Finite Group Databasebeta

← #101714

Subgroups — #101714

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29141C211yesmaximal, characteristic
2#23247C211yesmaximal, characteristic
3#9898C211yesmaximal, characteristic
4#9691C211yesmaximal, characteristic
5#8587C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2024C414yes—
11#1984—22no—
12#1956—21no—
13#1939—21no—
14#1939—21no—
15#1842C2211yescharacteristic
16#1842C2211yescharacteristic
17#1834—21no—
18#1803—22no—
19C8×C4⋊C4—21no—
20C8×C4⋊C4—21no—
21C4×C4⋊C8C2211yescharacteristic
22C4×C4⋊C8C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C2×#333—21no—
25C2×#333—21no—
26C2×#333—21no—
27#1296—21no—
28#1296—21no—
29#1296—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8C2×C412yes—
33C2×C4⋊Q8D411yescharacteristic
34C2×C4×Q8D412yes—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×#103—24no—
48C2×C4×C8—21no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—22no—
52#391C2×C412yes—
53#389—21no—
54#389—21no—
55#384—21no—
56#384—21no—
57#384—21no—
58#384—21no—
59#384D411yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C43C2311yescharacteristic
63#340—24no—
64#336—22no—
65#336—22no—
66#336—21no—
67#336—21no—
68#336C2×C412yes—
69#333—42no—
70#333—42no—
71#333—42no—
72C22×Q8—22no—
73C22×Q8C22⋊C412yes—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—81no—
88C4⋊Q8—81no—
89C4⋊Q8—82no—
90C4⋊Q8SD1612yes—
91C4⋊Q8Q1612yes—
92C4×Q8—42no—
93C4×Q8—42no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C22⋊C412yes—
106C2×C42—22no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—81no—
131C4⋊C8—81no—
132C4⋊C8—82no—
133C4⋊C8SD1612yes—
134C4⋊C8Q1612yes—
135#103—84no—
136C4×C8—41no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—41no—
141C4×C8—41no—
142C4×C8—82no—
143#95—22no—
144#95—22no—
145C2×Q8—22no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—82no—
151C2×Q8—82no—
152C2×Q8#10314yes—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4C4×D411yescharacteristic
159C22×C4#12011yescharacteristic
160C22×C4#12111yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4#12311yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—81no—
182C2×C8—81no—
183C2×C8—81no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—42no—
211C42—42no—
212C42C2×SD1611yescharacteristic
213C42C2×SD1611yescharacteristic
214C42C2×Q1611yescharacteristic
215C42C2×Q1611yescharacteristic
216C42#13511yescharacteristic
217C42#13511yescharacteristic
218C42#13511yescharacteristic
219C42#13511yescharacteristic
220C42AGL(1,ℤ/8)11yescharacteristic
221C42#13711yescharacteristic
222C42#13711yescharacteristic
223C42#13711yescharacteristic
224C23#38611yescharacteristic
225Q8—44no—
226Q8—82no—
227Q8—162no—
228Q8—162no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4C2×#10311yescharacteristic
248C2×C4#41611yescharacteristic
249C2×C4C4×SD1611yescharacteristic
250C2×C4C4×Q1611yescharacteristic
251C2×C4#44011yescharacteristic
252C2×C4#44111yescharacteristic
253C2×C4#44811yescharacteristic
254C2×C4#44911yescharacteristic
255C2×C4#45111yescharacteristic
256C2×C4#45211yescharacteristic
257C2×C4#45511yescharacteristic
258C2×C4#45511yescharacteristic
259C2×C4#46011yescharacteristic
260C2×C4#46211yescharacteristic
261C2×C4#46411yescharacteristic
262C2×C4#46411yescharacteristic
263C2×C4#46511yescharacteristic
264C2×C4#46511yescharacteristic
265C2×C4#46711yescharacteristic
266C2×C4#46711yescharacteristic
267C2×C4#47111yescharacteristic
268C2×C4#47111yescharacteristic
269C2×C4#47211yescharacteristic
270C2×C4#47311yescharacteristic
271C8—42no—
272C8—42no—
273C8—82no—
274C8—81no—
275C8—81no—
276C8—161no—
277C8—161no—
278C22#187811yescharacteristic
279C22#188011yescharacteristic
280C22#188511yescharacteristic
281C22#189211yescharacteristic
282C22#190711yescharacteristic
283C22#190811yescharacteristic
284C22#191411yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—41no—
293C4—82no—
294C4—162no—
295C4#164311yescharacteristic
296C4#164411yescharacteristic
297C4#164511yescharacteristic
298C4#164611yescharacteristic
299C4#166611yescharacteristic
300C4#167011yescharacteristic
301C4#167211yescharacteristic
302C4#167611yescharacteristic
303C4#167811yescharacteristic
304C4#168011yescharacteristic
305C4#168411yescharacteristic
306C4#168411yescharacteristic
307C2#906011yescharacteristic
308C2#921611yescharacteristic
309C2#922011yescharacteristic
310C2#926811yescharacteristic
311C2#927411yescharacteristic
312C2#939111yescharacteristic
313C2#943311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#291412
#232472
#98982
#96912
#85872
#70312