Finite Group Databasebeta

← #135296

Subgroups — #135296

Order 512.

534 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 92 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#20662C211yesmaximal, characteristic
2#12682C211yesmaximal, characteristic
3#10759C211yesmaximal, characteristic
4#7254C214yesmaximal, complemented
5C22×#131C2211yescharacteristic
6C2×#427C411yescharacteristic, complemented
7C22×#106C411yescharacteristic, complemented
8C2×#410—21no—
9#2311—21no—
10C2×#349—21no—
11C2×#349C2212yes—
12#2112—21no—
13#1809C412yescomplemented
14#1809C2212yes—
15#1807—21no—
16#1807—21no—
17#1736—21no—
18#1735—21no—
19C2×H(3,ℤ/4)C2212yes—
20C22×#41D411yescharacteristic
21C2×#131—24no—
22C2×#131—42no—
23C2×#131—42no—
24C22×M16C2×C411yescharacteristic
25C22×M16D411yescharacteristic
26C22×M16C2311yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C23×C8D411yescharacteristic
30C2×#117—21no—
31C2×#117—22no—
32C22×C4⋊C4—21no—
33C2×M32—22no—
34#427—42no—
35C2×#107C2×C412yes—
36C2×#106—21no—
37C2×#106—21no—
38C2×#106—42no—
39C2×#106C2×C412yes—
40#410—24no—
41#410—42no—
42C2×#99—41no—
43C2×#99—41no—
44#391—41no—
45#390—41no—
46#386—41no—
47#386—41no—
48#380—22no—
49#380—22no—
50#349—24no—
51#349—42no—
52C22⋊C16—42no—
53#340—22no—
54#340—22no—
55#340—42no—
56H(3,ℤ/4)—24no—
57C2×#41—22no—
58C2×#41—22no—
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—41no—
62C2×#41—41no—
63C22×Q8SD1611yescharacteristic
64C22×D4D811yescharacteristic
65C22×D4D811yescharacteristic
66C22×D4SD1611yescharacteristic
67C23×C4—41no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71#131—416no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16#4114yes—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94#117—24no—
95#117—42no—
96#117—42no—
97#117—42no—
98#117—42no—
99#117—42no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—42no—
111C2×C42—41no—
112M32—42no—
113M32—42no—
114C2×C16—42no—
115#107—42no—
116#106—42no—
117#106—42no—
118#106—42no—
119#99—44no—
120#99—44no—
121#95—42no—
122#95—42no—
123C24#10211yescharacteristic
124C24#10211yescharacteristic
125C24C2×D811yescharacteristic
126#41—44no—
127#41—44no—
128#41—44no—
129#41—44no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—41no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—162no—
161C22×C4#9912yes—
162C22×C4#10211yescharacteristic
163C22×C4#10211yescharacteristic
164C22×C4#10412yes—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4#12312yes—
167C22×C4C2×SD1611yescharacteristic
168C22×C4#13712yes—
169M16—44no—
170M16—44no—
171M16—44no—
172C2×C8—28no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8#9912yes—
185C2×C8#11712yes—
186C2×C8#12312yes—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C22⋊C4—44no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—162no—
198C22⋊C4—162no—
199C42—44no—
200C42—82no—
201C42—82no—
202C16—82no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23#32712yes—
214C23C2×#10211yescharacteristic
215C23#48012yes—
216Q8—44no—
217D4—44no—
218D4—44no—
219D4—44no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—164no—
247C2×C4—164no—
248C2×C4#32712yes—
249C2×C4#38111yescharacteristic
250C2×C4C2×#9911yescharacteristic
251C2×C4C2×#9911yescharacteristic
252C2×C4#41911yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#42111yescharacteristic
255C2×C4#48312yes—
256C8—22no—
257C8—22no—
258C8—22no—
259C8—41no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22—42no—
271C22—81no—
272C22—81no—
273C22C2×#32711yescharacteristic
274C22#151911yescharacteristic
275C22#182311yescharacteristic
276C4—22no—
277C4—22no—
278C4—41no—
279C4—162no—
280C4—164no—
281C4—162no—
282C4#150811yescharacteristic
283C4#151111yescharacteristic
284C4#181911yescharacteristic
285C4#182611yescharacteristic
286C2—21no—
287C2—21no—
288C2—42no—
289C2—42no—
290C2—81no—
291C2#871211yescharacteristic
292C2#1186712yes—
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC22×#41D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7254C2semidirect
6. C2×#427C4semidirect
7. C22×#106C4semidirect
13. #1809C4semidirect

Maximal subgroups

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

subgroupindex
#206622
#126822
#107592
#72542