Finite Group Databasebeta

← #98912

Subgroups — #98912

Order 512.

367 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 110 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#21266C211yesmaximal, characteristic
2#8807C211yesmaximal, characteristic, complemented
3C4×#334C211yesmaximal, characteristic, complemented
4#7574C212yesmaximal
5#7022C212yesmaximal, complemented
6C2×#433—21no—
7#2293C2211yescharacteristic
8#1954C411yescharacteristic
9#1954C411yescharacteristic
10#1939C411yescharacteristic
11#1939C411yescharacteristic
12#1923—21no—
13#1923—21no—
14#1831C2212yes—
15#1767C2211yescharacteristic
16C4×C4⋊C8C2212yes—
17#1750—21no—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21C2×#334—21no—
22#1277—21no—
23#1271—22no—
24#1270—22no—
25C2×C4×D4C2×C411yescharacteristic
26C2×#117—21no—
27#433—42no—
28#433—42no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×#97—21no—
40C2×C4×C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#390D411yescharacteristic
48#390Q811yescharacteristic
49#389D411yescharacteristic
50#389Q811yescharacteristic
51#388—21no—
52#388—21no—
53C4×C4⋊C4C2×C411yescharacteristic
54C4×C22⋊C4—21no—
55C43C2311yescharacteristic
56#336—21no—
57#336—21no—
58#336—21no—
59#336—22no—
60#336—22no—
61#336—22no—
62#335—42no—
63#334—24no—
64#334—42no—
65C22×D4C4⋊C411yescharacteristic
66C23×C4—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C4×D4—22no—
76C4×D4—22no—
77#117—41no—
78#117—41no—
79#117—41no—
80#117—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C4⋊C411yescharacteristic
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×Q811yescharacteristic
97C2×C42#4111yescharacteristic
98C2×C42#4111yescharacteristic
99C4⋊C8—24no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—41no—
112#97—41no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—41no—
119C4×C8—82no—
120C4×C8D812yes—
121C4×C8SD1612yes—
122#95—21no—
123#95—21no—
124#95—21no—
125C24—21no—
126C2×D4—22no—
127C2×D4—41no—
128C2×D4—41no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C2×C4⋊C411yescharacteristic
140C22×C4#11711yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4#12111yescharacteristic
143C22×C4C22⋊Q811yescharacteristic
144C22×C4C22⋊Q811yescharacteristic
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—44no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8#10212yes—
167C2×C8#10212yes—
168C2×C8#10412yes—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C42—21no—
184C42—21no—
185C42—22no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42#13112yes—
192C42#13112yes—
193C42C2×D811yescharacteristic
194C42C2×SD1611yescharacteristic
195C42AGL(1,ℤ/8)11yescharacteristic
196C42#13711yescharacteristic
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23#38011yescharacteristic
205D4—82no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4#40811yescharacteristic
233C2×C4#40811yescharacteristic
234C2×C4C2×#10211yescharacteristic
235C2×C4#41711yescharacteristic
236C2×C4C2×#10411yescharacteristic
237C2×C4#42111yescharacteristic
238C2×C4#42412yes—
239C2×C4#43312yes—
240C2×C4#45911yescharacteristic
241C2×C4#46111yescharacteristic
242C2×C4#47112yes—
243C2×C4#47411yescharacteristic
244C2×C4#47611yescharacteristic
245C8—22no—
246C8—22no—
247C8—22no—
248C8—41no—
249C8—84no—
250C8—81no—
251C8—81no—
252C8—162no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#179811yescharacteristic
261C22#180211yescharacteristic
262C22#180611yescharacteristic
263C22#181211yescharacteristic
264C22#181311yescharacteristic
265C22#181411yescharacteristic
266C22#181511yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4#148011yescharacteristic
279C4#148311yescharacteristic
280C4#149211yescharacteristic
281C4#149511yescharacteristic
282C2—161no—
283C2#856711yescharacteristic
284C2#860311yescharacteristic
285C2#860711yescharacteristic
286C2#867411yescharacteristic
287C2#868211yescharacteristic
288C2#869111yescharacteristic
289C2#869611yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8807C2semidirect
3. C4×#334C2semidirect
5. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#212662
#88072
C4×#3342
#75742
#70222