Finite Group Databasebeta

← #102148

Subgroups — #102148

Order 512.

615 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 142 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#33170C211yesmaximal, characteristic
2#10861C211yesmaximal, characteristic, complemented
3#9885C212yesmaximal, complemented
4#9680C212yesmaximal
5C8×C22⋊C8C211yesmaximal, characteristic, complemented
6C2×#492C411yescharacteristic, complemented
7C2×#465—21no—
8C2×#465—21no—
9C22×#106—21no—
10C22×C4×C8C2211yescharacteristic
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#1968—21no—
14#1968—21no—
15#1957C411yescharacteristic, complemented
16#1956C411yescharacteristic, complemented
17#1844C411yescharacteristic, complemented
18C4×C22⋊C8C2212yes—
19C2×#340—21no—
20C2×#327—24no—
21C2×C82—21no—
22C4⋊#95C2212yes—
23#1294—24no—
24#1273—21no—
25C22×SD16—21no—
26C22×SD16—21no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×#127C2×C411yescharacteristic
31C2×C22⋊Q8—21no—
32C2×#121—22no—
33C2×#121—21no—
34C22×C4⋊C4—22no—
35C22×C42C2311yescharacteristic
36#492—42no—
37#465—42no—
38#465—42no—
39#465—42no—
40#465—42no—
41C2×#106—24no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—42no—
45C2×#106—42no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8D414yes—
51C2×C4×C8—22no—
52C2×C4×C8—42no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57#391—21no—
58#390—21no—
59#390—22no—
60#389—22no—
61#384C2×C411yescharacteristic
62#384C2×C411yescharacteristic
63#380—22no—
64C2×#95—22no—
65C2×#95—22no—
66#340—42no—
67#336—22no—
68#327—48no—
69C82—42no—
70C22×Q8—21no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—41no—
74C23×C4—42no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C2×SD16—42no—
78C2×SD16—42no—
79C2×SD16—42no—
80C2×SD16—42no—
81C2×SD16—42no—
82C2×SD16—42no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—24no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8D812yes—
93C22×C8D812yes—
94C22×C8SD1612yes—
95C22×C8SD1612yes—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98#127—41no—
99#127—41no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102C22⋊Q8—41no—
103C22⋊Q8—41no—
104#121—41no—
105#121—41no—
106#121—44no—
107#121—41no—
108#121—41no—
109#121—44no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4112yes—
130#106—48no—
131#106—42no—
132#106—42no—
133#103—42no—
134#103—42no—
135#102—42no—
136#102—42no—
137C22⋊C8—24no—
138C22⋊C8SD1618yescomplemented
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—41no—
142C4×C8—41no—
143C4×C8—42no—
144C4×C8—42no—
145#95—42no—
146#95—44no—
147#95—44no—
148C24—41no—
149C24#12711yescharacteristic
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—44no—
163C2×D4—82no—
164C2×D4—82no—
165C2×D4—81no—
166C2×D4—82no—
167C2×D4—81no—
168C2×D4—82no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C4×D412yes—
187C22×C4#12112yes—
188C22×C4#12411yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4C2×SD1611yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×SD1614yes—
194C22×C4#13514yes—
195SD16—84no—
196SD16—84no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—28no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—44no—
209C2×C8—44no—
210C2×C8#9912yes—
211C2×C8#10212yes—
212C2×C8#10212yes—
213C2×C8#10212yes—
214C2×C8#10212yes—
215C2×C8#10412yes—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—84no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C23—21no—
240C23—22no—
241C23—21no—
242C23—81no—
243C23—81no—
244C23—82no—
245C23—82no—
246C23—81no—
247C23#39011yescharacteristic
248C23#49211yescharacteristic
249C23#49212yes—
250C23#49311yescharacteristic
251C23#49512yes—
252Q8—81no—
253Q8—81no—
254Q8—81no—
255Q8—81no—
256D4—81no—
257D4—81no—
258D4—81no—
259D4—81no—
260D4—164no—
261D4—161no—
262D4—162no—
263D4—161no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—162no—
291C2×C4—162no—
292C2×C4—162no—
293C2×C4—162no—
294C2×C4C2×#9911yescharacteristic
295C2×C4#41011yescharacteristic
296C2×C4C2×#10211yescharacteristic
297C2×C4C2×#10211yescharacteristic
298C2×C4C2×#10411yescharacteristic
299C2×C4#42111yescharacteristic
300C2×C4C4×SD1614yes—
301C2×C4#46514yes—
302C2×C4#48611yescharacteristic
303C2×C4#48611yescharacteristic
304C8—22no—
305C8—22no—
306C8—22no—
307C8—48no—
308C8#32714yescomplemented
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—161no—
318C22—162no—
319C22—161no—
320C22—161no—
321C22—162no—
322C22#170511yescharacteristic
323C22#170612yes—
324C22#170811yescharacteristic
325C22#195711yescharacteristic
326C22#196411yescharacteristic
327C22#196812yes—
328C22#197612yes—
329C22#198011yescharacteristic
330C4—24no—
331C4—24no—
332C4—21no—
333C4—21no—
334C4—164no—
335C4—161no—
336C4—161no—
337C4—161no—
338C4C2×#32711yescharacteristic
339C4#150211yescharacteristic
340C4#151711yescharacteristic
341C4#151911yescharacteristic
342C2—22no—
343C2—21no—
344C2—21no—
345C2—321no—
346C2#968011yescharacteristic
347C2#971712yes—
348C2#975611yescharacteristic
349C2#978112yes—
350C2#982011yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10861C2semidirect
3. #9885C2semidirect
5. C8×C22⋊C8C2semidirect
6. C2×#492C4semidirect
15. #1957C4semidirect
16. #1956C4semidirect
17. #1844C4semidirect
138. C22⋊C8SD16semidirect
308. C8#327semidirect

Maximal subgroups

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

subgroupindex
#331702
#108612
#98852
#96802
C8×C22⋊C82