Finite Group Databasebeta

← #134281

Subgroups — #134281

Order 512.

794 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#381C211yesmaximal, characteristic
2C2×#1317C212yesmaximal
3#8559C212yesmaximal
4#7187C212yesmaximal
5C23×C42C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—22no—
11C2×#323C2214yes—
12#1434—22no—
13C22⋊#95C411yescharacteristic
14C22⋊#95C2211yescharacteristic
15C4×#95—22no—
16#1317—24no—
17#1280—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C22×C22⋊C4C2×C411yescharacteristic
22C22×C22⋊C4C2×C411yescharacteristic
23C22×C22⋊C4C2×C411yescharacteristic
24C22×C22⋊C4C2311yescharacteristic
25C22×C42—21no—
26C22×C42—21no—
27C22×C42—22no—
28C22×C42—22no—
29C22×C42—21no—
30C22×C42—42no—
31C22×C42—42no—
32C22×C42D412yes—
33C2×C4⋊C8—42no—
34C2×C22⋊C8—24no—
35#381—22no—
36#381—22no—
37#381C814yescomplemented
38#381D412yes—
39#381Q812yes—
40C4×C22⋊C4—42no—
41C4×C22⋊C4—42no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95D412yes—
48#336—42no—
49#323—24no—
50#323—24no—
51C25C22×C411yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—44no—
65C23×C4—44no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C4⋊C412yes—
76C23×C4C2×C812yes—
77C23×C4M1612yes—
78C23×C4M1612yes—
79C23×C4C2×D412yes—
80C22×C8—44no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4C4⋊C412yes—
92C2×C22⋊C4C2×C812yes—
93C2×C22⋊C4C2×C812yes—
94C2×C22⋊C4M1612yes—
95C2×C22⋊C4C2×D411yescharacteristic
96C2×C22⋊C4C2×Q811yescharacteristic
97C2×C22⋊C4#4112yes—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—48no—
104C2×C42—48no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—44no—
109C2×C42—42no—
110C2×C42—44no—
111C4⋊C8—84no—
112C22⋊C8—44no—
113C22⋊C8—44no—
114#95—28no—
115#95—24no—
116#95—24no—
117#95—42no—
118#95—44no—
119#95—44no—
120#95—42no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—42no—
126C24—42no—
127C24C2×C22⋊C411yescharacteristic
128C24C2×C4⋊C411yescharacteristic
129C24#11711yescharacteristic
130C24C22×C811yescharacteristic
131C24C2×M1611yescharacteristic
132C24C2×M1611yescharacteristic
133C24C2×M1611yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
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—48no—
170C22×C4—44no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊C814yes—
200C22×C4#9912yes—
201C22×C4#10012yes—
202C22×C4C4⋊C814yescomplemented
203C22×C4#12112yes—
204C22×C4C22⋊Q812yes—
205C2×C8—84no—
206C2×C8—84no—
207C2×C8—84no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—48no—
218C42—48no—
219C42—44no—
220C42—82no—
221C42—84no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—44no—
247C23—44no—
248C23—42no—
249C23—42no—
250C23#38011yescharacteristic
251C23C2×C22⋊C811yescharacteristic
252C23#40711yescharacteristic
253C23C2×#9911yescharacteristic
254C23#41011yescharacteristic
255C23C2×#10011yescharacteristic
256C23#41311yescharacteristic
257C23C2×C4⋊C811yescharacteristic
258C23#42311yescharacteristic
259C23#43111yescharacteristic
260C23#43211yescharacteristic
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4#35214yes—
313C8—164no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22#145911yescharacteristic
345C22#146011yescharacteristic
346C22#146211yescharacteristic
347C22#146411yescharacteristic
348C22#179711yescharacteristic
349C22#179911yescharacteristic
350C22#180011yescharacteristic
351C22C2×#35212yes—
352C22#212012yes—
353C4—24no—
354C4—44no—
355C4—44no—
356C4—42no—
357C4—44no—
358C4—42no—
359C4—42no—
360C4—82no—
361C4—82no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—41no—
370C2#855911yescharacteristic
371C2#1147212yes—
372C2#1150011yescharacteristic
373C2#1150111yescharacteristic
374C2#1279412yes—
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
37. #381C8semidirect
202. C22×C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3812
C2×#13172
#85592
#71872