Finite Group Databasebeta

← #102437

Subgroups — #102437

Order 512.

646 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 98 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#33106C212yesmaximal, complemented
2#30239C211yesmaximal, characteristic
3#11346C211yesmaximal, characteristic, complemented
4#7664C212yesmaximal, complemented
5#7020C211yesmaximal, characteristic, complemented
6C2×#480—22no—
7C2×#466—22no—
8#2856C2212yes—
9C2×C4×C4⋊C4C2211yescharacteristic
10#2093—22no—
11#2011—22no—
12#1982—22no—
13#1947—22no—
14#1925—24no—
15#1874—22no—
16#1797C2212yes—
17C2×#336—22no—
18C2×#336C2211yescharacteristic
19C2×#336C2211yescharacteristic
20#1270—24no—
21C22×D8—22no—
22C23×C8—21no—
23C23×C8D412yes—
24C2×#127D412yes—
25C2×#121—22no—
26C2×#121—22no—
27C2×#121—22no—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32#480—44no—
33#480—44no—
34#466—44no—
35#466—44no—
36C2×#107—24no—
37C2×#107—22no—
38C2×C4⋊C8—22no—
39C2×#102—24no—
40C2×#102—24no—
41C2×#102—24no—
42C2×C22⋊C8—22no—
43#390—22no—
44#390—22no—
45#390—22no—
46#384D412yes—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C2×#95—21no—
50#340—24no—
51#336—22no—
52#336—22no—
53#336—44no—
54#336D414yes—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—42no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4#4111yescharacteristic
64C2×D8—44no—
65C2×D8—82no—
66C2×D8—82no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—24no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8—41no—
76C22×C8—42no—
77C22×C8D814yes—
78#127—44no—
79#121—44no—
80#121—42no—
81#121—42no—
82#121—42no—
83#121—44no—
84#121—42no—
85#121—84no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4D812yescomplemented
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C42—42no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42C2×D412yes—
104C2×C42C2×D412yes—
105#107—48no—
106#107—44no—
107C4⋊C8—44no—
108#102—48no—
109#102—48no—
110#102—48no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—42no—
114C24—42no—
115C24#12111yescharacteristic
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—42no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—84no—
128C2×D4—82no—
129C2×D4—82no—
130C2×D4—84no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4#12012yes—
151C22×C4#12011yescharacteristic
152C22×C4#12112yes—
153C22×C4#12711yescharacteristic
154C22×C4C2×D812yes—
155C22×C4C2×D811yescharacteristic
156C22×C4#13511yescharacteristic
157C22×C4AGL(1,ℤ/8)12yes—
158C22×C4AGL(1,ℤ/8)12yes—
159D8—88no—
160C2×C8—24no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—42no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—82no—
170C2×C8—81no—
171C2×C8—82no—
172C2×C8—81no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C22⋊C4—84no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C42—24no—
187C42—42no—
188C42—82no—
189C42—81no—
190C42—81no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—82no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23—82no—
200C23—82no—
201C23—82no—
202C23#39211yescharacteristic
203C23#45912yes—
204C23#46611yescharacteristic
205C23#46911yescharacteristic
206C23#47212yes—
207D4—84no—
208D4—84no—
209D4—84no—
210D4—84no—
211D4—164no—
212D4—164no—
213D4—164no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4#44712yes—
243C2×C4#44712yes—
244C2×C4#44912yes—
245C2×C4#45511yescharacteristic
246C2×C4#45611yescharacteristic
247C2×C4#45812yes—
248C2×C4#49311yescharacteristic
249C2×C4#49611yescharacteristic
250C8—44no—
251C8—84no—
252C8—82no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—22no—
260C22—22no—
261C22—162no—
262C22—162no—
263C22—162no—
264C22—162no—
265C22—162no—
266C22—162no—
267C22—162no—
268C22#166712yes—
269C22#168712yes—
270C22#199911yescharacteristic
271C22#200711yescharacteristic
272C22#201011yescharacteristic
273C22#201111yescharacteristic
274C22#201711yescharacteristic
275C22#201811yescharacteristic
276C22#202211yescharacteristic
277C4—22no—
278C4—22no—
279C4—21no—
280C4—21no—
281C4—42no—
282C4—82no—
283C4—82no—
284C4—162no—
285C4#160712yes—
286C4#161012yes—
287C2—22no—
288C2—22no—
289C2—322no—
290C2#997011yescharacteristic
291C2#997711yescharacteristic
292C2#1002811yescharacteristic
293C2#1003211yescharacteristic
294C2#1025511yescharacteristic
295C2#1026011yescharacteristic
296C2#1032111yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radical#30239C2
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33106C2semidirect
3. #11346C2semidirect
4. #7664C2semidirect
5. #7020C2semidirect
94. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#331062
#302392
#113462
#76642
#70202