Finite Group Databasebeta

← #351616

Subgroups — #351616

Order 512.

655 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 150 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#29925C212yesmaximal, complemented
2#29806C211yesmaximal, characteristic
3#20548C211yesmaximal, characteristic, complemented
4#9734C212yesmaximal, complemented
5#9616C212yesmaximal, complemented
6#9572C212yesmaximal, complemented
7#9571C212yesmaximal, complemented
8#9141C212yesmaximal, complemented
9#8599C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2856C2212yes—
14#2283C2211yescharacteristic
15#2092—22no—
16#2073—22no—
17#2054—22no—
18#2034—22no—
19#2033—22no—
20#1999—22no—
21#1982C2212yes—
22#1982C2212yes—
23#1957C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1834C2211yescharacteristic
27#1797C2212yes—
28#1770C2212yes—
29C8×C4⋊C4C2212yes—
30#1768—22no—
31#1753—22no—
32C2×#335C2212yes—
33C2×#331C2212yes—
34C2×#331C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×D8—22no—
40C22×M16—21no—
41C2×#127C2312yes—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C22×C42C2311yescharacteristic
46#493—24no—
47#431—24no—
48C2×#107—22no—
49C2×#106—22no—
50#423—24no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#102—22no—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×#102—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8D412yes—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#390—22no—
69#390—22no—
70#390—22no—
71#384C2312yes—
72#383—21no—
73#382D411yescharacteristic
74#382D411yescharacteristic
75#381—21no—
76#380D412yes—
77C4×C4⋊C4C2312yes—
78C4×C22⋊C4—21no—
79#376—21no—
80#340—22no—
81#340—22no—
82#340—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#335—24no—
87#331—24no—
88#331—24no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—42no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C2×D8—44no—
97C2×D8—82no—
98C2×D8—82no—
99C2×M16—24no—
100C2×M16—82no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106#127—22no—
107#127—22no—
108#127—22no—
109#127—22no—
110#121—82no—
111#121—82no—
112#121—82no—
113#121—82no—
114#121—84no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42C2411yescharacteristic
131#107—82no—
132#106—82no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8#4114yes—
141#102—82no—
142#102—82no—
143#102—82no—
144#102—82no—
145#102—82no—
146#102—82no—
147C22⋊C8—24no—
148C22⋊C8—82no—
149#97—82no—
150C4×C8—22no—
151C4×C8—82no—
152C4×C8#4114yes—
153#95—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—22no—
159C24—42no—
160C24#12111yescharacteristic
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4—84no—
178C2×D4—84no—
179C2×D4—84no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C2×#4111yescharacteristic
199D8—88no—
200M16—84no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8#12314yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C22⋊C4—84no—
226C22⋊C4—84no—
227C22⋊C4—84no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—22no—
232C42—22no—
233C42—42no—
234C42—42no—
235C42C2×#4112yes—
236C42C2×#4112yes—
237C23—21no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23C2×#12111yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—82no—
257D4—82no—
258D4—162no—
259D4—162no—
260D4—168no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×#12311yescharacteristic
285C2×C4#52511yescharacteristic
286C2×C4#52912yes—
287C2×C4#53212yes—
288C2×C4#57612yes—
289C2×C4#57612yes—
290C2×C4#57611yescharacteristic
291C2×C4#57712yes—
292C2×C4#57811yescharacteristic
293C8—24no—
294C8—42no—
295C8—44no—
296C8—82no—
297C8—82no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—162no—
304C22—162no—
305C22—162no—
306C22—162no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22#285411yescharacteristic
311C22#304512yes—
312C22#305811yescharacteristic
313C22#306111yescharacteristic
314C22#306812yes—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4#309612yes—
323C4#312211yescharacteristic
324C4#312611yescharacteristic
325C4#317512yes—
326C4#317612yes—
327C2—81no—
328C2—322no—
329C2#1953812yes—
330C2#1954212yes—
331C2#3311611yescharacteristic
332C2#3312511yescharacteristic
333C2#3313811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#29806C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29925C2semidirect
3. #20548C2semidirect
4. #9734C2semidirect
5. #9616C2semidirect
6. #9572C2semidirect
7. #9571C2semidirect
8. #9141C2semidirect
9. #8599C2semidirect

Maximal subgroups

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

subgroupindex
#299252
#298062
#205482
#97342
#96162
#95722
#95712
#91412
#85992