Finite Group Databasebeta

← #141578

Subgroups — #141578

Order 512.

366 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 72 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#28705C211yesmaximal, characteristic
2#11787C212yesmaximal
3#11569C212yesmaximal
4#8612C211yesmaximal, characteristic
5C8×#99C211yesmaximal, characteristic
6C2×#500C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8#2563C2212yes—
9#2076—21no—
10#2076—21no—
11#1987C412yescomplemented
12#1810—21no—
13#1808—21no—
14#1808—21no—
15C4×#107C411yescharacteristic, complemented
16C4×#99C2211yescharacteristic
17C8×C22⋊C4—21no—
18#1465C2212yes—
19#1459C2211yescharacteristic
20#1351—24no—
21#1324—24no—
22#1293—21no—
23#1282—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4⋊Q8C2×C412yes—
27C2×C4×D4C2311yescharacteristic
28#500—41no—
29#500—41no—
30C8×D4—41no—
31C8×D4—41no—
32C2×#108—41no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×#99—21no—
38#407—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8D411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×C4×C8—41no—
43C2×C4×C8C2×C411yescharacteristic
44#393—22no—
45#393—22no—
46#390—22no—
47#382—22no—
48#380—22no—
49#380—22no—
50C4×C4⋊C4C2×C412yes—
51C4×C22⋊C4—21no—
52#342D412yes—
53#341—41no—
54#340—22no—
55#340—22no—
56#340—21no—
57#336—41no—
58#324—22no—
59#324D412yes—
60#323—21no—
61#323—22no—
62C22×Q8—22no—
63C22×D4#4111yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C2×M16—42no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4×D4—41no—
81C4×D4—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4C2×D411yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4C2×D411yescharacteristic
90C2×C22⋊C4#4111yescharacteristic
91C2×C42—22no—
92C2×C42—41no—
93C2×C42C22×C411yescharacteristic
94#108—82no—
95#107—42no—
96#107—42no—
97#107—41no—
98#107—41no—
99C4⋊C8—41no—
100#99—24no—
101C22⋊C8—22no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—82no—
110#95—22no—
111#95—41no—
112#95—42no—
113#95—42no—
114C24#12111yescharacteristic
115C24#12411yescharacteristic
116C2×Q8—82no—
117C2×Q8—162no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—41no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C4×D411yescharacteristic
134C22×C4C4×D411yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12711yescharacteristic
137M16—82no—
138M16—82no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—82no—
154C2×C8#9911yescharacteristic
155C2×C8#9911yescharacteristic
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C22⋊C4—24no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C22⋊C4—42no—
166C22⋊C4—162no—
167C22⋊C4—162no—
168C42—21no—
169C42—21no—
170C42—44no—
171C42—81no—
172C42—81no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23#39011yescharacteristic
183C23#47312yes—
184Q8—164no—
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—44no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4C2×#9911yescharacteristic
212C2×C4#41011yescharacteristic
213C2×C4#44312yes—
214C8—21no—
215C8—41no—
216C8—41no—
217C8—41no—
218C8—82no—
219C8—162no—
220C8#35412yes—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22#196411yescharacteristic
234C22#197411yescharacteristic
235C22#197911yescharacteristic
236C4—22no—
237C4—21no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—84no—
242C4—162no—
243C4—162no—
244C4C2×#35411yescharacteristic
245C4#212911yescharacteristic
246C2—21no—
247C2—21no—
248C2—42no—
249C2—81no—
250C2#981411yescharacteristic
251C2#1322411yescharacteristic
252C2#1323411yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#500C4semidirect
11. #1987C4semidirect
15. C4×#107C4semidirect

Maximal subgroups

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

subgroupindex
#287052
#117872
#115692
#86122
C8×#992