Finite Group Databasebeta

← #213611

Subgroups — #213611

Order 512.

1438 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1438 conjugacy classes of subgroups — too many to draw. Only its 230 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#36624C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3#9120C218yesmaximal, complemented
4#8722C212yesmaximal, complemented
5#8337C212yesmaximal, complemented
6C2×#532C2211yescharacteristic
7C2×#525C412yes—
8C23×C42C2211yescharacteristic
9C2×#407—22no—
10C2×C4×M16C2214yes—
11#2857C412yes—
12#2856C412yes—
13#2854—22no—
14#2850C2214yes—
15#2847—22no—
16C4×#121—22no—
17C4×#121—22no—
18#2297C412yes—
19#2285C2211yescharacteristic
20#1749C2214yes—
21#1749C2214yes—
22C4×C22⋊C8—22no—
23C4×C22⋊C8—22no—
24#1728—22no—
25C2×#329C2218yescomplemented
26#1295C2218yescomplemented
27C22×#41C2×C412yes—
28C24×C4C2×C411yescharacteristic
29C24×C4D412yes—
30C22×M16—24no—
31#532—44no—
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×#127C2×C411yescharacteristic
34C2×#124C2312yes—
35#525—48no—
36C2×#123—22no—
37C2×C22⋊Q8C2×C414yes—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121C2×C414yes—
42C2×#120—22no—
43C2×C4×D4—24no—
44C2×C4×D4—22no—
45C2×#117C2×C412yes—
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#407—216no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C4×M16—48no—
58C2×#97C2318yes—
59C2×C4×C8—24no—
60#391C2×C414yes—
61#390—22no—
62#390—22no—
63#390C2×C414yes—
64#389—22no—
65#389—22no—
66#388—24no—
67#388—24no—
68#386—24no—
69#386—24no—
70#386—24no—
71#384C2×C412yes—
72#382C2312yes—
73#380—22no—
74#380—22no—
75#380—22no—
76C4×C4⋊C4—22no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79#336—24no—
80#336—24no—
81#329—48no—
82C25C22⋊C411yescharacteristic
83C2×#41—44no—
84C2×#41—44no—
85C2×#41—44no—
86C22×Q8C22×C412yes—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4C22×C412yes—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4—22no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—44no—
97C23×C4—44no—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C2×M16—216no—
109C2×M16—48no—
110C2×M16—48no—
111C22×C8—28no—
112C4⋊Q8—42no—
113#127—42no—
114#124—28no—
115#123—48no—
116C22⋊Q8—48no—
117#121—48no—
118#121—48no—
119#121—48no—
120#121—48no—
121#120—48no—
122C4×D4—48no—
123C4×D4—416no—
124#117—44no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4C22×C414yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C42—24no—
137C2×C42—28no—
138C2×C42—22no—
139C2×C42—42no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42—44no—
144C2×C42C22⋊C411yescharacteristic
145C2×C42C22⋊C411yescharacteristic
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22×C411yescharacteristic
149C2×C42C22×C412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2411yescharacteristic
154C22⋊C8—216no—
155C22⋊C8—216no—
156#97—216no—
157C4×C8—48no—
158#95—24no—
159#95—24no—
160#95—24no—
161C24—22no—
162C24—24no—
163C24—22no—
164C24—41no—
165C24—41no—
166C24—42no—
167C24C2×C22⋊C411yescharacteristic
168C24C2×C22⋊C412yes—
169#41—816no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—24no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—48no—
179C2×D4—44no—
180C2×D4—48no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—48no—
184C2×D4—44no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—432no—
216C22×C4—416no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4#10014yes—
221C22×C4#10418yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C23×C411yescharacteristic
228C22×C4C22×D412yes—
229M16—432no—
230C2×C8—216no—
231C2×C8—48no—
232C2×C8—48no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C42—24no—
245C42—22no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—42no—
250C42—42no—
251C42—41no—
252C42—41no—
253C42—44no—
254C42H(5,𝔽2)14yes—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—44no—
266C23—48no—
267C23—48no—
268C23—44no—
269C23—44no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23C2×#10012yes—
276C23C2×#10414yes—
277C23C22×C22⋊C411yescharacteristic
278Q8—84no—
279D4—84no—
280D4—88no—
281D4—816no—
282D4—88no—
283C2×C4—216no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4C2×#10014yes—
315C2×C4#41312yes—
316C2×C4C2×#10414yes—
317C2×C4C2×#10414yes—
318C2×C4#42114yes—
319C2×C4#51812yes—
320C2×C4#53414yes—
321C8—416no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—416no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—82no—
336C22—84no—
337C22—82no—
338C22—84no—
339C22—82no—
340C22#230811yescharacteristic
341C22C22×#10011yescharacteristic
342C22C2×#41311yescharacteristic
343C22C22×#10412yes—
344C22C2×#42112yes—
345C4—24no—
346C4—24no—
347C4—24no—
348C4—24no—
349C4—22no—
350C4—22no—
351C4—24no—
352C4—84no—
353C4—82no—
354C2—44no—
355C2—42no—
356C2—162no—
357C2#1763811yescharacteristic
358C2#1764712yes—
359C2#1767311yescharacteristic
360C2#2175211yescharacteristic
361C2#2177312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. #9120C2semidirect
4. #8722C2semidirect
5. #8337C2semidirect
25. C2×#329C22semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C4×#4072
#91202
#87222
#83372