Finite Group Databasebeta

← #101450

Subgroups — #101450

Order 512.

381 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 104 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#29273C211yesmaximal, characteristic
2#22307C211yesmaximal, characteristic
3#9643C211yesmaximal, characteristic
4#8591C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7018C211yesmaximal, characteristic
8C4×C4⋊Q8C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2099C412yes—
11#2058C412yes—
12#1987—21no—
13#1984—22no—
14#1948—21no—
15#1939—21no—
16#1919—22no—
17#1916—22no—
18#1842C2211yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22C8×C4⋊C4—21no—
23#1744C2211yescharacteristic
24C2×#332—21no—
25C2×Q8⋊C8—21no—
26C2×Q8⋊C8—21no—
27#1296—21no—
28#1271—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8D411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33C2×#106—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4×C8—22no—
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—21no—
45#391—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#389C2×C412yes—
50#384C2×C411yescharacteristic
51#384D411yescharacteristic
52C4×C4⋊C4—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D412yes—
56C43C2311yescharacteristic
57#340—22no—
58#340—22no—
59#340—22no—
60#336—22no—
61#336—22no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336C2×C412yes—
69#336D412yes—
70#332—42no—
71Q8⋊C8—42no—
72Q8⋊C8—42no—
73C22×Q8—21no—
74C22×Q8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—42no—
84C4⋊Q8SD1612yes—
85C4⋊Q8Q1612yes—
86C4×Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×Q8—82no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4C22⋊C412yes—
102C2×C4⋊C4C22⋊C412yes—
103C2×C42—22no—
104C2×C42—41no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112#106—44no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—81no—
124C4⋊C8—82no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—82no—
129C4×C8SD1612yes—
130#95—22no—
131#95—22no—
132#95—41no—
133#95—41no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—81no—
143C2×Q8—81no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D411yescharacteristic
150C22×C4C4×D411yescharacteristic
151C22×C4#12011yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4#12311yescharacteristic
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—42no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8—81no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—81no—
187C4⋊C4—82no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4#10314yes—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42—42no—
199C42—42no—
200C42—81no—
201C42—81no—
202C42C2×SD1611yescharacteristic
203C42C2×SD1611yescharacteristic
204C42C2×Q1611yescharacteristic
205C42#13511yescharacteristic
206C42#13511yescharacteristic
207C42#13511yescharacteristic
208C42AGL(1,ℤ/8)11yescharacteristic
209C42#13711yescharacteristic
210C23#38611yescharacteristic
211Q8—42no—
212Q8—42no—
213Q8—82no—
214Q8—82no—
215Q8—81no—
216Q8—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4C2×#10311yescharacteristic
244C2×C4#41611yescharacteristic
245C2×C4C4×SD1611yescharacteristic
246C2×C4#44011yescharacteristic
247C2×C4#44811yescharacteristic
248C2×C4#44911yescharacteristic
249C2×C4#46011yescharacteristic
250C2×C4#46411yescharacteristic
251C2×C4#46511yescharacteristic
252C2×C4#46511yescharacteristic
253C2×C4#46711yescharacteristic
254C2×C4#46711yescharacteristic
255C2×C4#47111yescharacteristic
256C2×C4#47311yescharacteristic
257C8—42no—
258C8—82no—
259C8—82no—
260C8—81no—
261C8—161no—
262C8—161no—
263C22#188011yescharacteristic
264C22#188111yescharacteristic
265C22#189111yescharacteristic
266C22#189211yescharacteristic
267C22#190411yescharacteristic
268C22#190811yescharacteristic
269C22#191211yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—44no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—161no—
284C4#166611yescharacteristic
285C4#167011yescharacteristic
286C4#168011yescharacteristic
287C4#168411yescharacteristic
288C2#905011yescharacteristic
289C2#922011yescharacteristic
290C2#930211yescharacteristic
291C2#930311yescharacteristic
292C2#937411yescharacteristic
293C2#948411yescharacteristic
294C2#952711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292732
#223072
#96432
#85912
#76522
#70402
#70182