Finite Group Databasebeta

← #138016

Subgroups — #138016

Order 512.

413 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 88 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#33419C211yesmaximal, characteristic, complemented
2#33132C211yesmaximal, characteristic
3#11705C212yesmaximal
4#10814C211yesmaximal, characteristic, complemented
5#7262C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2624—21no—
9#2164—24no—
10#2150—21no—
11#2112C2212yes—
12C4×M32—21no—
13#1972C411yescharacteristic, complemented
14#1972C411yescharacteristic, complemented
15#1944C411yescharacteristic, complemented
16#1944C411yescharacteristic, complemented
17#1934—21no—
18#1934—21no—
19#1895—21no—
20#1807C2211yescharacteristic
21#1740—21no—
22#1735C2212yes—
23C2×H(3,ℤ/4)—21no—
24#1347—24no—
25C22×D8C2×C411yescharacteristic
26C22×M16D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#121—21no—
29C2×#117—22no—
30C2×#117D411yescharacteristic
31C2×M32—22no—
32#466—22no—
33#466—22no—
34C2×#107C2×C411yescharacteristic
35C2×#106C2×C411yescharacteristic
36C2×#106C2×C411yescharacteristic
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40#405—22no—
41#405—22no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#399—41no—
47#397—41no—
48#394—41no—
49#394—41no—
50#390—21no—
51#390—21no—
52#389—21no—
53#389—21no—
54#380—21no—
55C4⋊C16—24no—
56C22⋊C16—22no—
57C22⋊C16D414yes—
58#346—22no—
59C4×C16—22no—
60#342—41no—
61#342—41no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340—22no—
67H(3,ℤ/4)—24no—
68C22×D4—21no—
69C22×D4—41no—
70C23×C4C2×D411yescharacteristic
71C2×D8—22no—
72C2×D8—41no—
73C2×D8—41no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D412yes—
84C22×C8#4112yes—
85#121—42no—
86#121—42no—
87#117—22no—
88#117—22no—
89#117—24no—
90#117—42no—
91#117—42no—
92#117—42no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4SD1611yescharacteristic
100C2×C4⋊C4SD1611yescharacteristic
101C2×C22⋊C4—42no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4D811yescharacteristic
109C2×C22⋊C4D811yescharacteristic
110C2×C42—42no—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113M32—42no—
114M32—42no—
115C2×C16—24no—
116#107—22no—
117#106—41no—
118#106—41no—
119#102—42no—
120#102—42no—
121#102—41no—
122#102—41no—
123#97—22no—
124#97—22no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129#95—41no—
130#95—41no—
131#95—41no—
132C24—41no—
133C24C2×D811yescharacteristic
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—82no—
141C2×D4—161no—
142C2×D4—161no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—82no—
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4#12112yes—
157C22×C4C2×SD1611yescharacteristic
158C22×C4#13512yes—
159D8—82no—
160M16—44no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8#10412yes—
173C2×C8C4×D412yes—
174C2×C8#12011yescharacteristic
175C2×C8#12311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—44no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—82no—
186C4⋊C4—161no—
187C4⋊C4—161no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—44no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—161no—
194C22⋊C4—161no—
195C22⋊C4—161no—
196C22⋊C4—161no—
197C22⋊C4—161no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—44no—
203C42—82no—
204C16—44no—
205C23—21no—
206C23—21no—
207C23—41no—
208C23—41no—
209C23—81no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23C2×#10211yescharacteristic
217C23#46612yes—
218D4—81no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—164no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—84no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—162no—
248C2×C4—162no—
249C2×C4#38611yescharacteristic
250C2×C4#41011yescharacteristic
251C2×C4#41011yescharacteristic
252C2×C4#41611yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#42111yescharacteristic
255C2×C4#46512yes—
256C8—22no—
257C8—21no—
258C8—22no—
259C8—21no—
260C8—41no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—81no—
267C22—81no—
268C22—161no—
269C22—161no—
270C22—161no—
271C22—161no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22#150211yescharacteristic
276C22#151711yescharacteristic
277C22#189311yescharacteristic
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—161no—
284C4—161no—
285C4—164no—
286C4—161no—
287C4#150011yescharacteristic
288C4#152111yescharacteristic
289C4#187111yescharacteristic
290C4#189911yescharacteristic
291C2—21no—
292C2—21no—
293C2—42no—
294C2—321no—
295C2#911411yescharacteristic
296C2#1307411yescharacteristic
297C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33419C2semidirect
4. #10814C2semidirect
5. #7262C2semidirect
13. #1972C4semidirect
14. #1972C4semidirect
15. #1944C4semidirect
16. #1944C4semidirect

Maximal subgroups

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

subgroupindex
#334192
#331322
#117052
#108142
#72622