Finite Group Databasebeta

← #139364

Subgroups — #139364

Order 512.

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

normal subgroups only

This group has 384 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#33487C211yesmaximal, characteristic
2#28553C211yesmaximal, characteristic
3#11896C212yesmaximal, complemented
4#8612C211yesmaximal, characteristic
5#7264C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2417—21no—
9#2168—24no—
10#2163—21no—
11#2112C2212yes—
12C2×C22⋊C16—21no—
13#2087C411yescharacteristic
14#2076C411yescharacteristic
15#2056C411yescharacteristic
16#2001C411yescharacteristic
17#1933—21no—
18#1892—21no—
19#1872—21no—
20#1810C2212yes—
21C2×#340C2211yescharacteristic
22#1465—21no—
23#1350—24no—
24#1293—21no—
25C22×Q16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—21no—
30C2×C4×D4D411yescharacteristic
31C22×C4⋊C4—21no—
32C2×M32—22no—
33C22×C16—22no—
34#467—22no—
35#467—22no—
36C8×D4—22no—
37C8×D4—22no—
38C2×#108—22no—
39C2×#107C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44#407—22no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#398—41no—
49#396—41no—
50#391—21no—
51#389—21no—
52#386—41no—
53#380—21no—
54#380—41no—
55C4⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16D414yes—
59#346—22no—
60#341—22no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#324—41no—
67#323—41no—
68C22×Q8—21no—
69C22×D4SD1611yescharacteristic
70C23×C4—41no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×Q16—22no—
74C2×Q16—81no—
75C2×M16—42no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—81no—
79C22×C8—81no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D412yes—
84C22×C8C2×D411yescharacteristic
85C22×C8#4112yes—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—42no—
89C22⋊Q8—81no—
90C4×D4—22no—
91C4×D4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4D811yescharacteristic
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4D811yescharacteristic
106C2×C22⋊C4SD1611yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108M32—82no—
109C2×C16—24no—
110C2×C16—82no—
111#108—82no—
112#107—22no—
113C4⋊C8—22no—
114#103—42no—
115#103—41no—
116#103—41no—
117#103—81no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C4×C8—21no—
123C4×C8—21no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128C24#10211yescharacteristic
129C24C2×SD1611yescharacteristic
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—41no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4—161no—
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4#12112yes—
157C22×C4C2×D811yescharacteristic
158C22×C4#13512yes—
159Q16—82no—
160M16—82no—
161M16—82no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—81no—
172C2×C8—82no—
173C2×C8C4×D412yes—
174C2×C8#12011yescharacteristic
175C2×C8#12311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—161no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C22⋊C4—161no—
192C42—21no—
193C42—21no—
194C16—44no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23#36212yes—
204C23C2×#10211yescharacteristic
205C23#46512yes—
206Q8—81no—
207Q8—81no—
208Q8—81no—
209Q8—81no—
210D4—44no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—164no—
240C2×C4#38611yescharacteristic
241C2×C4#41611yescharacteristic
242C2×C4#46612yes—
243C8—21no—
244C8—21no—
245C8—22no—
246C8—41no—
247C8—41no—
248C8—162no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22#189311yescharacteristic
262C22C2×#36211yescharacteristic
263C22#214811yescharacteristic
264C4—22no—
265C4—21no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—82no—
270C4—161no—
271C4—161no—
272C4—161no—
273C4#188411yescharacteristic
274C4#191311yescharacteristic
275C2—21no—
276C2—21no—
277C2—42no—
278C2—42no—
279C2#938411yescharacteristic
280C2#1306611yescharacteristic
281C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11896C2semidirect
5. #7264C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#285532
#118962
#86122
#72642