Finite Group Databasebeta

← #135170

Subgroups — #135170

Order 512.

1116 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 134 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×C4⋊#95C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7189C214yesmaximal, complemented
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C22×#95—22no—
8#1779—22no—
9C2×#324—22no—
10C2×#324C2214yes—
11C4⋊#95—22no—
12C4⋊#95C412yes—
13C4⋊#95C2212yes—
14C24×C4—21no—
15C24×C4—21no—
16C24×C4C2×C411yescharacteristic
17C24×C4D412yes—
18C22×C4⋊C4—21no—
19C22×C4⋊C4—24no—
20C22×C4⋊C4—24no—
21C22×C4⋊C4—42no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4—41no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4—42no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42—22no—
33C22×C42—42no—
34C22×C42—42no—
35C22×C42C814yescomplemented
36C22×C42C2×C412yes—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—42no—
39C2×C22⋊C8—42no—
40C2×#95—24no—
41C2×#95—24no—
42C2×#95—48no—
43#324—28no—
44#324—44no—
45C25C22⋊C411yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—44no—
68C23×C4—44no—
69C23×C4—42no—
70C23×C4—44no—
71C23×C4—44no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C412yes—
76C23×C4C2×C812yes—
77C23×C4M1612yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C22×C8—44no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—48no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4—48no—
96C2×C4⋊C4—48no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4M1614yes—
100C2×C4⋊C4#4114yes—
101C2×C42—24no—
102C2×C42—22no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—48no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—48no—
109C2×C42C2×C814yes—
110C22⋊C8—44no—
111C22⋊C8—44no—
112C22⋊C8—84no—
113C22⋊C8—84no—
114#95—48no—
115#95—48no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24C22⋊C812yes—
125C24#9911yescharacteristic
126C24#9911yescharacteristic
127C24#10112yes—
128C24C2×C22⋊C411yescharacteristic
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—28no—
134C22×C4—28no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—216no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—416no—
181C22×C4—416no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—88no—
190C22×C4—88no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4C22⋊C812yes—
196C22×C4#9912yes—
197C22×C4#10012yes—
198C22×C4#11712yes—
199C22×C4#12314yes—
200C22×C4C22×C811yescharacteristic
201C22×C4C2×M1612yes—
202C22×C4C2×M1611yescharacteristic
203C2×C8—84no—
204C2×C8—84no—
205C2×C8—84no—
206C4⋊C4—44no—
207C4⋊C4—48no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C42—44no—
213C42—44no—
214C42—48no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23#32412yes—
244C23#32412yes—
245C23#35312yes—
246C23#35412yes—
247C23#35512yes—
248C23#35612yes—
249C23#38111yescharacteristic
250C23C2×C22⋊C811yescharacteristic
251C23#40711yescharacteristic
252C23C2×#9911yescharacteristic
253C23C2×#9911yescharacteristic
254C23C2×#10011yescharacteristic
255C23C2×#10111yescharacteristic
256C2×C4—28no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—28no—
275C2×C4—24no—
276C2×C4—432no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—88no—
294C2×C4—88no—
295C2×C4—84no—
296C2×C4—88no—
297C2×C4—88no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#32314yes—
302C2×C4#43112yes—
303C2×C4#43212yes—
304C8—164no—
305C22—28no—
306C22—28no—
307C22—28no—
308C22—28no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22#132414yes—
334C22#132514yes—
335C22C2×#32311yescharacteristic
336C22C2×#32411yescharacteristic
337C22#146211yescharacteristic
338C22#146311yescharacteristic
339C22#181711yescharacteristic
340C22#181911yescharacteristic
341C22#182111yescharacteristic
342C22C2×#35311yescharacteristic
343C22C2×#35411yescharacteristic
344C22C2×#35511yescharacteristic
345C22C2×#35611yescharacteristic
346C4—24no—
347C4—22no—
348C4—22no—
349C4—48no—
350C4—44no—
351C4—82no—
352C4—84no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—24no—
357C2—24no—
358C2—21no—
359C2#869811yescharacteristic
360C2C2×#132411yescharacteristic
361C2C2×#132511yescharacteristic
362C2#1150511yescharacteristic
363C2#1150611yescharacteristic
364C2#1285611yescharacteristic
365C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×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. #7189C2semidirect
35. C22×C42C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#952
#82992
#71892