Finite Group Databasebeta

← #116568

Subgroups — #116568

Order 512.

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

normal subgroups only

This group has 635 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#17537C212yesmaximal
4C4×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
5#8071C212yesmaximal, complemented
6C2×#529C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×C4×M16C2211yescharacteristic
10C4×#117C2212yes—
11#1985C411yescharacteristic, complemented
12#1985C411yescharacteristic, complemented
13#1955C411yescharacteristic, complemented
14#1955C411yescharacteristic, complemented
15#1870—22no—
16#1750—22no—
17C2×H(3,ℤ/4)—22no—
18#1569—24no—
19#1564—22no—
20#1509—24no—
21#1508—22no—
22#1507—22no—
23#1500—24no—
24#1300—24no—
25C22×M16D412yes—
26#529—22no—
27#529—22no—
28#529—22no—
29#529—22no—
30C2×#126—21no—
31C2×#125Q811yescharacteristic
32C2×#124Q811yescharacteristic
33C2×#123—21no—
34C2×C22⋊Q8D411yescharacteristic
35C2×#121D411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4C2×C411yescharacteristic
39C2×#117—22no—
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—
58C2×#97—21no—
59C2×#97C2×C411yescharacteristic
60C2×#97C2×C411yescharacteristic
61C2×C4×C8D412yes—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66#376D414yes—
67C43—22no—
68H(3,ℤ/4)—28no—
69#336—22no—
70#336—22no—
71#335—24no—
72#334—24no—
73#328—28no—
74#327—28no—
75C22×Q8C4⋊C411yescharacteristic
76C22×D4—21no—
77C22×D4C4⋊C411yescharacteristic
78C23×C4—21no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4#4111yescharacteristic
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8C22⋊C412yes—
89C22×C8C22⋊C412yes—
90#126—42no—
91#126—42no—
92#125—22no—
93#125—22no—
94#124—22no—
95#124—22no—
96#123—42no—
97#123—42no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100#121—22no—
101#121—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—42no—
107C4×D4—42no—
108#117—28no—
109#117—44no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C4⋊C411yescharacteristic
120C2×C4⋊C4C4⋊C411yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—42no—
127C2×C42—24no—
128C2×C42—21no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×Q811yescharacteristic
136C2×C42#4112yes—
137C2×C42#4112yes—
138C2×C42#4111yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C22⋊C8—24no—
144C22⋊C8—44no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—44no—
148#97—22no—
149#97—22no—
150#97—24no—
151#97—41no—
152#97—41no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—42no—
156#95—24no—
157#95—24no—
158C24—21no—
159C24#12111yescharacteristic
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—44no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4#12012yes—
182C22×C4#12112yes—
183C22×C4#12112yes—
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12312yes—
188C22×C4#12312yes—
189C22×C4#12412yes—
190M16—48no—
191M16—84no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—44no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8#10414yes—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—48no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C22⋊C4—48no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—28no—
221C42—44no—
222C42—41no—
223C42—41no—
224C42—84no—
225C42—84no—
226C42C4×D412yes—
227C42C4×D412yes—
228C42#12011yescharacteristic
229C42#12112yes—
230C42C22⋊Q812yes—
231C42C22⋊Q812yes—
232C42#12311yescharacteristic
233C42#12411yescharacteristic
234C42#12612yes—
235C42#12711yescharacteristic
236C23—22no—
237C23—41no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23#38011yescharacteristic
244C23#39412yes—
245Q8—82no—
246D4—82no—
247D4—84no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4#38512yes—
273C2×C4#38611yescharacteristic
274C2×C4#38812yes—
275C2×C4#39011yescharacteristic
276C2×C4#39211yescharacteristic
277C2×C4#39311yescharacteristic
278C2×C4#39612yes—
279C2×C4#39711yescharacteristic
280C2×C4#39911yescharacteristic
281C2×C4C2×#10412yes—
282C2×C4#42112yes—
283C8—24no—
284C8—42no—
285C8—84no—
286C8—81no—
287C8—81no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—82no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#143911yescharacteristic
299C22#180612yes—
300C4—22no—
301C4—21no—
302C4—22no—
303C4—21no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—88no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4#187112yes—
312C4#196211yescharacteristic
313C4#196311yescharacteristic
314C2—22no—
315C2—42no—
316C2—161no—
317C2#1105311yescharacteristic
318C2#1110211yescharacteristic
319C2#1111011yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#121D4
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×H(3,ℤ/4)C2semidirect
5. #8071C2semidirect
11. #1985C4semidirect
12. #1985C4semidirect
13. #1955C4semidirect
14. #1955C4semidirect

Maximal subgroups

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

subgroupindex
#213762
#177952
#175372
C4×H(3,ℤ/4)2
#80712