Finite Group Databasebeta

← #128882

Subgroups — #128882

Order 512.

548 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 118 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#14241C211yesmaximal, characteristic
2C2×#1354C212yesmaximal
3#10802C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5#7251C212yesmaximal
6C2×#395C2211yescharacteristic
7C2×#391C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C4×C22⋊C8—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324C2212yes—
14#1446C411yescharacteristic
15#1444C411yescharacteristic
16#1441C2211yescharacteristic
17#1436—21no—
18#1436—21no—
19#1436—21no—
20C4⋊#95—21no—
21C4×#95C411yescharacteristic
22#1354C418yescomplemented
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C22×M16—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×C4×C8—41no—
37#395—22no—
38#395C2×C414yes—
39#391—41no—
40#391—41no—
41C4×C4⋊C4—41no—
42C4×C4⋊C4—41no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56#343—22no—
57#343—42no—
58#343C2×C414yes—
59#336—41no—
60#336—41no—
61#336—41no—
62#324—22no—
63#324C2×C414yes—
64C22×Q8—21no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—81no—
68C22×Q8—81no—
69C22×Q8C4212yes—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C2×M16—24no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C22×C8—42no—
88C22×C8—42no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4SD1612yes—
102C2×C4⋊C4Q1612yes—
103C2×C4⋊C4C22×C412yes—
104C2×C4⋊C4#4112yes—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—41no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—81no—
116C2×C42—81no—
117C2×C42—81no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C4×C8—82no—
121#95—24no—
122#95—21no—
123#95—24no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95C4212yes—
136C24C2×C22⋊C411yescharacteristic
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8—84no—
142C2×Q8—82no—
143C2×Q8—82no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
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—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4#10312yes—
204C22×C4#10312yes—
205C22×C4C2×C4211yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D412yes—
208C22×C4C4×D412yes—
209C22×C4C2×SD1611yescharacteristic
210C22×C4C2×Q1611yescharacteristic
211C22×C4#13512yes—
212M16—44no—
213M16—44no—
214M16—82no—
215M16—82no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—82no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—81no—
233C42—81no—
234C42—81no—
235C42—84no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23C4×C22⋊C411yescharacteristic
241C23C2×#9911yescharacteristic
242C23#41011yescharacteristic
243C23C2×#10311yescharacteristic
244C23#41611yescharacteristic
245C23#42111yescharacteristic
246C23#42111yescharacteristic
247Q8—84no—
248Q8—84no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
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—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4#35612yes—
320C2×C4C4×SD1612yes—
321C2×C4C4×Q1612yes—
322C8—84no—
323C8—84no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22#151611yescharacteristic
337C22#151811yescharacteristic
338C22#152011yescharacteristic
339C22#152111yescharacteristic
340C22C4×#9911yescharacteristic
341C22#175911yescharacteristic
342C22C4×#10311yescharacteristic
343C22#212111yescharacteristic
344C22#212211yescharacteristic
345C22C2×#35611yescharacteristic
346C22#213211yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—44no—
351C4—41no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C4—82no—
360C4—161no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2#822111yescharacteristic
366C2#1185111yescharacteristic
367C2#1185411yescharacteristic
368C2#1190611yescharacteristic
369C2#1190811yescharacteristic
370C2#1262211yescharacteristic
371C2C4×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
22. #1354C4semidirect

Maximal subgroups

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

subgroupindex
#142412
C2×#13542
#108022
C4×#3242
#72512