Finite Group Databasebeta

← #102169

Subgroups — #102169

Order 512.

524 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 134 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#33207C211yesmaximal, characteristic
2#10862C211yesmaximal, characteristic
3#9884C212yesmaximal
4#9679C212yesmaximal
5#8144C211yesmaximal, characteristic
6C2×#498C411yescharacteristic, complemented
7C22×#107—21no—
8C22×C4×C8C2211yescharacteristic
9#2857C2212yes—
10#1990C412yescomplemented
11#1941—21no—
12#1941—21no—
13#1844C411yescharacteristic, complemented
14#1807—21no—
15#1807—21no—
16C4×C22⋊C8C2212yes—
17C2×#340—21no—
18C2×#328—24no—
19C2×#322—21no—
20C4⋊#95C2211yescharacteristic
21C4⋊#95C2211yescharacteristic
22#1294—24no—
23#1273—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4⋊Q8C2×C412yes—
27C2×C22⋊Q8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C42C2311yescharacteristic
31#498—42no—
32C2×#107—24no—
33C2×#107—42no—
34C2×#107—42no—
35C2×#106—21no—
36C2×#106—21no—
37C2×#106—21no—
38C2×#106—21no—
39C2×C22⋊C8D414yes—
40C2×C4×C8—22no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—22no—
47#389—22no—
48#389—22no—
49#384C2×C412yes—
50#380—22no—
51#380—22no—
52C2×#95—22no—
53C2×#95—22no—
54#340—22no—
55#340—22no—
56#340—42no—
57#336—22no—
58#328—48no—
59#322—42no—
60C22×Q8—22no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D412yes—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—24no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8SD1612yes—
75C22×C8SD1612yes—
76C22×C8Q1612yes—
77C22×C8Q1612yes—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C22⋊Q8—44no—
81C22⋊Q8—44no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42#4112yes—
106#107—48no—
107#106—42no—
108#106—42no—
109#106—42no—
110#106—42no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—41no—
116C4×C8—41no—
117C4×C8—42no—
118C4×C8—42no—
119#95—42no—
120#95—42no—
121#95—44no—
122#95—44no—
123C24#12711yescharacteristic
124C2×Q8—44no—
125C2×Q8—82no—
126C2×Q8—82no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C4×D412yes—
147C22×C4#12112yes—
148C22×C4#12411yescharacteristic
149C22×C4C2×SD1611yescharacteristic
150C22×C4C2×SD1611yescharacteristic
151C22×C4C2×Q1611yescharacteristic
152C22×C4C2×Q1611yescharacteristic
153C22×C4AGL(1,ℤ/8)14yes—
154C22×C4#13714yes—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—24no—
162C2×C8—24no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8#9912yes—
169C2×C8#10312yes—
170C2×C8#10312yes—
171C2×C8#10312yes—
172C2×C8#10312yes—
173C2×C8#10412yes—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—84no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—42no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23#39011yescharacteristic
201C23#49211yescharacteristic
202C23#49411yescharacteristic
203C23#49612yes—
204C23#49712yes—
205Q8—164no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—161no—
233C2×C4—161no—
234C2×C4—162no—
235C2×C4—162no—
236C2×C4—161no—
237C2×C4—161no—
238C2×C4C2×#9911yescharacteristic
239C2×C4#41011yescharacteristic
240C2×C4C2×#10311yescharacteristic
241C2×C4C2×#10311yescharacteristic
242C2×C4C2×#10411yescharacteristic
243C2×C4#42111yescharacteristic
244C2×C4#44014yes—
245C2×C4#46814yes—
246C2×C4#48711yescharacteristic
247C2×C4#48711yescharacteristic
248C8—22no—
249C8—22no—
250C8—22no—
251C8—84no—
252C8#32814yescomplemented
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—22no—
260C22—22no—
261C22#171511yescharacteristic
262C22#171611yescharacteristic
263C22#172312yes—
264C22#195611yescharacteristic
265C22#196411yescharacteristic
266C22#197312yes—
267C22#197612yes—
268C22#198011yescharacteristic
269C4—24no—
270C4—24no—
271C4—21no—
272C4—21no—
273C4—162no—
274C4—162no—
275C4—162no—
276C4—162no—
277C4C2×#32811yescharacteristic
278C4#150311yescharacteristic
279C4#151811yescharacteristic
280C4#152011yescharacteristic
281C2—21no—
282C2—21no—
283C2—22no—
284C2#967911yescharacteristic
285C2#972212yes—
286C2#975511yescharacteristic
287C2#978612yes—
288C2#984111yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×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
6. C2×#498C4semidirect
10. #1990C4semidirect
13. #1844C4semidirect
252. C8#328semidirect

Maximal subgroups

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

subgroupindex
#332072
#108622
#98842
#96792
#81442