Finite Group Databasebeta

← #111433

Subgroups — #111433

Order 512.

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

normal subgroups only

This group has 892 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#60940C211yesmaximal, characteristic
2#17717C212yesmaximal, complemented
3#17384C212yesmaximal, complemented
4#7987C211yesmaximal, characteristic, complemented
5#7851C211yesmaximal, characteristic, complemented
6C2×#532C2212yes—
7#3245—22no—
8#3226—22no—
9#3205—22no—
10#3186C414yes—
11#3169—22no—
12#3100—24no—
13#2920C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#1553—24no—
18#1549—22no—
19#1545—22no—
20#1540—24no—
21#1471—24no—
22#1469—24no—
23#1460—24no—
24#1290—22no—
25#1273—22no—
26C22×#41—22no—
27#532—22no—
28#532—22no—
29#532—22no—
30#532—22no—
31#532—22no—
32#532—22no—
33#532—22no—
34#532—22no—
35C2×C4⋊Q8D412yes—
36C2×#127D412yes—
37C2×#124—22no—
38C2×#121—22no—
39C2×#121C2×C414yes—
40C2×C4×D4—22no—
41C2×C4×D4D414yes—
42C22×C42C2311yescharacteristic
43#500—22no—
44#493—22no—
45#486—24no—
46#480—28no—
47#474—28no—
48#466—28no—
49#459—28no—
50#447—28no—
51C4×D8—28no—
52#432—22no—
53#432D414yes—
54#431—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#431—22no—
61#431—22no—
62#423—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8D412yes—
65C2×C4⋊C8D412yes—
66#407—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8C2×C412yes—
69C2×C22⋊C8D412yes—
70C4×M16—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8D412yes—
74#332—28no—
75#331—28no—
76#323—216no—
77C2×#41—42no—
78C2×#41—42no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41—88no—
82C22×Q8—22no—
83C22×D4—24no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4C22⋊C414yes—
87C23×C4—24no—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4#4111yescharacteristic
91C2×D8—28no—
92C2×M16—42no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8C22⋊C414yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100#127—22no—
101#127—22no—
102#127—22no—
103#127—22no—
104#124—42no—
105#124—42no—
106#124—42no—
107#124—42no—
108#121—24no—
109#121—24no—
110#121—24no—
111#121—24no—
112#121—42no—
113#121—42no—
114#121—42no—
115#121—42no—
116C4×D4—24no—
117C4×D4—24no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C22⋊C414yes—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42#4111yescharacteristic
135C2×C42#4112yes—
136#107—28no—
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—24no—
140C4⋊C8—42no—
141C4⋊C8—42no—
142#102—216no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148#97—24no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—42no—
154C4×C8—41no—
155C4×C8—41no—
156C24—24no—
157C24C4×D411yescharacteristic
158#41—88no—
159#41—88no—
160#41—88no—
161#41—88no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—84no—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—48no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—88no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—88no—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×D412yes—
188C22×C4#12012yes—
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12312yes—
194C22×C4#12411yescharacteristic
195C22×C4#12711yescharacteristic
196D8—88no—
197M16—84no—
198C2×C8—24no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C42#12011yescharacteristic
214C42#12012yes—
215C42#12011yescharacteristic
216C42#12012yes—
217C42#12112yes—
218C42#12112yes—
219C42#12112yes—
220C42#12312yes—
221C42#12411yescharacteristic
222C42#12711yescharacteristic
223C23—22no—
224C23—41no—
225C23—41no—
226C23—48no—
227C23—44no—
228C23—84no—
229C23—84no—
230C23#38612yes—
231C23#39011yescharacteristic
232Q8—84no—
233Q8—84no—
234D4—48no—
235D4—84no—
236D4—84no—
237D4—84no—
238D4—88no—
239D4—88no—
240D4—84no—
241D4—84no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—88no—
255C2×C4—88no—
256C2×C4—84no—
257C2×C4#37911yescharacteristic
258C2×C4#38612yes—
259C2×C4#39011yescharacteristic
260C2×C4#39212yes—
261C2×C4#39211yescharacteristic
262C2×C4#39211yescharacteristic
263C2×C4#39412yes—
264C2×C4#39411yescharacteristic
265C2×C4#39411yescharacteristic
266C2×C4#47214yes—
267C8—44no—
268C8—82no—
269C8—84no—
270C8—82no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—82no—
277C22—84no—
278C22—84no—
279C22—84no—
280C22—168no—
281C22#143811yescharacteristic
282C22#197812yes—
283C4—22no—
284C4—21no—
285C4—22no—
286C4—22no—
287C4—21no—
288C4—84no—
289C4#179112yes—
290C4#201812yes—
291C4#204512yes—
292C4#205312yes—
293C2—22no—
294C2—42no—
295C2—164no—
296C2#1099611yescharacteristic
297C2#1102711yescharacteristic
298C2#1103811yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC22#1438
prime radical#60940C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17717C2semidirect
3. #17384C2semidirect
4. #7987C2semidirect
5. #7851C2semidirect

Maximal subgroups

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

subgroupindex
#609402
#177172
#173842
#79872
#78512