Finite Group Databasebeta

← #138516

Subgroups — #138516

Order 512.

525 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 96 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#29731C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#11887C212yesmaximal, complemented
4C2×#1314C211yesmaximal, characteristic
5#7230C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2848—21no—
9C2×#380—21no—
10C2×C22⋊C16C2212yes—
11#2076C412yescomplemented
12#2036—22no—
13#2001C412yescomplemented
14#1892—21no—
15#1852—24no—
16#1808—21no—
17#1808C2211yescharacteristic
18#1808C2212yes—
19#1348—24no—
20#1314—22no—
21C24×C4D411yescharacteristic
22C22×Q16C2×C411yescharacteristic, complemented
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C22⋊C4—41no—
29C22×C16—22no—
30#467—22no—
31#467—22no—
32C2×#107—22no—
33C2×#107—22no—
34C2×#107—21no—
35C2×#107C2×C411yescharacteristic, complemented
36C2×#103—21no—
37C2×#103—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8D412yes—
42#385—41no—
43#385—42no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—21no—
48#380—41no—
49#380—41no—
50#379—41no—
51C2×#95—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yescomplemented
57#340D414yes—
58C25D811yescharacteristic
59C22×Q8—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—42no—
65C23×C4—81no—
66C23×C4—81no—
67C23×C4C22⋊C412yes—
68C23×C4D811yescharacteristic
69C23×C4SD1612yes—
70C23×C4C2×D411yescharacteristic
71C2×Q16—22no—
72C2×Q16—81no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—81no—
76C22×C8—81no—
77C22×C8C22⋊C412yes—
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D412yes—
80C22×C8C2×D411yescharacteristic
81C22×C8#4112yes—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C22⋊Q8—82no—
87C22⋊Q8—82no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—82no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—82no—
103C2×C22⋊C4—82no—
104C2×C16—42no—
105C2×C16—82no—
106#107—22no—
107#107—82no—
108#107—82no—
109#107—81no—
110#103—42no—
111#103—82no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114#95—42no—
115#95—42no—
116#95—41no—
117#95—42no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—81no—
123C24—81no—
124C24#10212yes—
125C24SD3212yes—
126C24Q3212yes—
127C24C2×D811yescharacteristic
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—44no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—84no—
155C22×C4—84no—
156C22×C4—84no—
157C22×C4#10212yes—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4#12011yescharacteristic
160C22×C4#12311yescharacteristic
161C22×C4C2×SD1611yescharacteristic
162C22×C4AGL(1,ℤ/8)11yescharacteristic
163C22×C4#13711yescharacteristic
164Q16—82no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—81no—
173C2×C8C4×D412yes—
174C2×C8#12112yes—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—84no—
186C22⋊C4—84no—
187C22⋊C4—84no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—82no—
191C22⋊C4—81no—
192C22⋊C4—82no—
193C22⋊C4—81no—
194C16—82no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—81no—
208C23—81no—
209C23—82no—
210C23—82no—
211C23—82no—
212C23—82no—
213C23#35814yes—
214C23C2×#10211yescharacteristic
215C23#44711yescharacteristic
216C23#48011yescharacteristic
217C23C2×SD3211yescharacteristic
218C23C2×Q3211yescharacteristic
219C23#50911yescharacteristic
220C23#51011yescharacteristic
221Q8—82no—
222Q8—81no—
223Q8—81no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—168no—
253C2×C4#38611yescharacteristic
254C2×C4#41711yescharacteristic
255C2×C4#44811yescharacteristic
256C2×C4#48111yescharacteristic
257C8—22no—
258C8—22no—
259C8—42no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—24no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—44no—
272C22—44no—
273C22—44no—
274C22—42no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#187411yescharacteristic
280C22C2×#35811yescharacteristic
281C22#213911yescharacteristic
282C22#218711yescharacteristic
283C22#218811yescharacteristic
284C22#223511yescharacteristic
285C22#223611yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—42no—
290C4—162no—
291C4—164no—
292C4—161no—
293C4—161no—
294C4#191111yescharacteristic
295C4#191311yescharacteristic
296C2—21no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—44no—
303C2#920311yescharacteristic
304C2#1299311yescharacteristic
305C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11887C2semidirect
5. #7230C2semidirect
11. #2076C4semidirect
13. #2001C4semidirect
22. C22×Q16C2×C4semidirect
35. C2×#107C2×C4semidirect
56. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#297312
#219402
#118872
C2×#13142
#72302