Finite Group Databasebeta

← #138341

Subgroups — #138341

Order 512.

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

normal subgroups only

This group has 725 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#32930C211yesmaximal, characteristic
2C2×#1807C211yesmaximal, characteristic, complemented
3#12679C211yesmaximal, characteristic, complemented
4C2×#1347C212yesmaximal, complemented
5#7258C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C2×#466C412yescomplemented
8C2×#466C2211yescharacteristic
9C22×#106—21no—
10#2847—21no—
11C2×#380—21no—
12C2×C22⋊C16—22no—
13C2×C22⋊C16C2212yes—
14#1893—22no—
15#1893—21no—
16#1808C412yescomplemented
17#1808C2211yescharacteristic
18#1807—24no—
19C2×#340C2212yes—
20#1347—24no—
21C24×C4D411yescharacteristic
22C22×D8C2×C411yescharacteristic
23C23×C8—22no—
24C23×C8—22no—
25C23×C8—44no—
26C23×C8C2×C412yes—
27C23×C8D411yescharacteristic
28C23×C8D412yes—
29C23×C8C2311yescharacteristic
30C2×#121—22no—
31C2×#121—21no—
32C2×#120—41no—
33C22×C4⋊C4—22no—
34C22×C22⋊C4—41no—
35C22×C16—22no—
36#466—22no—
37#466—22no—
38#466—44no—
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C412yes—
41C2×#106—24no—
42C2×#106—44no—
43C2×#102—22no—
44C2×#102—21no—
45#386—41no—
46#386—42no—
47#380—24no—
48#380—22no—
49#380—21no—
50#380—42no—
51C2×#95—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54C22⋊C16—44no—
55#340—22no—
56#340—22no—
57#340—21no—
58#340D414yescomplemented
59C25D811yescharacteristic
60C22×D4—21no—
61C22×D4—42no—
62C22×D4—41no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4C22⋊C412yes—
71C23×C4D811yescharacteristic
72C23×C4SD1612yes—
73C23×C4C2×D411yescharacteristic
74C2×D8—22no—
75C2×D8—42no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—24no—
82C22×C8—24no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—48no—
87C22×C8—48no—
88C22×C8C22⋊C412yes—
89C22×C8C22×C411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D412yes—
92C22×C8#4112yes—
93#121—44no—
94#121—44no—
95#121—41no—
96#121—41no—
97#121—41no—
98#121—41no—
99#120—84no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—84no—
112C2×C16—44no—
113C2×C16—42no—
114#107—22no—
115#107—42no—
116#106—48no—
117#102—44no—
118#102—42no—
119#95—44no—
120#95—42no—
121#95—41no—
122C24—24no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—42no—
128C24—41no—
129C24#10212yes—
130C24C2×D811yescharacteristic
131C2×D4—41no—
132C2×D4—42no—
133C2×D4—41no—
134C2×D4—42no—
135C2×D4—84no—
136C2×D4—84no—
137C2×D4—84no—
138C2×D4—81no—
139C2×D4—81no—
140C2×D4—82no—
141C2×D4—82no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4#10212yes—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12311yescharacteristic
174C22×C4C2×SD1611yescharacteristic
175C22×C4AGL(1,ℤ/8)11yescharacteristic
176C22×C4#13711yescharacteristic
177D8—82no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—48no—
187C2×C8—48no—
188C2×C8#9914yes—
189C2×C8C4×D412yes—
190C2×C8#12112yes—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—84no—
198C22⋊C4—84no—
199C22⋊C4—88no—
200C22⋊C4—84no—
201C22⋊C4—82no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C16—82no—
205C23—22no—
206C23—22no—
207C23—24no—
208C23—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—48no—
216C23—48no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—81no—
222C23—81no—
223C23—81no—
224C23—82no—
225C23—82no—
226C23#36014yes—
227C23C2×#10211yescharacteristic
228C23#44711yescharacteristic
229C23#48011yescharacteristic
230C23#50812yes—
231C23#50912yes—
232D4—81no—
233D4—81no—
234D4—82no—
235D4—168no—
236D4—162no—
237D4—162no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—164no—
266C2×C4—164no—
267C2×C4#38611yescharacteristic
268C2×C4C2×#9912yes—
269C2×C4#41711yescharacteristic
270C2×C4#42112yes—
271C2×C4#44811yescharacteristic
272C2×C4#48111yescharacteristic
273C8—22no—
274C8—24no—
275C8—22no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—48no—
286C22—48no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—162no—
292C22—162no—
293C22—161no—
294C22—161no—
295C22—161no—
296C22#151912yes—
297C22#187411yescharacteristic
298C22#213811yescharacteristic
299C22#214011yescharacteristic
300C22C2×#36012yes—
301C22#218612yes—
302C22#223412yes—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—164no—
307C4—162no—
308C4—161no—
309C4#151312yes—
310C4#189711yescharacteristic
311C4#189811yescharacteristic
312C2—22no—
313C2—22no—
314C2—44no—
315C2—321no—
316C2#915511yescharacteristic
317C2#1188512yes—
318C2#1188812yes—
319C2#1299211yescharacteristic
320C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1807C2semidirect
3. #12679C2semidirect
4. C2×#1347C2semidirect
5. #7258C2semidirect
7. C2×#466C4semidirect
16. #1808C4semidirect
58. #340D4semidirect

Maximal subgroups

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

subgroupindex
#329302
C2×#18072
#126792
C2×#13472
#72582