Finite Group Databasebeta

← #504237

Subgroups — #504237

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C212yesmaximal
2#28814C211yesmaximal, characteristic
3#23184C211yesmaximal, characteristic
4#13475C212yesmaximal
5#13366C212yesmaximal
6#13253C211yesmaximal, characteristic
7#13203C211yesmaximal, characteristic
8#13177C211yesmaximal, characteristic
9#13159C211yesmaximal, characteristic
10#12938C212yesmaximal
11C4×#366C211yesmaximal, characteristic
12C2×#499C2212yes—
13C2×#494C2211yescharacteristic
14C2×C4×Q16C2212yes—
15#2552—21no—
16#2365—21no—
17#2229—28no—
18C2×#367C2212yes—
19C2×#366C2211yescharacteristic
20C2×#366C2211yescharacteristic
21C2×C4⋊C16C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×#358C2212yes—
26C2×#358C2212yes—
27C2×C4×C16C2211yescharacteristic
28#2059—22no—
29#1981—21no—
30#1969C2212yes—
31#1948—21no—
32#1942C2212yes—
33#1878—22no—
34#1845C2211yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38#1380C2211yescharacteristic
39#1380C2212yes—
40#1380C2211yescharacteristic
41#1374C2212yes—
42C22×Q16—21no—
43C22×Q16C2312yes—
44C2×C4⋊Q8—21no—
45C2×#125—22no—
46C2×C4×Q8—22no—
47C22×C16C2311yescharacteristic
48C22×C16C2312yes—
49C22×C16C2311yescharacteristic
50#499—24no—
51#499—42no—
52#494—22no—
53#494Q814yes—
54C4×Q16—24no—
55C4×Q16—42no—
56C2×#107—21no—
57C2×#107C2312yes—
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#106—22no—
63C2×C4⋊C8D411yescharacteristic
64C2×#103—22no—
65C2×#103—22no—
66C2×C4×C8C2312yes—
67C2×C4×C8C2311yescharacteristic
68#398—42no—
69#391—22no—
70#389—22no—
71#389—41no—
72#389—41no—
73#387—42no—
74#384—21no—
75#384—41no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—41no—
79C4×C4⋊C4D411yescharacteristic
80#367—24no—
81#366—22no—
82#366D414yes—
83C4⋊C16—24no—
84C4⋊C16—22no—
85#358—24no—
86#358—24no—
87#358—24no—
88#358—42no—
89C4×C16—22no—
90#340—21no—
91#340—22no—
92#340—21no—
93#336D412yes—
94C22×Q8—22no—
95C2×Q16—24no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16—81no—
102C2×Q16—81no—
103C2×Q16C2×Q814yes—
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2411yescharacteristic
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109#125—44no—
110#125—82no—
111C4×Q8—44no—
112C4×Q8—82no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4D811yescharacteristic
127C2×C4⋊C4D811yescharacteristic
128C2×C42—42no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C16—21no—
134C2×C16—21no—
135C2×C16—22no—
136C2×C16—21no—
137C2×C16—22no—
138C2×C16—22no—
139C2×C16—21no—
140C2×C16—21no—
141C2×C16C2×D412yes—
142#107—21no—
143#107—21no—
144#107—21no—
145#107—22no—
146#107—21no—
147#107—22no—
148#107—21no—
149#107—81no—
150#107C2×D412yes—
151#107C2×D412yes—
152#107#4112yes—
153#106—44no—
154C4⋊C8—22no—
155#103—44no—
156#103—82no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8C2×Q812yes—
160C4×C8#4112yes—
161#95—41no—
162#95—41no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95D812yes—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4C2×D812yes—
180C22×C4C2×D812yes—
181C22×C4C2×D811yescharacteristic
182C22×C4C2×D811yescharacteristic
183C22×C4C22×D411yescharacteristic
184Q16—24no—
185Q16—24no—
186Q16—42no—
187Q16—42no—
188Q16—84no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8C22⋊Q812yes—
195C2×C8C22×D411yescharacteristic
196C2×C8C22×Q811yescharacteristic
197C2×C8C2×#4111yescharacteristic
198C2×C8C2×#4111yescharacteristic
199C2×C8H(5,𝔽2)11yescharacteristic
200C2×C82−1+411yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—82no—
219C4⋊C4—161no—
220C4⋊C4—162no—
221C42—22no—
222C42—22no—
223C42—84no—
224C42—82no—
225C42—82no—
226C42C22⋊Q812yes—
227C42#13712yes—
228C16—22no—
229C16—42no—
230C16—41no—
231C23C22×D811yescharacteristic
232Q8—44no—
233Q8—44no—
234Q8—82no—
235Q8—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4#47412yes—
263C2×C4#47412yes—
264C2×C4#50812yes—
265C2×C4#51012yes—
266C2×C4C2×C22⋊Q811yescharacteristic
267C2×C4D4211yescharacteristic
268C2×C4#54811yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4#57611yescharacteristic
271C2×C4#57711yescharacteristic
272C8—22no—
273C8—42no—
274C8#53211yescharacteristic
275C8#54211yescharacteristic
276C8D4×Q811yescharacteristic
277C8#55411yescharacteristic
278C22C2×#47411yescharacteristic
279C22D4×D811yescharacteristic
280C22#335811yescharacteristic
281C22C2×#50811yescharacteristic
282C22C2×#51011yescharacteristic
283C22#341611yescharacteristic
284C22#341711yescharacteristic
285C4—22no—
286C4—42no—
287C4—42no—
288C4—82no—
289C4—82no—
290C4—84no—
291C4—82no—
292C4—162no—
293C4—161no—
294C4#222914yescomplemented
295C4#245211yescharacteristic
296C4#308111yescharacteristic
297C4#327811yescharacteristic
298C4#337511yescharacteristic
299C2#2646011yescharacteristic
300C2C2×#222911yescharacteristic
301C2#3363211yescharacteristic
302C2#3384411yescharacteristic
303C2#3384911yescharacteristic
304C2#3392711yescharacteristic
305C2#3394111yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
294. C4#2229semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
#288142
#231842
#134752
#133662
#132532
#132032
#131772
#131592
#129382
C4×#3662