Finite Group Databasebeta

← #142012

Subgroups — #142012

Order 512.

648 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 118 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#10847C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#423—21no—
7C2×#395—21no—
8C2×#389—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#343—21no—
12C2×#343C2212yes—
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
34#423—24no—
35#423—42no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40#395—42no—
41#395—42no—
42#389—44no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C412yes—
50C2×#95C2×C411yescharacteristic
51C2×#95D411yescharacteristic
52#343—24no—
53#343—42no—
54#336—41no—
55#324—22no—
56#324Q814yescomplemented
57C22×Q8—42no—
58C22×Q8—44no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—82no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—42no—
78C22×C8—42no—
79C4⋊Q8—48no—
80C4⋊Q8—44no—
81C4⋊Q8—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4D812yes—
99C2×C4⋊C4SD1612yes—
100C2×C4⋊C4C2×Q812yes—
101C2×C4⋊C4#4112yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—22no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42SD1612yes—
112C2×C42Q1612yes—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C4×C8—82no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—44no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—82no—
126C24C2×C22⋊C411yescharacteristic
127C2×Q8—48no—
128C2×Q8—84no—
129C2×Q8—88no—
130C2×Q8—88no—
131C2×Q8—84no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—24no—
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—41no—
159C22×C4—41no—
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—42no—
171C22×C4—42no—
172C22×C4#9911yescharacteristic
173C22×C4#9911yescharacteristic
174C22×C4#10212yes—
175C22×C4#10212yes—
176C22×C4#10312yes—
177C22×C4#10312yes—
178C22×C4#10412yes—
179C22×C4C4×Q812yes—
180C22×C4C22⋊Q812yes—
181C22×C4#12411yescharacteristic
182C22×C4C4⋊Q811yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185C22×C4C2×SD1611yescharacteristic
186C22×C4C2×Q1611yescharacteristic
187M16—44no—
188M16—44no—
189M16—82no—
190M16—82no—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—88no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—22no—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#39111yescharacteristic
220C23C2×#9911yescharacteristic
221C23#41011yescharacteristic
222C23C2×#10211yescharacteristic
223C23C2×#10311yescharacteristic
224C23C2×#10411yescharacteristic
225C23#42111yescharacteristic
226Q8—816no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#32812yes—
283C2×C4#32812yes—
284C2×C4AΣL(1,𝔽16)12yes—
285C2×C4#35612yes—
286C2×C4#48611yescharacteristic
287C2×C4#48711yescharacteristic
288C2×C4#49811yescharacteristic
289C2×C4#50011yescharacteristic
290C8—84no—
291C8—84no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22C2×#32811yescharacteristic
305C22#150211yescharacteristic
306C22#151811yescharacteristic
307C22#151911yescharacteristic
308C22#198611yescharacteristic
309C22#199011yescharacteristic
310C22#199611yescharacteristic
311C22C2×AΣL(1,𝔽16)11yescharacteristic
312C22#212311yescharacteristic
313C22C2×#35611yescharacteristic
314C22#213411yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—84no—
326C4—84no—
327C4#135414yes—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#988411yescharacteristic
333C2C2×#135411yescharacteristic
334C2#1172311yescharacteristic
335C2#1188011yescharacteristic
336C2#1188311yescharacteristic
337C2#1329611yescharacteristic
338C2#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
56. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#151282
#108472
C4×#3242
#72512