Finite Group Databasebeta

← #504321

Subgroups — #504321

Order 512.

496 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#23184C211yesmaximal, characteristic
3#13529C212yesmaximal
4#13475C212yesmaximal
5#13253C211yesmaximal, characteristic
6#13203C211yesmaximal, characteristic
7#13176C211yesmaximal, characteristic
8#13159C211yesmaximal, characteristic
9#12938C212yesmaximal
10#12814C212yesmaximal
11C4×#367C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C4×Q16C2212yes—
14#2571—21no—
15#2365—21no—
16C2×#367C2211yescharacteristic
17C2×#367C2211yescharacteristic
18C2×#366C2212yes—
19C2×C4⋊C16C2211yescharacteristic
20C2×C4⋊C16C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×C4×C16C2211yescharacteristic
26#2085—22no—
27#1981—21no—
28#1969C2212yes—
29#1948—21no—
30#1942C2212yes—
31#1878—22no—
32#1846C2212yes—
33#1845C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C4×#107C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1380C2211yescharacteristic
38#1380C2212yes—
39#1380C2211yescharacteristic
40#1374C2212yes—
41C22×Q16—21no—
42C22×Q16C2312yes—
43C2×C4⋊Q8—21no—
44C2×C4×Q8—22no—
45C22×C16C2311yescharacteristic
46C22×C16C2312yes—
47C22×C16C2311yescharacteristic
48#494—22no—
49#494Q814yes—
50C4×Q16—24no—
51C4×Q16—42no—
52C2×#107—21no—
53C2×#107C2312yes—
54C2×#107C2311yescharacteristic
55C2×#107C2311yescharacteristic
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#106—22no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—22no—
61C2×#103—22no—
62C2×C4×C8C2312yes—
63C2×C4×C8C2311yescharacteristic
64#400—42no—
65#391—22no—
66#389—22no—
67#389—41no—
68#389—41no—
69#387—42no—
70#384—21no—
71#382—22no—
72#382—41no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4D411yescharacteristic
77#367—22no—
78#367—22no—
79#366—42no—
80C4⋊C16—22no—
81C4⋊C16—42no—
82#358—24no—
83#358—42no—
84#358—42no—
85#358—42no—
86C4×C16—22no—
87#340—21no—
88#340—22no—
89#340—21no—
90#336D412yes—
91C22×Q8—22no—
92C2×Q16—24no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—81no—
99C2×Q16—81no—
100C2×Q16C2×Q814yes—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—24no—
105C4⋊Q8—42no—
106C4×Q8—44no—
107C4×Q8—82no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4D811yescharacteristic
120C2×C4⋊C4D811yescharacteristic
121C2×C42—42no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C16—21no—
127C2×C16—21no—
128C2×C16—21no—
129C2×C16—22no—
130C2×C16—21no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—21no—
134C2×C16#4112yes—
135#107—21no—
136#107—22no—
137#107—21no—
138#107—21no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—22no—
143#107—21no—
144#107—81no—
145#107#4112yes—
146#106—82no—
147C4⋊C8—22no—
148#103—44no—
149#103—82no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8C2×Q812yes—
153C4×C8#4112yes—
154#95—41no—
155#95—41no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95D812yes—
162C2×Q8—24no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4C2×D812yes—
175C22×C4C2×D812yes—
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C22×D411yescharacteristic
179Q16—24no—
180Q16—24no—
181Q16—42no—
182Q16—42no—
183Q16—84no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8C22⋊Q812yes—
190C2×C8C22×Q811yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C2×C8C2×#4111yescharacteristic
193C2×C8C2×#4111yescharacteristic
194C2×C8H(5,𝔽2)11yescharacteristic
195C2×C8H(5,𝔽2)11yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C4⋊C4—161no—
213C4⋊C4—162no—
214C42—22no—
215C42—22no—
216C42—84no—
217C42—82no—
218C42—82no—
219C42C22⋊Q812yes—
220C42#13712yes—
221C16—22no—
222C16—41no—
223C16—42no—
224C23C22×D811yescharacteristic
225Q8—44no—
226Q8—44no—
227Q8—82no—
228Q8—82no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#47412yes—
256C2×C4#47412yes—
257C2×C4#50812yes—
258C2×C4#51012yes—
259C2×C4C2×C22⋊Q811yescharacteristic
260C2×C4#54611yescharacteristic
261C2×C4#55011yescharacteristic
262C2×C4C2×#13711yescharacteristic
263C2×C4#57611yescharacteristic
264C2×C4#57711yescharacteristic
265C8—22no—
266C8—42no—
267C8#52911yescharacteristic
268C8#54011yescharacteristic
269C8#55411yescharacteristic
270C8#55411yescharacteristic
271C22C2×#47411yescharacteristic
272C22#329411yescharacteristic
273C22#339111yescharacteristic
274C22C2×#50811yescharacteristic
275C22C2×#51011yescharacteristic
276C22#341611yescharacteristic
277C22#341711yescharacteristic
278C4—22no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—84no—
283C4—161no—
284C4—161no—
285C4—162no—
286C4—161no—
287C4#222914yes—
288C4#245411yescharacteristic
289C4#308111yescharacteristic
290C4#333311yescharacteristic
291C4#340011yescharacteristic
292C2#2652611yescharacteristic
293C2C2×#222911yescharacteristic
294C2#3363211yescharacteristic
295C2#3386911yescharacteristic
296C2#3389611yescharacteristic
297C2#3396111yescharacteristic
298C2#3396511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#288282
#231842
#135292
#134752
#132532
#132032
#131762
#131592
#129382
#128142
C4×#3672