Finite Group Databasebeta

← #480682

Subgroups — #480682

Order 512.

936 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C2×#2499C211yesmaximal, characteristic
2#17357C212yesmaximal
3#11556C218yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11506C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×#117C2211yescharacteristic
8#2920—22no—
9#2499—22no—
10#2499—22no—
11#2499C414yes—
12#2499C2214yes—
13C2×#397—22no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#387—22no—
17C2×#384C411yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20#1464—22no—
21#1464C2214yes—
22#1463C2214yes—
23#1462—22no—
24#1458—22no—
25#1458C2214yes—
26#1324C2218yescomplemented
27#1319C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—42no—
32C2×C22⋊Q8—41no—
33C2×C22⋊Q8—41no—
34C2×C22⋊Q8C2×C414yes—
35C2×#117—42no—
36C2×#117—41no—
37C2×#117—41no—
38C2×#117C2×C412yes—
39C2×#117D412yes—
40C2×#117D412yes—
41C2×#117C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4D412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D412yes—
53#432—24no—
54#432—42no—
55#431—24no—
56#431—42no—
57#423—42no—
58#407—24no—
59C4×M16—42no—
60#397—24no—
61#397—24no—
62#397—24no—
63#397—24no—
64#391—21no—
65#391—21no—
66#391—22no—
67#391C2×C412yes—
68#391C2×C414yes—
69#391C2312yes—
70#387—24no—
71#387—24no—
72#387—24no—
73#387—24no—
74#384—21no—
75#384—21no—
76#384C2×C414yes—
77#382—22no—
78#382C2×C412yes—
79#382C2312yes—
80C4×C22⋊C4—22no—
81C4×C22⋊C4C2×C414yes—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86#324—24no—
87#324C2318yes—
88#323—24no—
89#323—24no—
90C25C22⋊C411yescharacteristic
91C22×Q8—21no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8C22×C412yes—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C2×M16—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117#117—42no—
118#117—42no—
119#117—42no—
120#117—42no—
121#117—42no—
122#117—42no—
123#117—42no—
124#117—42no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2411yescharacteristic
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C2×D412yes—
156C2×C22⋊C4C2×D412yes—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C22⋊C8—28no—
174C22⋊C8—44no—
175#97—44no—
176C4×C8—44no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184C24—21no—
185C24—21no—
186C24—41no—
187C24—41no—
188C24—42no—
189C24—42no—
190C24C2×C22⋊C411yescharacteristic
191C24C2×C22⋊C412yes—
192C2×Q8—24no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—84no—
199C2×Q8—84no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#9912yes—
241C22×C4#9911yescharacteristic
242C22×C4#9911yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250M16—88no—
251C2×C8—48no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—24no—
279C42—42no—
280C42—42no—
281C42—42no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—84no—
298C23—84no—
299C23C2×#9911yescharacteristic
300C23C2×#9911yescharacteristic
301C23C22×C22⋊C411yescharacteristic
302Q8—84no—
303Q8—84no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4C2×#9912yes—
349C2×C4C2×#9912yes—
350C2×C4#41312yes—
351C2×C4#56812yes—
352C8—88no—
353C22—24no—
354C22—24no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22—84no—
367C22—84no—
368C22#212912yes—
369C22#213212yes—
370C22#287911yescharacteristic
371C22C22×#9911yescharacteristic
372C22C2×#41311yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—82no—
383C4—82no—
384C4—82no—
385C2—21no—
386C2—21no—
387C2—22no—
388C2—22no—
389C2—82no—
390C2#1739211yescharacteristic
391C2C2×#212911yescharacteristic
392C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11556C2semidirect
4. #11511C2semidirect
5. #11506C2semidirect
26. #1324C22semidirect
27. #1319C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#173572
#115562
#115112
#115062