Finite Group Databasebeta

← #139010

Subgroups — #139010

Order 512.

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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#33795C211yesmaximal, characteristic
2#18068C211yesmaximal, characteristic
3#11983C211yesmaximal, characteristic
4C2×#1364C211yesmaximal, characteristic
5#9799C211yesmaximal, characteristic
6#7288C211yesmaximal, characteristic
7#7209C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2962—21no—
11#2254—21no—
12#2241—21no—
13#2235C414yes—
14#2225—22no—
15#2162C2211yescharacteristic
16C2×C4⋊C16C2211yescharacteristic
17#1966—21no—
18#1591—21no—
19#1584—22no—
20C8×Q16—21no—
21#1567C2211yescharacteristic
22C2×#335C2211yescharacteristic
23#1546—21no—
24#1481—21no—
25C8×M16C2211yescharacteristic
26#1365—21no—
27#1364—21no—
28#1364—21no—
29#1362—21no—
30#533—21no—
31C2×C4⋊Q8—21no—
32C22×C16—21no—
33#500—21no—
34#500—21no—
35#500—21no—
36#500—21no—
37#500D411yescharacteristic
38#494D411yescharacteristic
39#487—21no—
40#467—21no—
41#467C2×C412yes—
42#446—22no—
43#446—42no—
44C8×Q8—22no—
45C4×Q16—21no—
46C4×Q16D412yes—
47#431D411yescharacteristic
48C2×#107—21no—
49C2×#107C2×C412yes—
50#424—41no—
51#423—21no—
52C2×C4⋊C8—21no—
53C2×#101—41no—
54C2×#101—41no—
55#405—41no—
56C4×M16D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#367—24no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62#358—24no—
63C22⋊C16C2×C412yes—
64C4×C16—21no—
65#335—21no—
66#335—21no—
67#335—21no—
68#335D411yescharacteristic
69#335D411yescharacteristic
70#332—42no—
71Q8⋊C8—22no—
72Q8⋊C8—22no—
73#322—21no—
74C82—21no—
75C22×Q8—41no—
76C2×Q16—21no—
77C2×Q16C22⋊C412yes—
78C2×M16—41no—
79C2×M16D811yescharacteristic
80C2×M16D811yescharacteristic
81C22×C8—41no—
82C22×C8C22×C411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90#125—41no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C4×Q8—22no—
94C4×Q8—41no—
95#117—41no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C42C2×D411yescharacteristic
99C2×C16—22no—
100C2×C16—41no—
101C2×C16—81no—
102#107—21no—
103#107—22no—
104#107—22no—
105#107—41no—
106#107—41no—
107#107C22⋊C412yes—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116#103—22no—
117#103—22no—
118#101—82no—
119#101—82no—
120C22⋊C8—21no—
121#97—21no—
122#97—42no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—42no—
126C4×C8C2×D411yescharacteristic
127C4×C8C2×D411yescharacteristic
128C4×C8C2×D411yescharacteristic
129C4×C8#4111yescharacteristic
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×Q8—81no—
138C22×C4—42no—
139C22×C4C4×D411yescharacteristic
140C22×C4C2×D811yescharacteristic
141C22×C4#13511yescharacteristic
142Q16—42no—
143Q16—41no—
144Q16—41no—
145M16—21no—
146M16—21no—
147M16—21no—
148M16—81no—
149M16—81no—
150M16SD3212yes—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—81no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8C2×C22⋊C411yescharacteristic
163C2×C8C4×D411yescharacteristic
164C2×C8#12111yescharacteristic
165C2×C8#12311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C22⋊C4—42no—
177C42—42no—
178C42#12011yescharacteristic
179C42#12111yescharacteristic
180C42AGL(1,ℤ/8)11yescharacteristic
181C42AGL(1,ℤ/8)11yescharacteristic
182C16—42no—
183C16—81no—
184C23C4×D811yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—161no—
188Q8—161no—
189Q8—161no—
190Q8—161no—
191Q8—161no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#38611yescharacteristic
201C2×C4#44211yescharacteristic
202C2×C4#44711yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#45911yescharacteristic
205C2×C4#46311yescharacteristic
206C2×C4C2×SD3211yescharacteristic
207C2×C4#50811yescharacteristic
208C2×C4#50911yescharacteristic
209C2×C4#51011yescharacteristic
210C8—21no—
211C8—21no—
212C8—21no—
213C8—22no—
214C8—41no—
215C8—81no—
216C8—81no—
217C8—81no—
218C8—161no—
219C8#47111yescharacteristic
220C8#47311yescharacteristic
221C22—21no—
222C22—21no—
223C22#187911yescharacteristic
224C22C4×SD3211yescharacteristic
225C22#217511yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—82no—
230C4—82no—
231C4—162no—
232C4#190811yescharacteristic
233C4#191511yescharacteristic
234C4#218511yescharacteristic
235C4#218611yescharacteristic
236C4#220711yescharacteristic
237C4#221111yescharacteristic
238C2—21no—
239C2—21no—
240C2#927111yescharacteristic
241C2#1301311yescharacteristic
242C2#1303111yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C4×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#337952
#180682
#119832
C2×#13642
#97992
#72882
#72092