Finite Group Databasebeta

← #116567

Subgroups — #116567

Order 512.

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

normal subgroups only

This group has 611 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#21376C211yesmaximal, characteristic
2#17795C211yesmaximal, characteristic, complemented
3#17642C212yesmaximal
4C4×#338C211yesmaximal, characteristic, complemented
5#8087C212yesmaximal, complemented
6C2×#529C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×C4×M16C2212yes—
10C2×C4×M16C2211yescharacteristic
11#1985C411yescharacteristic
12#1985C411yescharacteristic
13#1955C411yescharacteristic
14#1955C411yescharacteristic
15#1870—22no—
16#1750—22no—
17C2×#338—22no—
18#1569—24no—
19#1564—22no—
20#1529—24no—
21#1528—22no—
22#1527—22no—
23#1524—24no—
24#1302—24no—
25C22×M16—22no—
26C22×M16D412yes—
27#529—22no—
28#529—22no—
29#529—22no—
30#529—22no—
31C2×#126—21no—
32C2×#125D411yescharacteristic
33C2×#124D411yescharacteristic
34C2×#123—21no—
35C2×C22⋊Q8Q811yescharacteristic
36C2×#121Q811yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4C2×C411yescharacteristic
40C2×#117—21no—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432D414yes—
46#431—22no—
47#431—22no—
48#423—22no—
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C4⋊C8C2×C411yescharacteristic
51#407—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C4×M16—22no—
56C4×M16—22no—
57C4×M16—22no—
58C4×M16—22no—
59C4×M16—22no—
60C4×M16—22no—
61C4×M16—22no—
62C2×#97—21no—
63C2×#97C2×C411yescharacteristic
64C2×#97C2×C411yescharacteristic
65C2×#97D414yes—
66C2×C4×C8—22no—
67C2×C4×C8D412yes—
68#338—28no—
69#336—22no—
70#336—22no—
71#335—24no—
72#334—24no—
73#330—28no—
74#329—28no—
75C22×Q8C4⋊C411yescharacteristic
76C22×D4—21no—
77C22×D4C4⋊C411yescharacteristic
78C23×C4—21no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×Q811yescharacteristic
81C23×C4#4111yescharacteristic
82C2×M16—28no—
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—42no—
86C2×M16—44no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C22×C8—24no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8C22⋊C412yes—
94C22×C8C22⋊C412yes—
95#126—42no—
96#126—42no—
97#125—22no—
98#125—22no—
99#124—22no—
100#124—22no—
101#123—42no—
102#123—42no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105#121—22no—
106#121—22no—
107C4×Q8—22no—
108C4×Q8—22no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—42no—
112C4×D4—42no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C4⋊C411yescharacteristic
122C2×C4⋊C4C4⋊C411yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C42—21no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42#4112yes—
135C2×C42#4112yes—
136C2×C42#4111yescharacteristic
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—24no—
140C4⋊C8—42no—
141C22⋊C8—24no—
142C22⋊C8—44no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—44no—
146#97—22no—
147#97—22no—
148#97—24no—
149#97—24no—
150#97—24no—
151#97—41no—
152#97—41no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—42no—
156C4×C8—42no—
157C4×C8—42no—
158C24—21no—
159C24C22⋊Q811yescharacteristic
160C2×Q8—22no—
161C2×Q8—81no—
162C2×D4—22no—
163C2×D4—42no—
164C2×D4—82no—
165C2×D4—81no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4#11711yescharacteristic
180C22×C4#12112yes—
181C22×C4#12112yes—
182C22×C4#12111yescharacteristic
183C22×C4#12112yes—
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q812yes—
187C22×C4C22⋊Q811yescharacteristic
188C22×C4#12612yes—
189M16—48no—
190M16—48no—
191M16—48no—
192M16—84no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—28no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—44no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8#10414yes—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C42—41no—
223C42—41no—
224C42C4×D412yes—
225C42C4×D412yes—
226C42#12012yes—
227C42#12011yescharacteristic
228C42#12312yes—
229C42#12311yescharacteristic
230C42#12312yes—
231C42#12412yes—
232C42#12411yescharacteristic
233C42#12711yescharacteristic
234C23—22no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23#38011yescharacteristic
242C23#39612yes—
243Q8—82no—
244D4—82no—
245D4—84no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4#38512yes—
268C2×C4#38611yescharacteristic
269C2×C4#38812yes—
270C2×C4#39011yescharacteristic
271C2×C4#39211yescharacteristic
272C2×C4#39311yescharacteristic
273C2×C4#39412yes—
274C2×C4#39711yescharacteristic
275C2×C4#39911yescharacteristic
276C2×C4C2×#10412yes—
277C2×C4#42112yes—
278C8—24no—
279C8—48no—
280C8—42no—
281C8—84no—
282C8—81no—
283C8—81no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—82no—
290C22—84no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#143911yescharacteristic
295C22#180612yes—
296C4—22no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4#187112yes—
307C4#196211yescharacteristic
308C4#196311yescharacteristic
309C2—22no—
310C2—42no—
311C2—161no—
312C2#1105311yescharacteristic
313C2#1110311yescharacteristic
314C2#1110911yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#121Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17795C2semidirect
4. C4×#338C2semidirect
5. #8087C2semidirect

Maximal subgroups

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

subgroupindex
#213762
#177952
#176422
C4×#3382
#80872