Finite Group Databasebeta

← #142206

Subgroups — #142206

Order 512.

612 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 100 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#15128C211yesmaximal, characteristic
2#10802C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#395—21no—
7C2×#389—21no—
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324C2212yes—
14#1446—21no—
15#1444C412yes—
16#1441C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic
19#1280—21no—
20C23×Q8—21no—
21C22×M16—22no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36#395—42no—
37#395—42no—
38#389—44no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95C2×C412yes—
46C2×#95C2×C411yescharacteristic
47C2×#95D411yescharacteristic
48#343—22no—
49#343—42no—
50#343Q814yescomplemented
51#336—41no—
52#336—41no—
53#336—41no—
54#324—24no—
55C22×Q8—42no—
56C22×Q8—44no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—82no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C2×M16—24no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C22×C8—42no—
76C22×C8—42no—
77C4⋊Q8—48no—
78C4⋊Q8—44no—
79C4⋊Q8—44no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4D812yes—
97C2×C4⋊C4SD1612yes—
98C2×C4⋊C4C2×Q812yes—
99C2×C4⋊C4#4112yes—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—41no—
108C2×C42—42no—
109C2×C42—42no—
110C4⋊C8—82no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—44no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—82no—
121C24C2×C22⋊C411yescharacteristic
122C2×Q8—48no—
123C2×Q8—84no—
124C2×Q8—88no—
125C2×Q8—88no—
126C2×Q8—84no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4#9911yescharacteristic
171C22×C4#9911yescharacteristic
172C22×C4#10212yes—
173C22×C4#10212yes—
174C22×C4C4×Q812yes—
175C22×C4C22⋊Q812yes—
176C22×C4#12411yescharacteristic
177C22×C4C4⋊Q811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×SD1611yescharacteristic
180C22×C4#13712yes—
181M16—44no—
182M16—44no—
183M16—82no—
184M16—82no—
185C2×C8—44no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—48no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—88no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23#39111yescharacteristic
212C23C2×#9911yescharacteristic
213C23#41011yescharacteristic
214C23C2×#10211yescharacteristic
215C23#41911yescharacteristic
216C23#42111yescharacteristic
217C23#42111yescharacteristic
218Q8—816no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4AΣL(1,𝔽16)12yes—
275C2×C4#35612yes—
276C2×C4#47412yes—
277C2×C4#47512yes—
278C8—84no—
279C8—162no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22#151111yescharacteristic
293C22#151511yescharacteristic
294C22#151711yescharacteristic
295C22#151911yescharacteristic
296C22#198811yescharacteristic
297C22#199511yescharacteristic
298C22#199611yescharacteristic
299C22C2×AΣL(1,𝔽16)11yescharacteristic
300C22#212311yescharacteristic
301C22C2×#35611yescharacteristic
302C22#213411yescharacteristic
303C4—22no—
304C4—22no—
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—84no—
313C4—81no—
314C4—84no—
315C2—21no—
316C2—21no—
317C2—21no—
318C2—21no—
319C2#991211yescharacteristic
320C2#1181511yescharacteristic
321C2#1181811yescharacteristic
322C2#1188011yescharacteristic
323C2#1188311yescharacteristic
324C2#1329311yescharacteristic
325C2#1330211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
50. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#151282
#108022
#85602
#72512