Finite Group Databasebeta

← #508458

Subgroups — #508458

Order 512.

513 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2228C211yesmaximal, characteristic
2C2×#2226C211yesmaximal, characteristic
3#33319C211yesmaximal, characteristic, complemented
4#21282C211yesmaximal, characteristic
5#13507C212yesmaximal, complemented
6#13506C212yesmaximal, complemented
7#13482C212yesmaximal
8#13480C212yesmaximal
9#13277C211yesmaximal, characteristic, complemented
10#13275C211yesmaximal, characteristic, complemented
11#12880C211yesmaximal, characteristic, complemented
12C2×#501—21no—
13C2×#500C2211yescharacteristic
14C2×#499C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2660—21no—
17#2319—21no—
18#2228—24no—
19#2226—24no—
20C2×#367C2212yes—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2212yes—
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28#1970C2212yes—
29#1942C2212yes—
30#1936—22no—
31#1847—22no—
32#1846C2212yes—
33#1845C2212yes—
34C4×#107C2211yescharacteristic
35#1762—21no—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2212yes—
42#1374C2212yes—
43C22×D8C2311yescharacteristic
44C2×C4⋊Q8—21no—
45C2×#125—21no—
46C2×C4×D4—21no—
47C22×C16C2311yescharacteristic
48C22×C16C2312yes—
49C22×C16C2311yescharacteristic
50#501—44no—
51#500—22no—
52#500—22no—
53#499—22no—
54#499—22no—
55C4×D8—22no—
56C4×D8D414yes—
57C2×#107—22no—
58C2×#107—21no—
59C2×#107C2311yescharacteristic
60C2×#107C2312yes—
61C2×#107C2311yescharacteristic
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#106—22no—
65C2×#106—21no—
66C2×#102—21no—
67C2×#102—22no—
68C2×#97D412yes—
69C2×#97D411yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#398—41no—
73#398—41no—
74#395—42no—
75#390—22no—
76#389—22no—
77#388—42no—
78#384—22no—
79#382—22no—
80#381—41no—
81#380—41no—
82C4×C4⋊C4—21no—
83#376D411yescharacteristic
84#367—24no—
85#367—22no—
86#366—22no—
87C4⋊C16—22no—
88C4⋊C16—22no—
89#357—24no—
90#357—22no—
91#357—22no—
92#340—21no—
93#340—22no—
94C22×Q8—41no—
95C22×D4—21no—
96C23×C4—41no—
97C2×D8—21no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8C2×D412yes—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D412yes—
103C22×C8C2411yescharacteristic
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106#125—42no—
107#125—42no—
108C4×D4—24no—
109C4×D4—42no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C16—21no—
128C2×C16—22no—
129C2×C16—21no—
130C2×C16—22no—
131C2×C16—24no—
132C2×C16—22no—
133#107—21no—
134#107—21no—
135#107—21no—
136#107—22no—
137#107—21no—
138#107—21no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—44no—
143#107—42no—
144#107C2×D412yes—
145#106—44no—
146#106—42no—
147#102—24no—
148#102—44no—
149#97—24no—
150#97—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8C2×D412yes—
154C4×C8#4112yes—
155#95—41no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95D811yescharacteristic
160#95D811yescharacteristic
161#95D812yes—
162C24—41no—
163C2×Q8—82no—
164C2×Q8—161no—
165C2×Q8—161no—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—41no—
169C2×D4—42no—
170C22×C4—41no—
171C22×C4—44no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—162no—
177C22×C4C2×D812yes—
178C22×C4C2×D812yes—
179C22×C4C2×D811yescharacteristic
180C22×C4C2×D811yescharacteristic
181C22×C4C22×D411yescharacteristic
182D8—22no—
183D8—22no—
184D8—42no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8#12112yes—
189C2×C8#13714yes—
190C2×C8C22×D411yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C2×C82−1+412yes—
193C2×C82−1+412yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—162no—
208C4⋊C4—162no—
209C4⋊C4—162no—
210C22⋊C4—44no—
211C22⋊C4—162no—
212C22⋊C4—162no—
213C42—22no—
214C42—22no—
215C42—161no—
216C42#12112yes—
217C42AGL(1,ℤ/8)12yes—
218C16—41no—
219C16—42no—
220C16—41no—
221C23—42no—
222C23—81no—
223C23—82no—
224C23—81no—
225C23—82no—
226C23C22×D811yescharacteristic
227Q8—164no—
228D4—42no—
229D4—42no—
230D4—82no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—164no—
254C2×C4#45912yes—
255C2×C4#45912yes—
256C2×C4C2×#12111yescharacteristic
257C2×C4#53612yes—
258C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
259C2×C4C2×#13712yes—
260C8—24no—
261C8—22no—
262C8#53611yescharacteristic
263C8#53611yescharacteristic
264C8#54112yes—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—162no—
269C22—162no—
270C22C2×#45911yescharacteristic
271C22#322612yes—
272C22#341511yescharacteristic
273C22#341612yes—
274C22#341611yescharacteristic
275C4—22no—
276C4—22no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—162no—
281C4—161no—
282C4—161no—
283C4—162no—
284C4#262111yescharacteristic
285C4#303911yescharacteristic
286C4#323912yes—
287C2—162no—
288C2#2986411yescharacteristic
289C2#3358611yescharacteristic
290C2#3358911yescharacteristic
291C2#3380712yes—
292C2#3381012yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33319C2semidirect
5. #13507C2semidirect
6. #13506C2semidirect
9. #13277C2semidirect
10. #13275C2semidirect
11. #12880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#22262
#333192
#212822
#135072
#135062
#134822
#134802
#132772
#132752
#128802