Finite Group Databasebeta

← #141949

Subgroups — #141949

Order 512.

736 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 120 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#14130C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7187C214yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C2×#389—21no—
7C2×#385C412yes—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323C2212yes—
13C2×#323C2212yes—
14#1444—21no—
15#1442—21no—
16C4⋊#95—21no—
17C22⋊#95C2212yes—
18#1280—21no—
19#1280—21no—
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C412yes—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—44no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C22⋊C4C2311yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C4×C8—41no—
38#389—44no—
39#385—44no—
40#380—22no—
41#380C814yescomplemented
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95D411yescharacteristic
49C2×#95D411yescharacteristic
50#336—41no—
51#336—41no—
52#323—22no—
53#323—24no—
54#323Q814yes—
55C25C22×C411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—44no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—82no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×C812yes—
73C23×C4M1612yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—48no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—82no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4C2×C814yes—
102C2×C22⋊C4C2×Q812yes—
103C2×C22⋊C4#4112yes—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—42no—
109C2×C42—42no—
110C4⋊C8—82no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—82no—
116#95—24no—
117#95—41no—
118#95—41no—
119#95—44no—
120#95—44no—
121#95—82no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24C2×C22⋊C411yescharacteristic
131C24C4×Q812yes—
132C24C22×C811yescharacteristic
133C24C2×M1611yescharacteristic
134C24#13112yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—48no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—88no—
186C22×C4—88no—
187C22×C4—84no—
188C22×C4—84no—
189C22×C4C22⋊C812yes—
190C22×C4C22⋊C812yes—
191C22×C4#9911yescharacteristic
192C22×C4#9911yescharacteristic
193C22×C4#10012yes—
194C22×C4#10112yes—
195C22×C4C22⋊Q812yes—
196C22×C4#12411yescharacteristic
197C22×C4C4⋊Q811yescharacteristic
198C2×C8—44no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C4⋊C4—48no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—88no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—81no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#39111yescharacteristic
249C23C2×C22⋊C811yescharacteristic
250C23#40811yescharacteristic
251C23C2×#9911yescharacteristic
252C23#41011yescharacteristic
253C23C2×#10011yescharacteristic
254C23C2×#10111yescharacteristic
255C23C8×Q812yes—
256C23#44612yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—88no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—88no—
308C2×C4#32412yes—
309C2×C4#32412yes—
310C2×C4AΣL(1,𝔽16)12yes—
311C2×C4#35212yes—
312C8—84no—
313C8—162no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22C2×#32411yescharacteristic
343C22#146011yescharacteristic
344C22#147011yescharacteristic
345C22#147111yescharacteristic
346C22#198411yescharacteristic
347C22#199611yescharacteristic
348C22#199711yescharacteristic
349C22C2×AΣL(1,𝔽16)11yescharacteristic
350C22C2×#35211yescharacteristic
351C22#212211yescharacteristic
352C22#212311yescharacteristic
353C4—22no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—42no—
359C4—84no—
360C4—81no—
361C4—84no—
362C4#132614yes—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—42no—
369C2#987911yescharacteristic
370C2C2×#132611yescharacteristic
371C2#1145411yescharacteristic
372C2#1155811yescharacteristic
373C2#1156111yescharacteristic
374C2#1329311yescharacteristic
375C2#1329511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
41. #380C8semidirect

Maximal subgroups

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

subgroupindex
#141302
#85572
C4×#3232
#71872