Finite Group Databasebeta

← #492797

Subgroups — #492797

Order 512.

394 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33952C212yesmaximal
2Q162C211yesmaximal, characteristic, complemented
3#20006C211yesmaximal, characteristic
4#12521C211yesmaximal, characteristic, complemented
5#12461C212yesmaximal, complemented
6#12448C211yesmaximal, characteristic, complemented
7#12447C211yesmaximal, characteristic
8#12411C212yesmaximal
9#12410C212yesmaximal, complemented
10#12054C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic
12Q8×Q16—21no—
13Q8×Q16C2212yes—
14#3380C2212yes—
15#3338—21no—
16#2255—22no—
17#2252C2212yes—
18#2252C2212yes—
19#2229—22no—
20#2227C2212yes—
21#2226C2212yes—
22#2224—22no—
23#2183C2212yes—
24#2175—22no—
25C4×SD32C2211yescharacteristic
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1676—21no—
29#1676C2212yes—
30#1676C2212yes—
31#1675—21no—
32#1633—22no—
33#1632—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1365C2212yes—
41Q8⋊C16C2212yes—
42#1332C2211yescharacteristic
43#1331C2211yescharacteristic
44C8×C16C2211yescharacteristic
45Q82—22no—
46#554—22no—
47C2×SD32D411yescharacteristic
48#500—22no—
49#500—21no—
50#500—21no—
51#500—41no—
52#500C2312yes—
53#500C2312yes—
54#499—22no—
55#494—21no—
56#494—21no—
57#484—42no—
58#478—42no—
59#477—22no—
60#477—22no—
61#477—22no—
62#477—42no—
63#477—42no—
64#476—22no—
65#467—41no—
66C8×Q8—21no—
67C8×Q8C2312yes—
68C4×Q16—22no—
69C4×Q16—41no—
70C4×Q16D411yescharacteristic
71C4×Q16C2311yescharacteristic
72C4×D8—22no—
73C4×D8C2311yescharacteristic
74C8×D4—21no—
75C2×#107—41no—
76#367—22no—
77#367D411yescharacteristic
78#366—22no—
79#366—22no—
80#366—22no—
81#366—42no—
82C4⋊C16—22no—
83C4⋊C16C2312yes—
84#358—22no—
85#358D411yescharacteristic
86#357—22no—
87#357D411yescharacteristic
88#346—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#333—22no—
96Q8⋊C8—21no—
97Q8⋊C8D412yes—
98D4⋊C8—21no—
99C82C2311yescharacteristic
100C2×Q16—81no—
101C2×Q16—81no—
102C2×Q16D811yescharacteristic
103C2×Q16C2×D411yescharacteristic
104C2×D8C2×D411yescharacteristic
105C22×C8—41no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—82no—
111#125—42no—
112C22⋊Q8—42no—
113C22⋊Q8—82no—
114C4×Q8—21no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8C2×D412yes—
118C4×D4—21no—
119C4×D4—42no—
120C2×C4⋊C4—42no—
121SD32Q1614yescomplemented
122C2×C16—21no—
123C2×C16—22no—
124C2×C16C2×D411yescharacteristic
125C2×C16C2×D411yescharacteristic
126#107—22no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—42no—
131#107—41no—
132#107—41no—
133#107—41no—
134#107—41no—
135#107—42no—
136#107—42no—
137#107D811yescharacteristic
138#107C2×D411yescharacteristic
139#107C2×D411yescharacteristic
140#106—42no—
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—41no—
145C4⋊C8C2×D412yes—
146#103—21no—
147#103—42no—
148#103—42no—
149#103—42no—
150#103D812yes—
151#102—21no—
152#102—42no—
153C22⋊C8—41no—
154C4×C8—22no—
155C4×C8—41no—
156C4×C8C2×D411yescharacteristic
157C4×C8C2×D411yescharacteristic
158C4×C8C2411yescharacteristic
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—82no—
163C2×Q8C2×D812yes—
164C2×D4—21no—
165C22×C4—41no—
166C22×C4—82no—
167Q16—21no—
168Q16—22no—
169Q16—162no—
170Q16—162no—
171Q16C2×Q1612yes—
172D8—21no—
173D8C2×Q1612yes—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—82no—
179C2×C8C2×D811yescharacteristic
180C2×C8C2×D811yescharacteristic
181C2×C8C22×D411yescharacteristic
182C2×C8H(5,𝔽2)11yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4C2×D812yes—
196C22⋊C4—41no—
197C22⋊C4—82no—
198C42—22no—
199C42—41no—
200C42—82no—
201C42C22×D411yescharacteristic
202C16—22no—
203C16—21no—
204C16—42no—
205C16C2×Q1612yes—
206C23—41no—
207Q8—22no—
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—162no—
212Q8—162no—
213Q8—322no—
214D4—22no—
215D4—41no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—322no—
226C2×C4D4211yescharacteristic
227C2×C4C22×D811yescharacteristic
228C2×C4#57611yescharacteristic
229C8—22no—
230C8—21no—
231C8—21no—
232C8—42no—
233C8—81no—
234C8—81no—
235C8#51012yes—
236C8C22×Q1611yescharacteristic
237C8#57811yescharacteristic
238C22—42no—
239C22—81no—
240C22—81no—
241C22D4×D811yescharacteristic
242C4—21no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—81no—
247C4—81no—
248C4—162no—
249C4—162no—
250C4D4×Q1611yescharacteristic
251C4#329311yescharacteristic
252C4C2×#51011yescharacteristic
253C4#341711yescharacteristic
254C2—82no—
255C2D8×Q1611yescharacteristic
256C2#3384211yescharacteristic
257C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D8C2×D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q162C2semidirect
4. #12521C2semidirect
5. #12461C2semidirect
6. #12448C2semidirect
9. #12410C2semidirect
10. #12054C2semidirect
121. SD32Q16semidirect

Maximal subgroups

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

subgroupindex
#339522
Q1622
#200062
#125212
#124612
#124482
#124472
#124112
#124102
#120542
C8×SD322