Finite Group Databasebeta

← #137265

Subgroups — #137265

Order 512.

415 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 82 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#23696C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic, complemented
3#11667C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic, complemented
5#7301C212yesmaximal, complemented
6C2×C4×D8C2211yescharacteristic
7C2×#432—21no—
8C2×#431C2211yescharacteristic
9C4×#121—21no—
10#2163C2212yes—
11#2086C411yescharacteristic
12#2075C411yescharacteristic
13#2067—21no—
14#2054C411yescharacteristic
15#2033—21no—
16#1999C411yescharacteristic
17C4×#107C2211yescharacteristic
18C4×#102—21no—
19#1573—24no—
20#1568—21no—
21#1568C2212yes—
22#1333—24no—
23#1322—21no—
24#1314—21no—
25C22×D8C2×C411yescharacteristic
26C2×#121—21no—
27C2×C4×D4—21no—
28C2×C4×D4—41no—
29C22×C42D411yescharacteristic
30C4×D8—22no—
31C4×D8—22no—
32#432—22no—
33#432—22no—
34#432—41no—
35#432—41no—
36#431—22no—
37#431—22no—
38C2×#107—21no—
39C2×#107C2×C411yescharacteristic
40C2×C4⋊C8—22no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×#102—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C22⋊C8—41no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49C2×#97—41no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#390—21no—
53#389—21no—
54#388—41no—
55#386—41no—
56#380—21no—
57C4×C4⋊C4—21no—
58C4×C22⋊C4—41no—
59C4⋊C16—22no—
60C22⋊C16—22no—
61#346—22no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#335—22no—
67#335—22no—
68#334—22no—
69#334D414yescomplemented
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C2×D8—22no—
77C2×D8—42no—
78C22×C8—42no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83#121—42no—
84#121—42no—
85C4×D4—41no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—84no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42SD1611yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C16—42no—
109#107—22no—
110#107—42no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—24no—
114C4⋊C8—42no—
115#102—42no—
116#102—42no—
117#102—42no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—42no—
123#97—42no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8C2×D412yes—
127C4×C8#4112yes—
128#95—41no—
129#95—41no—
130#95—41no—
131C24—41no—
132C24#10211yescharacteristic
133C2×D4—22no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—44no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—82no—
141C2×D4—82no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166D8—24no—
167D8—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—44no—
175C2×C8—84no—
176C2×C8C4×D412yes—
177C2×C8#12112yes—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—82no—
185C22⋊C4—44no—
186C22⋊C4—82no—
187C22⋊C4—82no—
188C22⋊C4—82no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—82no—
198C42#12011yescharacteristic
199C42#12311yescharacteristic
200C42AGL(1,ℤ/8)11yescharacteristic
201C42#13711yescharacteristic
202C16—82no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—81no—
209C23—81no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#36112yes—
215C23C2×#10211yescharacteristic
216D4—42no—
217D4—42no—
218D4—81no—
219D4—88no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
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—81no—
248C2×C4—81no—
249C2×C4—162no—
250C2×C4—162no—
251C2×C4#38611yescharacteristic
252C2×C4#41711yescharacteristic
253C2×C4#44711yescharacteristic
254C2×C4#44811yescharacteristic
255C2×C4#48011yescharacteristic
256C2×C4#48111yescharacteristic
257C8—22no—
258C8—22no—
259C8—41no—
260C8—41no—
261C8—84no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22—161no—
276C22#187411yescharacteristic
277C22C2×#36111yescharacteristic
278C22#214811yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—82no—
288C4—161no—
289C4—161no—
290C4#188411yescharacteristic
291C4#189111yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—162no—
296C2#903511yescharacteristic
297C2#1300411yescharacteristic
298C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#1999C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17818C2semidirect
3. #11667C2semidirect
4. #11460C2semidirect
5. #7301C2semidirect
69. #334D4semidirect

Maximal subgroups

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

subgroupindex
#236962
#178182
#116672
#114602
#73012