Finite Group Databasebeta

← #101712

Subgroups — #101712

Order 512.

433 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29208C211yesmaximal, characteristic
2#23149C211yesmaximal, characteristic
3#9898C211yesmaximal, characteristic
4#9691C211yesmaximal, characteristic
5#8586C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×C8×Q8—21no—
9#2868C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2057C414yes—
12#1990—21no—
13#1984—21no—
14#1947—22no—
15#1939—21no—
16#1916—22no—
17#1845—21no—
18#1842C2211yescharacteristic
19#1842C2211yescharacteristic
20C8×C4⋊C4—21no—
21C4×C4⋊C8C2211yescharacteristic
22C4×C4⋊C8C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C2×#333—21no—
25C2×#333—21no—
26C2×#332—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×Q8—21no—
35C8×Q8—42no—
36C8×Q8—42no—
37C2×#107—24no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4×C8—22no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54#391—21no—
55#389—21no—
56#389C2×C412yes—
57#384—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384D411yescharacteristic
62C4×C4⋊C4—21no—
63C4×C4⋊C4D412yes—
64C43C2311yescharacteristic
65#340—24no—
66#336—22no—
67#336—22no—
68#336—21no—
69#336—21no—
70#336C2×C412yes—
71#333—42no—
72#333—42no—
73#332—42no—
74C22×Q8—21no—
75C22×Q8—22no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8—41no—
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4⋊Q8—81no—
90C4⋊Q8—81no—
91C4⋊Q8—82no—
92C4⋊Q8SD1612yes—
93C4⋊Q8Q1612yes—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C22⋊C412yes—
110C2×C42—22no—
111C2×C42C22×C411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118#107—84no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—44no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—81no—
137C4⋊C8—81no—
138C4⋊C8D812yes—
139C4⋊C8Q1612yes—
140C4×C8—44no—
141C4×C8—41no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—41no—
146C4×C8—41no—
147#95—22no—
148#95—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—82no—
158C2×Q8—82no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4#12011yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12311yescharacteristic
169C2×C8—24no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4#10314yes—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42C2×D811yescharacteristic
220C42C2×SD1611yescharacteristic
221C42C2×Q1611yescharacteristic
222C42C2×Q1611yescharacteristic
223C42#13511yescharacteristic
224C42#13511yescharacteristic
225C42#13511yescharacteristic
226C42#13511yescharacteristic
227C42AGL(1,ℤ/8)11yescharacteristic
228C42AGL(1,ℤ/8)11yescharacteristic
229C42#13711yescharacteristic
230C42#13711yescharacteristic
231C23#38611yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—42no—
235Q8—42no—
236Q8—162no—
237Q8—162no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C2×#10311yescharacteristic
257C2×C4#41611yescharacteristic
258C2×C4C4×D811yescharacteristic
259C2×C4C4×Q1611yescharacteristic
260C2×C4#44111yescharacteristic
261C2×C4#44211yescharacteristic
262C2×C4#44711yescharacteristic
263C2×C4#44911yescharacteristic
264C2×C4#45111yescharacteristic
265C2×C4#45211yescharacteristic
266C2×C4#45611yescharacteristic
267C2×C4#45611yescharacteristic
268C2×C4#45911yescharacteristic
269C2×C4#46211yescharacteristic
270C2×C4#46311yescharacteristic
271C2×C4#46411yescharacteristic
272C2×C4#46511yescharacteristic
273C2×C4#46511yescharacteristic
274C2×C4#46711yescharacteristic
275C2×C4#46711yescharacteristic
276C2×C4#47111yescharacteristic
277C2×C4#47111yescharacteristic
278C2×C4#47211yescharacteristic
279C2×C4#47311yescharacteristic
280C8—44no—
281C8—42no—
282C8—42no—
283C8—81no—
284C8—81no—
285C8—161no—
286C8—161no—
287C22#187811yescharacteristic
288C22#187911yescharacteristic
289C22#188611yescharacteristic
290C22#189211yescharacteristic
291C22#190711yescharacteristic
292C22#190811yescharacteristic
293C22#191411yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—44no—
301C4—41no—
302C4—82no—
303C4—162no—
304C4#164711yescharacteristic
305C4#164811yescharacteristic
306C4#164911yescharacteristic
307C4#165011yescharacteristic
308C4#166511yescharacteristic
309C4#166911yescharacteristic
310C4#167211yescharacteristic
311C4#167611yescharacteristic
312C4#167811yescharacteristic
313C4#167911yescharacteristic
314C4#168311yescharacteristic
315C4#168411yescharacteristic
316C2#905811yescharacteristic
317C2#921611yescharacteristic
318C2#921911yescharacteristic
319C2#926811yescharacteristic
320C2#927311yescharacteristic
321C2#939211yescharacteristic
322C2#943411yescharacteristic
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
#292082
#231492
#98982
#96912
#85862
#70312