Finite Group Databasebeta

← #120326

Subgroups — #120326

Order 512.

554 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 154 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#21964C211yesmaximal, characteristic
2C2×#1848C211yesmaximal, characteristic
3C2×#1309C211yesmaximal, characteristic
4#8041C212yesmaximal
5#7897C212yesmaximal
6C22×#108—22no—
7C2×#423C2211yescharacteristic
8C2×#423C2212yes—
9C2×#423C2211yescharacteristic
10C22×C4×C8C2211yescharacteristic
11#2285C2212yes—
12#1948—22no—
13#1947—22no—
14#1848—22no—
15#1848C414yescomplemented
16#1809—22no—
17#1309—22no—
18#1296—22no—
19#1296—22no—
20#1279—24no—
21C22×M16—21no—
22C22×M16—22no—
23C22×M16D412yes—
24C23×C8—21no—
25C2×#117—22no—
26C22×C42C2311yescharacteristic
27C2×#108—28no—
28C2×#108—44no—
29C2×#108—44no—
30C2×#107—22no—
31C2×#106—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36#423—22no—
37#423—21no—
38#423C2×C412yes—
39#423C2×C412yes—
40#423D412yes—
41#423Q814yes—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8—21no—
49C2×C4×C8C2×C412yes—
50C2×C4×C8D412yes—
51C2×C4×C8Q812yes—
52#384D412yes—
53#384D412yes—
54#382—22no—
55#380—22no—
56#380—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59#340—24no—
60C23×C4C2×D411yescharacteristic
61C23×C4#4112yes—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—24no—
65C2×M16—41no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—82no—
72C2×M16—82no—
73C2×M16C22⋊C414yes—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—42no—
84#117—82no—
85#117—82no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C22⋊C4—42no—
93C2×C42—42no—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×Q811yescharacteristic
99C2×C42C2×Q812yes—
100C2×C42#4111yescharacteristic
101C2×C42#4111yescharacteristic
102C2×C42#4112yes—
103#108—416no—
104#107—82no—
105#106—82no—
106C4⋊C8—24no—
107C4⋊C8—24no—
108C4⋊C8—24no—
109C4⋊C8—24no—
110C4⋊C8—24no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8C4⋊C414yes—
118#95—42no—
119#95—42no—
120C24#12411yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C4×Q812yes—
133C22×C4#12011yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12112yes—
136C22×C4C22⋊Q811yescharacteristic
137C22×C4C22⋊Q812yes—
138C22×C4#12311yescharacteristic
139C22×C4#12312yes—
140C22×C4#12512yes—
141C22×C4#12612yes—
142C22×C4C4⋊Q811yescharacteristic
143M16—44no—
144M16—44no—
145M16—82no—
146M16—82no—
147M16—84no—
148M16—84no—
149C2×C8—24no—
150C2×C8—24no—
151C2×C8—24no—
152C2×C8—24no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—82no—
164C2×C8—82no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C22⋊C4—84no—
176C42—82no—
177C42—82no—
178C42C2×C4⋊C411yescharacteristic
179C42#11711yescharacteristic
180C42C4×D412yes—
181C42#12111yescharacteristic
182C42#12112yes—
183C42C22⋊Q811yescharacteristic
184C42C22⋊Q812yes—
185C42C22⋊Q812yes—
186C42#12512yes—
187C42C4⋊Q812yes—
188C23—22no—
189C23—22no—
190C23—21no—
191C23—21no—
192C23#39111yescharacteristic
193C23#39411yescharacteristic
194C23#40011yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#38011yescharacteristic
222C2×C4#38611yescharacteristic
223C2×C4#38712yes—
224C2×C4#38912yes—
225C2×C4#39511yescharacteristic
226C2×C4#39612yes—
227C2×C4#39812yes—
228C2×C4#39811yescharacteristic
229C2×C4#45311yescharacteristic
230C2×C4#45311yescharacteristic
231C2×C4#45611yescharacteristic
232C2×C4#45611yescharacteristic
233C2×C4#47112yes—
234C2×C4#47112yes—
235C2×C4#47212yes—
236C2×C4#47312yes—
237C8—28no—
238C8—44no—
239C8—84no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22#144111yescharacteristic
249C22#203911yescharacteristic
250C22#203911yescharacteristic
251C22#205112yes—
252C22#205212yes—
253C4—21no—
254C4—21no—
255C4—22no—
256C4—22no—
257C4—162no—
258C4—162no—
259C4#188611yescharacteristic
260C4#188711yescharacteristic
261C4#195112yes—
262C4#195212yes—
263C4#206011yescharacteristic
264C4#206011yescharacteristic
265C4#206312yes—
266C4#206412yes—
267C2—41no—
268C2—41no—
269C2#1053011yescharacteristic
270C2#1053212yes—
271C2#1053411yescharacteristic
272C2#1121011yescharacteristic
273C2#1124912yes—
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1848C4semidirect

Maximal subgroups

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

subgroupindex
#219642
C2×#18482
C2×#13092
#80412
#78972