Finite Group Databasebeta

← #120327

Subgroups — #120327

Order 512.

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

normal subgroups only

This group has 510 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×#1837C211yesmaximal, characteristic
3C2×#1309C211yesmaximal, characteristic
4#8041C212yesmaximal
5#7897C212yesmaximal
6C2×#423C2211yescharacteristic
7C2×#423C2212yes—
8C2×#423C2211yescharacteristic
9C22×C4×C8C2211yescharacteristic
10#2285C2212yes—
11#1948—22no—
12#1947—22no—
13#1837—22no—
14#1837C414yescomplemented
15#1809—22no—
16C2×#341—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×#107—22no—
28C2×#106—22no—
29#423—22no—
30#423—22no—
31#423—22no—
32#423—22no—
33#423—22no—
34#423—21no—
35#423C2×C412yes—
36#423C2×C412yes—
37#423D412yes—
38#423Q814yes—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C412yes—
47C2×C4×C8D412yes—
48C2×C4×C8Q812yes—
49#384D412yes—
50#384D412yes—
51#382—22no—
52#380—22no—
53#380—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56#341—28no—
57#340—24no—
58C23×C4C2×D411yescharacteristic
59C23×C4#4112yes—
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—24no—
63C2×M16—41no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—82no—
70C2×M16—82no—
71C2×M16C22⋊C414yes—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—42no—
82#117—82no—
83#117—82no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C22⋊C4—42no—
91C2×C42—42no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×Q811yescharacteristic
97C2×C42C2×Q812yes—
98C2×C42#4111yescharacteristic
99C2×C42#4111yescharacteristic
100C2×C42#4112yes—
101#107—82no—
102#106—82no—
103C4⋊C8—24no—
104C4⋊C8—24no—
105C4⋊C8—24no—
106C4⋊C8—24no—
107C4⋊C8—24no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4×C8—22no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8C4⋊C414yes—
115#95—42no—
116#95—42no—
117C24#12411yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C4×Q812yes—
130C22×C4#12011yescharacteristic
131C22×C4#12111yescharacteristic
132C22×C4#12112yes—
133C22×C4C22⋊Q811yescharacteristic
134C22×C4C22⋊Q812yes—
135C22×C4#12311yescharacteristic
136C22×C4#12312yes—
137C22×C4#12512yes—
138C22×C4#12612yes—
139C22×C4C4⋊Q811yescharacteristic
140M16—44no—
141M16—44no—
142M16—82no—
143M16—82no—
144M16—84no—
145M16—84no—
146C2×C8—24no—
147C2×C8—24no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8—82no—
161C2×C8—82no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C22⋊C4—84no—
173C42—82no—
174C42—82no—
175C42C2×C4⋊C411yescharacteristic
176C42#11711yescharacteristic
177C42C4×D412yes—
178C42#12111yescharacteristic
179C42#12112yes—
180C42C22⋊Q811yescharacteristic
181C42C22⋊Q812yes—
182C42C22⋊Q812yes—
183C42#12512yes—
184C42C4⋊Q812yes—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23#39111yescharacteristic
190C23#39411yescharacteristic
191C23#40011yescharacteristic
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4#38011yescharacteristic
219C2×C4#38611yescharacteristic
220C2×C4#38712yes—
221C2×C4#38912yes—
222C2×C4#39511yescharacteristic
223C2×C4#39612yes—
224C2×C4#39812yes—
225C2×C4#39811yescharacteristic
226C2×C4#45411yescharacteristic
227C2×C4#45411yescharacteristic
228C2×C4#45511yescharacteristic
229C2×C4#45511yescharacteristic
230C2×C4#47112yes—
231C2×C4#47112yes—
232C2×C4#47212yes—
233C2×C4#47312yes—
234C8—44no—
235C8—44no—
236C8—84no—
237C8—82no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22#144111yescharacteristic
246C22#204011yescharacteristic
247C22#204011yescharacteristic
248C22#205112yes—
249C22#205212yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—162no—
255C4—162no—
256C4#188511yescharacteristic
257C4#188811yescharacteristic
258C4#195112yes—
259C4#195212yes—
260C4#206111yescharacteristic
261C4#206111yescharacteristic
262C4#206312yes—
263C4#206412yes—
264C2—41no—
265C2—41no—
266C2#1053111yescharacteristic
267C2#1053312yes—
268C2#1053511yescharacteristic
269C2#1121111yescharacteristic
270C2#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
14. #1837C4semidirect

Maximal subgroups

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

subgroupindex
#219642
C2×#18372
C2×#13092
#80412
#78972