Finite Group Databasebeta

← #111476

Subgroups — #111476

Order 512.

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

normal subgroups only

This group has 940 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#21424C212yesmaximal, complemented
3#17463C212yesmaximal, complemented
4#7987C211yesmaximal, characteristic, complemented
5#7850C211yesmaximal, 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×C4×M16C2211yescharacteristic
17#1997—22no—
18#1965—22no—
19#1900—24no—
20C4×#100—24no—
21#1484—24no—
22#1480—24no—
23#1478—24no—
24#1290—22no—
25#1273—22no—
26C22×#41—22no—
27C22×M16D412yes—
28#532—22no—
29#532—22no—
30#532—22no—
31#532—22no—
32#532—22no—
33#532—22no—
34#532—22no—
35#532—22no—
36C2×C4⋊Q8D412yes—
37C2×#127D412yes—
38C2×#124—22no—
39C2×#121—22no—
40C2×#121C2×C414yes—
41C2×C4×D4—22no—
42C2×C4×D4D414yes—
43C22×C42C2311yescharacteristic
44#500—22no—
45#493—22no—
46#486—24no—
47#480—28no—
48#474—28no—
49#466—28no—
50#459—28no—
51#447—28no—
52C4×D8—28no—
53#432—22no—
54#432—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#431—22no—
61#423—22no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8D412yes—
64C2×#100—28no—
65#407—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8C2×C412yes—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16D414yes—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8D412yes—
75#343—28no—
76D4⋊C8—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—24no—
93C2×M16—24no—
94C2×M16—44no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C22⋊C414yes—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102#127—22no—
103#127—22no—
104#127—22no—
105#127—22no—
106#124—42no—
107#124—42no—
108#124—42no—
109#124—42no—
110#121—24no—
111#121—24no—
112#121—24no—
113#121—24no—
114#121—42no—
115#121—42no—
116#121—42no—
117#121—42no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4C22⋊C414yes—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42#4111yescharacteristic
137C2×C42#4112yes—
138#107—28no—
139C4⋊C8—24no—
140C4⋊C8—24no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143#102—216no—
144#100—416no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#97—24no—
150#97—42no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—24no—
154C4×C8—41no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158C24—24no—
159C24C4×D411yescharacteristic
160#41—88no—
161#41—88no—
162#41—88no—
163#41—88no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—84no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—48no—
170C2×D4—48no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—84no—
178C2×D4—88no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—88no—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D412yes—
190C22×C4#12012yes—
191C22×C4#12111yescharacteristic
192C22×C4#12112yes—
193C22×C4#12112yes—
194C22×C4#12111yescharacteristic
195C22×C4#12312yes—
196C22×C4#12411yescharacteristic
197C22×C4#12711yescharacteristic
198D8—88no—
199M16—28no—
200M16—44no—
201M16—48no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—44no—
207C2×C8—44no—
208C2×C8—42no—
209C2×C8—44no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C42#12011yescharacteristic
219C42#12012yes—
220C42#12011yescharacteristic
221C42#12012yes—
222C42#12112yes—
223C42#12112yes—
224C42#12112yes—
225C42#12312yes—
226C42#12411yescharacteristic
227C42#12711yescharacteristic
228C23—22no—
229C23—41no—
230C23—41no—
231C23—48no—
232C23—44no—
233C23—84no—
234C23—84no—
235C23#38612yes—
236C23#39011yescharacteristic
237Q8—84no—
238Q8—84no—
239D4—48no—
240D4—48no—
241D4—84no—
242D4—84no—
243D4—88no—
244D4—88no—
245D4—84no—
246D4—84no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—88no—
260C2×C4—88no—
261C2×C4—84no—
262C2×C4#37911yescharacteristic
263C2×C4#38612yes—
264C2×C4#39011yescharacteristic
265C2×C4#39212yes—
266C2×C4#39211yescharacteristic
267C2×C4#39211yescharacteristic
268C2×C4#39412yes—
269C2×C4#39411yescharacteristic
270C2×C4#39411yescharacteristic
271C2×C4#47214yes—
272C8—44no—
273C8—44no—
274C8—84no—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—82no—
282C22—84no—
283C22—84no—
284C22—84no—
285C22—168no—
286C22#143811yescharacteristic
287C22#197812yes—
288C4—22no—
289C4—22no—
290C4—21no—
291C4—22no—
292C4—21no—
293C4—84no—
294C4#179112yes—
295C4#201012yes—
296C4#201812yes—
297C4#204512yes—
298C2—22no—
299C2—42no—
300C2—164no—
301C2#1099611yescharacteristic
302C2#1102711yescharacteristic
303C2#1104111yescharacteristic
subgroupG/H
centerC2×C4#392
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. #21424C2semidirect
3. #17463C2semidirect
4. #7987C2semidirect
5. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#609402
#214242
#174632
#79872
#78502