Finite Group Databasebeta

← #128313

Subgroups — #128313

Order 512.

790 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 150 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#13785C211yesmaximal, characteristic
2C2×#1318C212yesmaximal
3C4×#324C212yesmaximal, complemented
4#7189C212yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C2×#390C411yescharacteristic
7C2×#390C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×#376C411yescharacteristic
10C4×C22⋊C8—22no—
11C2×#324C2214yes—
12#1437—22no—
13C4⋊#95C411yescharacteristic
14C4⋊#95C2211yescharacteristic
15C22⋊#95—22no—
16#1318C418yescomplemented
17#1280—22no—
18C23×D4C2×C411yescharacteristic
19C24×C4—21no—
20C2×C4×D4—22no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C2×C4×D4C814yescomplemented
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2311yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—22no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34C2×C22⋊C8—24no—
35C2×C4×C8—42no—
36#390—22no—
37#390—22no—
38#390C814yescomplemented
39#390C2×C414yes—
40C4×C4⋊C4—42no—
41#376—42no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95D412yes—
47#336—42no—
48#324—24no—
49#324C2×C418yescomplemented
50C25—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—41no—
55C22×D4—41no—
56C22×D4—41no—
57C22×D4—41no—
58C22×D4C4212yes—
59C22×D4C2×C812yes—
60C22×D4C2×C812yes—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—24no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—82no—
68C23×C4—82no—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D412yes—
74C22×C8—44no—
75C4×D4—48no—
76C4×D4—48no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C2×C812yes—
85C2×C4⋊C4C2×C812yes—
86C2×C4⋊C4M1612yes—
87C2×C4⋊C4C22×C412yes—
88C2×C4⋊C4#4112yes—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—82no—
95C2×C22⋊C4—82no—
96C2×C22⋊C4—82no—
97C2×C22⋊C4—82no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42—82no—
109C2×C42C4212yes—
110C2×C42C2×C812yes—
111C2×C42C2×C812yes—
112C2×C42M1612yes—
113C22⋊C8—28no—
114C22⋊C8—44no—
115C4×C8—84no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—82no—
120#95—82no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—42no—
125C24—42no—
126C24—42no—
127C24—42no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24C2×C22⋊C411yescharacteristic
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C4×C814yescomplemented
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—48no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—88no—
188C22×C4—88no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4C22⋊C812yes—
192C22×C4C22⋊C814yes—
193C22×C4C22⋊C812yes—
194C22×C4#9912yes—
195C22×C4#10012yes—
196C22×C4C2×C4211yescharacteristic
197C22×C4#11711yescharacteristic
198C22×C4C4×D414yes—
199C22×C4C22×C811yescharacteristic
200C22×C4C22×C811yescharacteristic
201C22×C4C2×M1611yescharacteristic
202C22×C4C2×M1611yescharacteristic
203C2×C8—48no—
204C2×C8—84no—
205C2×C8—84no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C22⋊C4—48no—
213C22⋊C4—88no—
214C22⋊C4—88no—
215C42—24no—
216C42—24no—
217C42—42no—
218C42—42no—
219C42—44no—
220C42—42no—
221C42—42no—
222C42—82no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—24no—
228C23—44no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23—44no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—82no—
241C23—82no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23C4×C22⋊C411yescharacteristic
251C23C2×C22⋊C811yescharacteristic
252C23C2×C22⋊C811yescharacteristic
253C23C2×#9911yescharacteristic
254C23#41011yescharacteristic
255C23C2×#10011yescharacteristic
256C23#41311yescharacteristic
257D4—44no—
258D4—44no—
259D4—44no—
260D4—44no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—816no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4#32314yes—
313C2×C4C2×C4×C811yescharacteristic
314C2×C4C4×M1611yescharacteristic
315C2×C4#43111yescharacteristic
316C2×C4#43111yescharacteristic
317C8—88no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—48no—
331C22—44no—
332C22—84no—
333C22—84no—
334C22—84no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22C2×#32311yescharacteristic
342C22#145811yescharacteristic
343C22#145911yescharacteristic
344C22#146011yescharacteristic
345C22C4×C22⋊C811yescharacteristic
346C22C4×#9911yescharacteristic
347C22C4×#10011yescharacteristic
348C22#212711yescharacteristic
349C22#212811yescharacteristic
350C22#213311yescharacteristic
351C22#213411yescharacteristic
352C4—24no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—44no—
358C4—42no—
359C4—44no—
360C4—82no—
361C4—82no—
362C4—82no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—84no—
368C2C4×#32311yescharacteristic
369C2#1143211yescharacteristic
370C2#1143311yescharacteristic
371C2#1147911yescharacteristic
372C2#1148011yescharacteristic
373C2#1262811yescharacteristic
374C2#1263211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#324C2semidirect
4. #7189C2semidirect
16. #1318C4semidirect
23. C2×C4×D4C8semidirect
38. #390C8semidirect
49. #324C2×C4semidirect
148. C2×D4C4×C8semidirect

Maximal subgroups

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

subgroupindex
#137852
C2×#13182
C4×#3242
#71892