Finite Group Databasebeta

← #504322

Subgroups — #504322

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic
2#23184C211yesmaximal, characteristic
3#13530C212yesmaximal
4#13475C212yesmaximal
5#13253C211yesmaximal, characteristic
6#13203C211yesmaximal, characteristic
7#13176C211yesmaximal, characteristic
8#13159C211yesmaximal, characteristic
9#12936C212yesmaximal
10#12812C212yesmaximal
11C4×#366C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C4×Q16C2212yes—
14#2552—21no—
15#2365—21no—
16C2×#366C2211yescharacteristic
17C2×#366C2211yescharacteristic
18C2×#366C2212yes—
19C2×C4⋊C16C2211yescharacteristic
20C2×C4⋊C16C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×C4×C16C2211yescharacteristic
26#2083—22no—
27#1981—21no—
28#1969C2212yes—
29#1947—21no—
30#1942C2212yes—
31#1878—22no—
32#1845C2212yes—
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×#125—41no—
45C2×C4×Q8—22no—
46C22×C16C2311yescharacteristic
47C22×C16C2312yes—
48C22×C16C2311yescharacteristic
49#494—22no—
50#494Q814yes—
51C4×Q16—24no—
52C4×Q16—42no—
53C2×#107—21no—
54C2×#107—22no—
55C2×#107C2312yes—
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×C4⋊C8D411yescharacteristic
61C2×#103—22no—
62C2×#103—22no—
63C2×C4×C8C2312yes—
64C2×C4×C8C2311yescharacteristic
65#398—42no—
66#391—22no—
67#389—22no—
68#389—41no—
69#389—41no—
70#387—42no—
71#384—22no—
72#384—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4D411yescharacteristic
77#366—22no—
78#366—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—
106#125—84no—
107C4×Q8—44no—
108C4×Q8—82no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4D811yescharacteristic
123C2×C4⋊C4D811yescharacteristic
124C2×C42—42no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C16—21no—
130C2×C16—21no—
131C2×C16—21no—
132C2×C16—22no—
133C2×C16—21no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—21no—
137C2×C16#4112yes—
138#107—21no—
139#107—22no—
140#107—21no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—21no—
145#107—22no—
146#107—21no—
147#107—81no—
148#107—82no—
149#107#4112yes—
150C4⋊C8—22no—
151#103—44no—
152#103—82no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8C2×Q812yes—
156C4×C8#4112yes—
157#95—41no—
158#95—41no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95D812yes—
163C2×Q8—24no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4C2×D812yes—
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D812yes—
178C22×C4C2×D811yescharacteristic
179C22×C4C22×D411yescharacteristic
180Q16—24no—
181Q16—24no—
182Q16—42no—
183Q16—42no—
184Q16—84no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8C22⋊Q812yes—
191C2×C8C22×Q811yescharacteristic
192C2×C8C2×#4111yescharacteristic
193C2×C8C2×#4111yescharacteristic
194C2×C8C2×#4111yescharacteristic
195C2×C8H(5,𝔽2)11yescharacteristic
196C2×C8H(5,𝔽2)11yescharacteristic
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—84no—
212C4⋊C4—84no—
213C4⋊C4—82no—
214C4⋊C4—162no—
215C4⋊C4—161no—
216C4⋊C4Q3214yes—
217C42—22no—
218C42—22no—
219C42—84no—
220C42—82no—
221C42—82no—
222C42C22⋊Q812yes—
223C42#13712yes—
224C16—22no—
225C16—41no—
226C16—42no—
227C23C22×D811yescharacteristic
228Q8—44no—
229Q8—44no—
230Q8—82no—
231Q8—82no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—82no—
242C2×C4—81no—
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—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4#47412yes—
258C2×C4#47412yes—
259C2×C4C2×Q3214yes—
260C2×C4C2×Q3212yes—
261C2×C4#50912yes—
262C2×C4C2×C22⋊Q811yescharacteristic
263C2×C4#54611yescharacteristic
264C2×C4#55011yescharacteristic
265C2×C4C2×#13711yescharacteristic
266C2×C4#57611yescharacteristic
267C2×C4#57711yescharacteristic
268C8—22no—
269C8—42no—
270C8#52911yescharacteristic
271C8#54011yescharacteristic
272C8#55411yescharacteristic
273C8#55411yescharacteristic
274C22C2×#47411yescharacteristic
275C22#329411yescharacteristic
276C22#339111yescharacteristic
277C22C22×Q3211yescharacteristic
278C22C2×#50911yescharacteristic
279C22#341611yescharacteristic
280C22#341711yescharacteristic
281C4—24no—
282C4—22no—
283C4—42no—
284C4—82no—
285C4—84no—
286C4—161no—
287C4—161no—
288C4—161no—
289C4—162no—
290C4#222714yes—
291C4#245411yescharacteristic
292C4#308111yescharacteristic
293C4#333311yescharacteristic
294C4#340011yescharacteristic
295C2#2652611yescharacteristic
296C2C2×#222711yescharacteristic
297C2#3363211yescharacteristic
298C2#3386611yescharacteristic
299C2#3389711yescharacteristic
300C2#3396211yescharacteristic
301C2#3396411yescharacteristic
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
#288272
#231842
#135302
#134752
#132532
#132032
#131762
#131592
#129362
#128122
C4×#3662