#1834—21no—
29#1834—21no—
30#1803—21no—
31#1803—21no—
32#1803—21no—
33#1803C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2211yescharacteristic
36#1744—21no—
37C42×C8C2211yescharacteristic
38C2×#334C2214yes—
39C2×#333C2214yes—
40#1296C2214yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50#477—44no—
51#477—44no—
52C2×#107—22no—
53C2×#107—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×C4×C8—21no—
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D412yes—
67C2×C4×C8D411yescharacteristic
68C2×C4×C8C2312yes—
69#384—21no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—21no—
75#384—21no—
76#384C2312yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4D411yescharacteristic
81C4×C4⋊C4D411yescharacteristic
82C4×C4⋊C4C2311yescharacteristic
83C43—21no—
84C43C2311yescharacteristic
85#340—24no—
86#340—24no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336—21no—
95#334—44no—
96#333—44no—
97C22×Q8—22no—
98C22×Q8—41no—
99C22×Q8C2×D411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—44no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—82no—
111C4⋊Q8—82no—
112C4⋊Q8—82no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8SD1614yes—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42C2411yescharacteristic
150#107—44no—
151#107—44no—
152C4⋊C8—22no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—22no—
156C4⋊C8—44no—
157C4⋊C8—42no—
158C4⋊C8—42no—
159C4⋊C8—42no—
160C4⋊C8—42no—
161#103—44no—
162#103—44no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—42no—
167C4×C8—42no—
168C4×C8Q1614yes—
169#95—21no—
170#95—21no—
171#95—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—44no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×Q8C2×SD1612yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8C2×Q1614yes—
212C4⋊C4—21no—
213C4⋊C4—24no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4C2×SD1612yes—
235C42—22no—
236C42—24no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42C2×SD1612yes—
245C42C2×Q1612yes—
246C42#13512yes—
247C42#13712yes—
248C42#13712yes—
249C42#13712yes—
250C42H(5,𝔽2)12yes—
251C422−1+412yes—
252C23C2×#12111yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—41no—
256Q8—41no—
257Q8—164no—
258Q8—164no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#46212yes—
284C2×C4#46212yes—
285C2×C4#46512yes—
286C2×C4#46512yes—
287C2×C4#47012yes—
288C2×C4#47012yes—
289C2×C4#47112yes—
290C2×C4#47212yes—
291C2×C4#53511yescharacteristic
292C2×C4#53611yescharacteristic
293C2×C4#53712yes—
294C2×C4#54212yes—
295C2×C4C22×SD1611yescharacteristic
296C2×C4C22×Q1611yescharacteristic
297C2×C4C2×#13511yescharacteristic
298C2×C4C2×#13711yescharacteristic
299C2×C4C2×#13711yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4#57512yes—
302C8—24no—
303C8—42no—
304C8—82no—
305C8—82no—
306C8—82no—
307C22#262011yescharacteristic
308C22C2×#46211yescharacteristic
309C22#304011yescharacteristic
310C22C2×#46511yescharacteristic
311C22C2×#47011yescharacteristic
312C22C2×#47111yescharacteristic
313C22C2×#47211yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—42no—
318C4—42no—
319C4—81no—
320C4—81no—
321C4—162no—
322C4—162no—
323C4#167214yes—
324C4#169614yes—
325C4#320611yescharacteristic
326C4#321011yescharacteristic
327C4#322911yescharacteristic
328C4#323111yescharacteristic
329C4#324912yes—
330C4#325212yes—
331C2C2×#167211yescharacteristic
332C2C2×#169611yescharacteristic
333C2#1927411yescharacteristic
334C2#1930211yescharacteristic
335C2#2983011yescharacteristic
336C2#2984511yescharacteristic
337C2#2985811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332892
#331982
#213392
#105892
#104822
#98962
#96912
#88052