Finite Group Databasebeta

← #213542

Subgroups — #213542

Order 512.

1356 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1356 conjugacy classes of subgroups — too many to draw. Only its 218 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×#2854C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#9084C218yesmaximal, complemented
4#8704C212yesmaximal, complemented
5#8307C212yesmaximal
6C22×#123C412yes—
7C22×#121C2211yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#432C2214yes—
10#2854—22no—
11#2854—22no—
12#2854—22no—
13#2854—22no—
14C2×#390C412yes—
15C2×#389C412yes—
16C2×#388C2214yes—
17C2×#386C412yes—
18C2×#380C2211yescharacteristic
19#1817C2214yes—
20#1797—22no—
21#1797—22no—
22#1779C2214yes—
23#1749—22no—
24#1749—22no—
25C2×#323C2218yes—
26#1280C2218yes—
27C23×D4C2×C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2311yescharacteristic
32C2×#123—24no—
33C2×#123—22no—
34C2×#123—22no—
35C2×#123—48no—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—44no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C414yes—
41C22×C4⋊C4C2×C412yes—
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2×C414yes—
44C22×C22⋊C4C2×C414yes—
45C22×C22⋊C4C2312yes—
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2312yes—
53#432—48no—
54C2×C4⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8C2318yes—
58C2×#97—24no—
59#390—22no—
60#390—22no—
61#390—22no—
62#390—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#388—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#386—22no—
72#386—22no—
73#386—24no—
74#380—22no—
75#380—22no—
76C2×#95C2×C414yes—
77C2×#95C2312yes—
78#336—24no—
79#336—24no—
80#323—48no—
81C25C22×C411yescharacteristic
82C25C22×C412yes—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—42no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4—44no—
90C22×D4—44no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—44no—
94C23×C4—44no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C22×C414yes—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
112C22×C8—28no—
113#123—48no—
114#123—416no—
115#123—48no—
116#121—48no—
117#121—48no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—48no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—48no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—44no—
137C2×C42—28no—
138C2×C42—44no—
139C2×C42—44no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C412yes—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C4⋊C8—48no—
152C22⋊C8—216no—
153C22⋊C8—48no—
154C22⋊C8—48no—
155#97—48no—
156#95—24no—
157#95—24no—
158#95—24no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—24no—
163C24—22no—
164C24—24no—
165C24—28no—
166C24—42no—
167C24—42no—
168C24—44no—
169C24—42no—
170C24C2×C22⋊C411yescharacteristic
171C24C2×C22⋊C411yescharacteristic
172C24C23×C411yescharacteristic
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—88no—
183C2×D4—816no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
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—22no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—416no—
219C22×C4—416no—
220C22×C4—416no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4#10014yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C22×D412yes—
232C22×C4H(5,𝔽2)14yes—
233C2×C8—48no—
234C2×C8—48no—
235C2×C8—48no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C42—24no—
249C42—24no—
250C42—24no—
251C42—24no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—28no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—48no—
269C23—48no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23—42no—
276C23—42no—
277C23—48no—
278C23—84no—
279C23—84no—
280C23—88no—
281C23—88no—
282C23C2×#10012yes—
283C23#41312yes—
284C23C22×C22⋊C411yescharacteristic
285C23#51812yes—
286C23#56814yes—
287D4—88no—
288D4—88no—
289D4—88no—
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—88no—
324C2×C4—84no—
325C2×C4#41014yes—
326C2×C4C2×#10014yes—
327C2×C4#53414yes—
328C8—88no—
329C22—22no—
330C22—24no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—416no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—82no—
345C22—82no—
346C22—84no—
347C22—82no—
348C22—84no—
349C22—168no—
350C22#230811yescharacteristic
351C22#287912yes—
352C22C2×#41012yes—
353C22C22×#10011yescharacteristic
354C22C2×#41311yescharacteristic
355C22#297814yes—
356C4—24no—
357C4—24no—
358C4—24no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—84no—
364C2—24no—
365C2—42no—
366C2—42no—
367C2—162no—
368C2#1739912yes—
369C2#1740312yes—
370C2#2174311yescharacteristic
371C2#2174711yescharacteristic
372C2#2175211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #9084C2semidirect
4. #8704C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#207152
#90842
#87042
#83072