Finite Group Databasebeta

← #116875

Subgroups — #116875

Order 512.

625 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 132 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#17819C211yesmaximal, characteristic, complemented
3#17451C212yesmaximal
4#8121C212yesmaximal, complemented
5#8001C211yesmaximal, characteristic, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2212yes—
11#1818C412yes—
12#1797—22no—
13#1780C412yes—
14#1750—22no—
15#1574—24no—
16#1568—22no—
17#1489—24no—
18#1487—22no—
19#1486—22no—
20#1477—24no—
21#1307—24no—
22#1292—22no—
23C22×#41C2×C411yescharacteristic
24C22×M16—22no—
25C4×#41—22no—
26C4×#41—22no—
27C4×#41—22no—
28C4×#41—22no—
29C2×C4×Q8D411yescharacteristic
30C2×C4×Q8Q811yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×C4×D4D411yescharacteristic
34C2×C4×D4Q811yescharacteristic
35C2×#117—21no—
36C2×#117C2×C411yescharacteristic
37C22×C42—21no—
38C22×C42C2311yescharacteristic
39#432—22no—
40#432—22no—
41#432—22no—
42#432—22no—
43#431—22no—
44#431—22no—
45#431—22no—
46#431D414yes—
47#423—22no—
48#423—22no—
49#423—22no—
50#423—22no—
51#423D414yes—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D412yes—
56#407—22no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8C2×C412yes—
59C2×C22⋊C8C2×C412yes—
60C4×M16—22no—
61C2×#97—21no—
62C2×C4×C8D412yes—
63#336—24no—
64#335—24no—
65#334—24no—
66Q8⋊C8—28no—
67D4⋊C8—28no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—44no—
71C22×Q8C4⋊C411yescharacteristic
72C22×D4—21no—
73C22×D4C4⋊C411yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22×C411yescharacteristic
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C2×M16—24no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—82no—
84C2×M16—82no—
85C22×C8—22no—
86C22×C8—42no—
87C22×C8C22⋊C414yes—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—42no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100#117—22no—
101#117—22no—
102#117—41no—
103#117—41no—
104#117—41no—
105#117—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C4⋊C411yescharacteristic
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42#4112yes—
125C2×C42#4112yes—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—48no—
138C22⋊C8—42no—
139#97—41no—
140#97—41no—
141#97—42no—
142C4×C8—22no—
143C4×C8—24no—
144C4×C8—22no—
145C24—21no—
146C24#11711yescharacteristic
147#41—28no—
148#41—44no—
149#41—44no—
150#41—44no—
151C2×Q8—22no—
152C2×Q8—42no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4#12112yes—
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12312yes—
187C22×C4#12412yes—
188C22×C4#12612yes—
189M16—84no—
190M16—82no—
191M16—82no—
192M16—84no—
193C2×C8—24no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—84no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—41no—
212C42—41no—
213C42—42no—
214C42—42no—
215C42#12012yes—
216C42#12011yescharacteristic
217C42#12112yes—
218C42#12112yes—
219C42C22⋊Q812yes—
220C42C22⋊Q812yes—
221C42#12312yes—
222C42#12311yescharacteristic
223C42#12411yescharacteristic
224C42#12711yescharacteristic
225C23—21no—
226C23—21no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23#38011yescharacteristic
237C23#38812yes—
238Q8—24no—
239Q8—42no—
240D4—24no—
241D4—42no—
242D4—48no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#38512yes—
272C2×C4#38611yescharacteristic
273C2×C4#39011yescharacteristic
274C2×C4#39211yescharacteristic
275C2×C4#39311yescharacteristic
276C2×C4#39412yes—
277C2×C4#39612yes—
278C2×C4#39711yescharacteristic
279C2×C4#39911yescharacteristic
280C2×C4#47112yes—
281C2×C4#47211yescharacteristic
282C2×C4#47311yescharacteristic
283C8—44no—
284C8—82no—
285C8—84no—
286C8—84no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22#143911yescharacteristic
302C22#181511yescharacteristic
303C22#181611yescharacteristic
304C4—21no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—81no—
314C4#189112yes—
315C4#205112yes—
316C4#205212yes—
317C2—21no—
318C2—21no—
319C2—42no—
320C2—82no—
321C2#1107111yescharacteristic
322C2#1110511yescharacteristic
323C2#1110911yescharacteristic
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. #17819C2semidirect
4. #8121C2semidirect
5. #8001C2semidirect

Maximal subgroups

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

subgroupindex
#207212
#178192
#174512
#81212
#80012