Finite Group Databasebeta

← #138499

Subgroups — #138499

Order 512.

471 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 98 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#29806C211yesmaximal, characteristic
2#17814C211yesmaximal, characteristic, complemented
3#11961C212yesmaximal, complemented
4#11459C211yesmaximal, characteristic, complemented
5#7217C212yesmaximal, complemented
6C2×#493C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9#2856—21no—
10#2162C2212yes—
11#2054—21no—
12#2054C412yes—
13#1999—21no—
14#1999C412yes—
15#1957—21no—
16#1834C2211yescharacteristic
17#1571—24no—
18#1567—21no—
19#1567C2212yes—
20#1361—24no—
21#1322—21no—
22#1314—21no—
23C22×D8—21no—
24C22×D8C2×C412yes—
25C22×M16—41no—
26C2×#127—21no—
27C2×#121—22no—
28C2×#121—41no—
29C2×#121—41no—
30C22×C42D411yescharacteristic
31#493—22no—
32#493—22no—
33#431—22no—
34#431—22no—
35#423—22no—
36#423—22no—
37#423—41no—
38#423—41no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×#102—22no—
44C2×#102—22no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#390—22no—
50#390—41no—
51#390—41no—
52#384—21no—
53C4⋊C16—22no—
54C22⋊C16—22no—
55C4×C16—22no—
56#340—21no—
57#340C2×C412yes—
58#335—24no—
59#335—22no—
60#335D414yescomplemented
61C22×D4—22no—
62C22×D4—42no—
63C22×D4—41no—
64C22×D4—41no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×D8—24no—
69C2×D8—42no—
70C2×D8—82no—
71C2×D8—82no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—82no—
75C2×M16—82no—
76C22×C8—42no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#127—41no—
82#127—41no—
83#127—41no—
84#127—41no—
85#121—82no—
86#121—82no—
87#121—84no—
88#121—82no—
89#121—82no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42SD1611yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C16—42no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—24no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111#102—82no—
112#102—82no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2×D412yes—
118C4×C8#4112yes—
119C24—42no—
120C24#10211yescharacteristic
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—84no—
130C2×D4—84no—
131C2×D4—84no—
132C2×D4—81no—
133C2×D4—81no—
134C2×D4—81no—
135C2×D4—81no—
136C2×D4—81no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4—81no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4D1612yes—
156C22×C4SD3212yes—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C2×SD1611yescharacteristic
160D8—84no—
161D8—84no—
162M16—84no—
163M16—82no—
164M16—82no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—44no—
172C2×C8—84no—
173C2×C8C4×D412yes—
174C2×C8#12112yes—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—84no—
181C22⋊C4—84no—
182C22⋊C4—84no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42#12011yescharacteristic
190C42#12311yescharacteristic
191C42AGL(1,ℤ/8)11yescharacteristic
192C42#13711yescharacteristic
193C16—82no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—41no—
198C23—82no—
199C23—82no—
200C23—82no—
201C23—82no—
202C23—82no—
203C23—82no—
204C23—82no—
205C23#35712yes—
206C23C2×#10211yescharacteristic
207D4—82no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—162no—
212D4—168no—
213D4—161no—
214D4—161no—
215D4—161no—
216D4—161no—
217D4—162no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4#35712yes—
240C2×C4#38611yescharacteristic
241C2×C4#41711yescharacteristic
242C2×C4#44711yescharacteristic
243C2×C4#44811yescharacteristic
244C2×C4#48011yescharacteristic
245C2×C4#48111yescharacteristic
246C2×C4C2×D1611yescharacteristic
247C2×C4C2×SD3211yescharacteristic
248C2×C4#50911yescharacteristic
249C2×C4#51011yescharacteristic
250C8—22no—
251C8—22no—
252C8—41no—
253C8—41no—
254C8—84no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—162no—
263C22—162no—
264C22—162no—
265C22—162no—
266C22—162no—
267C22—162no—
268C22—162no—
269C22#187411yescharacteristic
270C22C2×#35711yescharacteristic
271C22#213911yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—162no—
278C4#191011yescharacteristic
279C4#191411yescharacteristic
280C4#218411yescharacteristic
281C4#218511yescharacteristic
282C4#223211yescharacteristic
283C4#223311yescharacteristic
284C4#223812yes—
285C2—21no—
286C2—21no—
287C2—42no—
288C2—322no—
289C2#919711yescharacteristic
290C2#1299111yescharacteristic
291C2#1301011yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#29806C2
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. #17814C2semidirect
3. #11961C2semidirect
4. #11459C2semidirect
5. #7217C2semidirect
60. #335D4semidirect

Maximal subgroups

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

subgroupindex
#298062
#178142
#119612
#114592
#72172