Finite Group Databasebeta

← #122621

Subgroups — #122621

Order 512.

424 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 78 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#23638C211yesmaximal, characteristic
2C2×#1314C211yesmaximal, characteristic
3#8008C211yesmaximal, characteristic
4#7087C212yesmaximal, complemented
5#7085C212yesmaximal, complemented
6C2×#427C411yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8C2211yescharacteristic
9#2280—21no—
10C2×C22⋊C16—21no—
11#2077—21no—
12#2077—21no—
13#2076C411yescharacteristic
14#2076C2211yescharacteristic
15#2075C411yescharacteristic
16#2075C2211yescharacteristic
17#1735—21no—
18#1735C411yescharacteristic
19#1314—21no—
20#1314C2212yes—
21#1293—21no—
22#1293C2212yes—
23C24×C4D411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#117—21no—
28C2×#117—21no—
29C22×C16—41no—
30#427—41no—
31#427—41no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35#407—22no—
36#407—22no—
37#407—41no—
38#407—41no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—41no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43C2×C22⋊C8C2×C411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C2×C22⋊C8D411yescharacteristic
46C2×C22⋊C8C2311yescharacteristic
47#381—41no—
48#380—21no—
49#380—21no—
50#380—21no—
51#380—21no—
52#379—41no—
53C4×C22⋊C4—41no—
54C4×C22⋊C4—41no—
55C22⋊C16—22no—
56C22⋊C16—41no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C25SD1611yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4D811yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—82no—
80C2×M16—82no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22×C411yescharacteristic
91C22×C8C2×D411yescharacteristic
92#117—42no—
93#117—42no—
94#117—41no—
95#117—41no—
96#117—41no—
97#117—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C16—42no—
111C2×C16—82no—
112#107—41no—
113#107—41no—
114#106—42no—
115C22⋊C8—24no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C24—21no—
128C24—21no—
129C24—22no—
130C24—41no—
131C24—42no—
132C24—81no—
133C24#10211yescharacteristic
134C24#10211yescharacteristic
135C24C2×SD1611yescharacteristic
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—82no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—82no—
165C22×C4#9911yescharacteristic
166C22×C4#9911yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4#10411yescharacteristic
170C22×C4#10411yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×D811yescharacteristic
173M16—84no—
174M16—82no—
175M16—82no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—44no—
188C2×C8—81no—
189C2×C8—84no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C22⋊C4—82no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C22⋊C4—82no—
205C42—82no—
206C42—81no—
207C42—81no—
208C16—82no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—81no—
227C23—81no—
228C23—84no—
229C23—84no—
230C23#32711yescharacteristic
231C23#32711yescharacteristic
232C23#35912yes—
233C23C2×#10211yescharacteristic
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#32711yescharacteristic
270C2×C4#32711yescharacteristic
271C2×C4#35212yes—
272C2×C4#35412yes—
273C2×C4C2×#9911yescharacteristic
274C2×C4C2×#10411yescharacteristic
275C8—41no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8—84no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—81no—
298C22—84no—
299C22—81no—
300C22—84no—
301C22#134412yes—
302C22C2×#32711yescharacteristic
303C22C2×#35911yescharacteristic
304C22#214811yescharacteristic
305C4—21no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—161no—
310C4—161no—
311C4—161no—
312C4—161no—
313C4#134412yes—
314C4C2×#35211yescharacteristic
315C4C2×#35411yescharacteristic
316C2—21no—
317C2—21no—
318C2—22no—
319C2—41no—
320C2—42no—
321C2—81no—
322C2#1168811yescharacteristic
323C2C2×#134411yescharacteristic
324C2#1170911yescharacteristic
subgroupG/H
centerC2×C4#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#327
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7087C2semidirect
5. #7085C2semidirect

Maximal subgroups

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

subgroupindex
#236382
C2×#13142
#80082
#70872
#70852