Finite Group Databasebeta

← #101448

Subgroups — #101448

Order 512.

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

normal subgroups only

This group has 399 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#29272C211yesmaximal, characteristic
2#22238C211yesmaximal, characteristic
3#9643C211yesmaximal, characteristic
4#8589C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7018C211yesmaximal, characteristic
8C2×C8×Q8—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2099C412yes—
12#2057C412yes—
13#1988—21no—
14#1984—21no—
15#1947—21no—
16#1918—22no—
17#1916—22no—
18#1916—22no—
19#1842C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
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—
33C8×Q8—42no—
34C8×Q8—42no—
35C2×#107—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4×C8—22no—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#389—21no—
51#389C2×C412yes—
52#384C2×C411yescharacteristic
53#384D411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4D412yes—
58C43C2311yescharacteristic
59#340—22no—
60#340—22no—
61#340—22no—
62#336—24no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336C2×C412yes—
70#336D412yes—
71#332—42no—
72Q8⋊C8—42no—
73Q8⋊C8—42no—
74C22×Q8—21no—
75C22×Q8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—42no—
85C4⋊Q8SD1612yes—
86C4⋊Q8Q1612yes—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C412yes—
107C2×C42—22no—
108C2×C42—41no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116#107—44no—
117C4⋊C8—44no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—81no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—44no—
133C4×C8D812yes—
134#95—22no—
135#95—22no—
136#95—41no—
137#95—41no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C4×D411yescharacteristic
156C22×C4C4×D411yescharacteristic
157C22×C4#12011yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12111yescharacteristic
160C22×C4#12311yescharacteristic
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—81no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4#10314yes—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—42no—
206C42—42no—
207C42—81no—
208C42—81no—
209C42C2×D811yescharacteristic
210C42C2×SD1611yescharacteristic
211C42C2×Q1611yescharacteristic
212C42#13511yescharacteristic
213C42#13511yescharacteristic
214C42#13511yescharacteristic
215C42AGL(1,ℤ/8)11yescharacteristic
216C42AGL(1,ℤ/8)11yescharacteristic
217C23#38611yescharacteristic
218Q8—42no—
219Q8—42no—
220Q8—42no—
221Q8—42no—
222Q8—42no—
223Q8—42no—
224Q8—81no—
225Q8—81no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4C2×#10311yescharacteristic
253C2×C4#41611yescharacteristic
254C2×C4C4×D811yescharacteristic
255C2×C4#44211yescharacteristic
256C2×C4#44711yescharacteristic
257C2×C4#44911yescharacteristic
258C2×C4#45911yescharacteristic
259C2×C4#46311yescharacteristic
260C2×C4#46511yescharacteristic
261C2×C4#46511yescharacteristic
262C2×C4#46711yescharacteristic
263C2×C4#46711yescharacteristic
264C2×C4#47111yescharacteristic
265C2×C4#47311yescharacteristic
266C8—42no—
267C8—82no—
268C8—82no—
269C8—81no—
270C8—161no—
271C8—161no—
272C22#187911yescharacteristic
273C22#188111yescharacteristic
274C22#189111yescharacteristic
275C22#189211yescharacteristic
276C22#190511yescharacteristic
277C22#190811yescharacteristic
278C22#191211yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—41no—
289C4—41no—
290C4—82no—
291C4—82no—
292C4—161no—
293C4#166511yescharacteristic
294C4#166911yescharacteristic
295C4#167911yescharacteristic
296C4#168311yescharacteristic
297C2#904911yescharacteristic
298C2#921911yescharacteristic
299C2#930011yescharacteristic
300C2#930411yescharacteristic
301C2#937411yescharacteristic
302C2#948411yescharacteristic
303C2#952811yescharacteristic
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
#292722
#222382
#96432
#85892
#76522
#70402
#70182