Finite Group Databasebeta

← #116874

Subgroups — #116874

Order 512.

603 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 124 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#20721C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic, complemented
3#17452C212yesmaximal
4#8125C212yesmaximal, complemented
5#7986C211yesmaximal, characteristic, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2212yes—
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#1818C412yes—
12#1797—22no—
13#1780C412yes—
14#1750—21no—
15#1750—21no—
16#1572—24no—
17#1567—21no—
18#1566—21no—
19#1491—22no—
20#1490—22no—
21#1488—24no—
22#1477—24no—
23#1308—24no—
24#1290—22no—
25C22×#41C2×C411yescharacteristic
26C4×#41—22no—
27C4×#41—22no—
28C4×#41—22no—
29C4×#41—22no—
30C2×C4×Q8D411yescharacteristic
31C2×C4×Q8Q811yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×C4×D4D411yescharacteristic
35C2×C4×D4Q811yescharacteristic
36C2×#117—21no—
37C2×#117C2×C411yescharacteristic
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432D414yes—
48#431—22no—
49#431—22no—
50#431—22no—
51#431D414yes—
52#423—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D412yes—
56#407—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2×C412yes—
61C2×C22⋊C8C2×C412yes—
62C4×M16—22no—
63C2×#97—22no—
64C2×#97—21no—
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67#336—22no—
68#336—22no—
69#335—24no—
70#334—24no—
71Q8⋊C8—28no—
72D4⋊C8—28no—
73C2×#41—22no—
74C2×#41—22no—
75C2×#41—44no—
76C22×Q8C4⋊C411yescharacteristic
77C22×D4—21no—
78C22×D4C4⋊C411yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4C22×C411yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—42no—
86C22×C8—22no—
87C22×C8—42no—
88C22×C8C22⋊C412yes—
89C22×C8C22⋊C412yes—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102#117—22no—
103#117—22no—
104#117—41no—
105#117—41no—
106#117—41no—
107#117—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C4⋊C411yescharacteristic
111C2×C4⋊C4C4⋊C411yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42#4112yes—
127C2×C42#4112yes—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—44no—
139C22⋊C8—44no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142#97—24no—
143#97—42no—
144#97—42no—
145#97—41no—
146#97—41no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—24no—
150C4×C8—21no—
151C4×C8—21no—
152C24—21no—
153C24#11711yescharacteristic
154#41—28no—
155#41—44no—
156#41—44no—
157#41—44no—
158C2×Q8—22no—
159C2×Q8—42no—
160C2×D4—22no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—81no—
178C22×C4—82no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4C4×D412yes—
182C22×C4C4×D412yes—
183C22×C4#12111yescharacteristic
184C22×C4#12112yes—
185C22×C4#12111yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4#12312yes—
189C22×C4#12412yes—
190C22×C4#12612yes—
191M16—84no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—84no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—42no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—41no—
214C42—41no—
215C42—42no—
216C42—42no—
217C42#12012yes—
218C42#12011yescharacteristic
219C42#12112yes—
220C42#12112yes—
221C42C22⋊Q812yes—
222C42C22⋊Q812yes—
223C42#12312yes—
224C42#12311yescharacteristic
225C42#12411yescharacteristic
226C42#12711yescharacteristic
227C23—22no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23#38011yescharacteristic
236C23#38812yes—
237Q8—24no—
238Q8—42no—
239D4—24no—
240D4—42no—
241D4—48no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4#38512yes—
266C2×C4#38611yescharacteristic
267C2×C4#39011yescharacteristic
268C2×C4#39211yescharacteristic
269C2×C4#39311yescharacteristic
270C2×C4#39412yes—
271C2×C4#39612yes—
272C2×C4#39711yescharacteristic
273C2×C4#39911yescharacteristic
274C8—42no—
275C8—42no—
276C8—82no—
277C8—84no—
278C8—84no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—44no—
285C22—42no—
286C22—82no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22#143911yescharacteristic
291C22#181412yes—
292C4—21no—
293C4—22no—
294C4—21no—
295C4—22no—
296C4—21no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—81no—
303C4#189112yes—
304C2—22no—
305C2—42no—
306C2—82no—
307C2#1107111yescharacteristic
308C2#1110411yescharacteristic
309C2#1111011yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17815C2semidirect
4. #8125C2semidirect
5. #7986C2semidirect

Maximal subgroups

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

subgroupindex
#207212
#178152
#174522
#81252
#79862