Finite Group Databasebeta

← #504068

Subgroups — #504068

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28737C211yesmaximal, characteristic
2Q8×#107C211yesmaximal, characteristic
3#13530C212yesmaximal
4#13481C212yesmaximal
5#13263C211yesmaximal, characteristic
6#13253C211yesmaximal, characteristic
7#13203C211yesmaximal, characteristic
8#13176C211yesmaximal, characteristic
9#13133C212yesmaximal
10#13133C212yesmaximal
11C4×#366C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2568—21no—
15Q8×C4⋊C4—21no—
16C2×#366C2211yescharacteristic
17C2×#366C2211yescharacteristic
18C2×#366C2212yes—
19C2×C4⋊C16C2211yescharacteristic
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×C4×C16C2211yescharacteristic
25#2056—22no—
26#1981—21no—
27#1969C2212yes—
28#1969C2212yes—
29#1947—22no—
30#1947—21no—
31#1845C2211yescharacteristic
32#1845C2212yes—
33C4×#107C2211yescharacteristic
34C4×#107C2212yes—
35C4×#107C2211yescharacteristic
36#1380C2211yescharacteristic
37#1380C2212yes—
38#1380C2211yescharacteristic
39#1374C2212yes—
40#1374C2212yes—
41C22×Q16—21no—
42C22×Q16C2312yes—
43C2×C4⋊Q8—21no—
44C2×C4×Q8—41no—
45C2×C4×Q8D411yescharacteristic
46C22×C16C2311yescharacteristic
47C22×C16C2312yes—
48C22×C16C2311yescharacteristic
49#494—22no—
50#494—22no—
51C8×Q8—22no—
52C8×Q8—22no—
53C2×#107—22no—
54C2×#107—21no—
55C2×#107C2312yes—
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×C4⋊C8D412yes—
61C2×C4⋊C8D411yescharacteristic
62C2×#103—22no—
63C2×#103—22no—
64C2×C4×C8C2312yes—
65C2×C4×C8C2311yescharacteristic
66#400—42no—
67#391—22no—
68#391—22no—
69#391—41no—
70#389—41no—
71#389—42no—
72#384—22no—
73#384—21no—
74#382—41no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—21no—
78#366—22no—
79#366—22no—
80#366—42no—
81C4⋊C16—22no—
82#358—24no—
83#358—24no—
84#358—42no—
85#358—42no—
86C4×C16—22no—
87#340—21no—
88#340—22no—
89#340—21no—
90C22×Q8—22no—
91C22×Q8D811yescharacteristic
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—42no—
99C2×Q16—81no—
100C2×Q16—81no—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—84no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4D811yescharacteristic
121C2×C4⋊C4D812yes—
122C2×C42—42no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C16—21no—
128C2×C16—21no—
129C2×C16—21no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—21no—
133C2×C16—22no—
134C2×C16—21no—
135C2×C16#4112yes—
136#107—21no—
137#107—22no—
138#107—21no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—22no—
143#107—21no—
144#107—21no—
145#107—81no—
146#107—82no—
147#107#4112yes—
148C4⋊C8—24no—
149C4⋊C8—22no—
150#103—82no—
151#103—82no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8#4112yes—
155C4×C8#4112yes—
156#95—42no—
157#95—41no—
158#95—41no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8Q3214yes—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4C2×D812yes—
179C22×C4C2×D811yescharacteristic
180C22×C4C2×D812yes—
181C22×C4C2×D811yescharacteristic
182C22×C4C22×D411yescharacteristic
183Q16—42no—
184Q16—42no—
185Q16—42no—
186Q16—42no—
187Q16—84no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8#12312yes—
194C2×C8C2×#4111yescharacteristic
195C2×C8C2×#4111yescharacteristic
196C2×C8C2×#4111yescharacteristic
197C2×C8C2×#4111yescharacteristic
198C2×C8H(5,𝔽2)11yescharacteristic
199C2×C82−1+411yescharacteristic
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—84no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—161no—
217C4⋊C4—162no—
218C42—22no—
219C42—22no—
220C42—84no—
221C42—82no—
222C42—82no—
223C42#12312yes—
224C42#13712yes—
225C16—22no—
226C16—41no—
227C16—42no—
228C23C22×D811yescharacteristic
229Q8—28no—
230Q8—82no—
231Q8—82no—
232Q8—82no—
233Q8—82no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#48012yes—
260C2×C4#48012yes—
261C2×C4C2×Q3214yes—
262C2×C4C2×Q3212yes—
263C2×C4#50912yes—
264C2×C4C2×#12311yescharacteristic
265C2×C4#54811yescharacteristic
266C2×C4#55011yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57611yescharacteristic
269C2×C4#57711yescharacteristic
270C8—22no—
271C8—42no—
272C8#52911yescharacteristic
273C8#54211yescharacteristic
274C8#55611yescharacteristic
275C8#55911yescharacteristic
276C22C2×#48011yescharacteristic
277C22#335811yescharacteristic
278C22#339111yescharacteristic
279C22C22×Q3211yescharacteristic
280C22C2×#50911yescharacteristic
281C22#341611yescharacteristic
282C22#341711yescharacteristic
283C4—24no—
284C4—22no—
285C4—42no—
286C4—82no—
287C4—161no—
288C4—162no—
289C4—161no—
290C4—161no—
291C4—162no—
292C4#223614yes—
293C4#244811yescharacteristic
294C4#309611yescharacteristic
295C4#337511yescharacteristic
296C4#340011yescharacteristic
297C2#2631611yescharacteristic
298C2C2×#223611yescharacteristic
299C2#3365111yescharacteristic
300C2#3393011yescharacteristic
301C2#3393411yescharacteristic
302C2#3396211yescharacteristic
303C2#3396411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ 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
#287372
Q8×#1072
#135302
#134812
#132632
#132532
#132032
#131762
#131332
C4×#3662