Finite Group Databasebeta

← #138248

Subgroups — #138248

Order 512.

1029 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 138 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#32930C211yesmaximal, characteristic
2C22×#340C211yesmaximal, characteristic, complemented
3#12679C211yesmaximal, characteristic, complemented
4C2×#1347C212yesmaximal, complemented
5#7258C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C23×C4⋊C4—21no—
8C2×#466C412yescomplemented
9C2×#466C2211yescharacteristic
10#2847—21no—
11C2×C22⋊C16—22no—
12C2×C22⋊C16C2212yes—
13#1893—22no—
14#1893—21no—
15#1808C412yescomplemented
16#1808C2211yescharacteristic
17C2×#340—24no—
18C2×#340—21no—
19C2×#340C2212yes—
20#1347—24no—
21C24×C4—41no—
22C24×C4D411yescharacteristic
23C22×D8C2×C411yescharacteristic, complemented
24C23×C8—22no—
25C23×C8—22no—
26C23×C8—44no—
27C23×C8C2×C412yes—
28C23×C8D412yes—
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×#121—22no—
32C2×#121—21no—
33C2×#120—41no—
34C22×C4⋊C4—28no—
35C22×C4⋊C4—24no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—44no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—41no—
40C22×C16—22no—
41#466—22no—
42#466—22no—
43#466—44no—
44C2×#107C2×C411yescharacteristic, complemented
45C2×#107C2×C412yescomplemented
46C2×#102—22no—
47C2×#102—21no—
48#386—41no—
49#386—42no—
50#380—22no—
51#380—21no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54C22⋊C16—44no—
55#340—24no—
56#340—22no—
57#340—21no—
58#340—22no—
59#340D414yescomplemented
60C25D811yescharacteristic
61C22×D4—21no—
62C22×D4—42no—
63C22×D4—41no—
64C23×C4—24no—
65C23×C4—24no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—48no—
71C23×C4—44no—
72C23×C4—42no—
73C23×C4—84no—
74C23×C4—82no—
75C23×C4—82no—
76C23×C4C22⋊C412yes—
77C23×C4D811yescharacteristic
78C23×C4SD1612yes—
79C23×C4C2×D411yescharacteristic
80C2×D8—22no—
81C2×D8—42no—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—24no—
87C22×C8—24no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—48no—
93C22×C8—48no—
94C22×C8C22⋊C412yes—
95C22×C8C22×C411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D412yes—
98C22×C8#4112yes—
99#121—44no—
100#121—44no—
101#121—41no—
102#121—41no—
103#121—41no—
104#121—41no—
105#120—82no—
106#120—82no—
107C2×C4⋊C4—216no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—416no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—41no—
122C2×C16—44no—
123C2×C16—42no—
124#107—22no—
125#107—42no—
126#102—44no—
127#102—42no—
128#95—42no—
129#95—41no—
130C24—24no—
131C24—24no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24#10212yes—
138C24D1612yes—
139C24SD3212yes—
140C24C2×D811yescharacteristic
141C2×D4—41no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—84no—
146C2×D4—84no—
147C2×D4—84no—
148C2×D4—81no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4—81no—
153C2×D4—81no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—28no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—416no—
174C22×C4—48no—
175C22×C4—44no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—816no—
179C22×C4—88no—
180C22×C4—816no—
181C22×C4—84no—
182C22×C4—88no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4#9914yes—
186C22×C4#10212yes—
187C22×C4#10414yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4C2×SD1611yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4#13711yescharacteristic
194D8—82no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—48no—
203C2×C8—48no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C4×D412yes—
207C2×C8#12112yes—
208C4⋊C4—28no—
209C4⋊C4—416no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C22⋊C4—84no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C22⋊C4—84no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C16—82no—
223C23—24no—
224C23—24no—
225C23—28no—
226C23—24no—
227C23—28no—
228C23—24no—
229C23—28no—
230C23—28no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—82no—
245C23—82no—
246C23#32714yes—
247C23#35714yes—
248C23#36014yes—
249C23C2×#10211yescharacteristic
250C23#44711yescharacteristic
251C23#48011yescharacteristic
252C23C2×D1611yescharacteristic
253C23C2×SD3211yescharacteristic
254C23#50911yescharacteristic
255C23#51011yescharacteristic
256D4—81no—
257D4—81no—
258D4—82no—
259D4—168no—
260D4—161no—
261D4—161no—
262D4—161no—
263D4—161no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—48no—
282C2×C4—832no—
283C2×C4—88no—
284C2×C4—88no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#32714yes—
294C2×C4#38611yescharacteristic
295C2×C4C2×#9912yes—
296C2×C4#41711yescharacteristic
297C2×C4C2×#10412yes—
298C2×C4#44811yescharacteristic
299C2×C4#48111yescharacteristic
300C8—22no—
301C8—22no—
302C8—42no—
303C22—28no—
304C22—28no—
305C22—28no—
306C22—28no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—21no—
315C22—24no—
316C22—21no—
317C22—24no—
318C22—21no—
319C22—24no—
320C22—21no—
321C22—162no—
322C22—162no—
323C22—161no—
324C22—161no—
325C22—161no—
326C22#134718yescomplemented
327C22C2×#32712yes—
328C22#187411yescharacteristic
329C22C2×#35711yescharacteristic
330C22#213911yescharacteristic
331C22C2×#36012yes—
332C22#218411yescharacteristic
333C22#218511yescharacteristic
334C22#223211yescharacteristic
335C22#223311yescharacteristic
336C4—24no—
337C4—21no—
338C4—21no—
339C4—88no—
340C4—162no—
341C4—161no—
342C4#150412yes—
343C4#189611yescharacteristic
344C4#189811yescharacteristic
345C2—21no—
346C2—28no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—321no—
351C2#914711yescharacteristic
352C2C2×#134712yes—
353C2#1170412yes—
354C2#1299111yescharacteristic
355C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#340C2semidirect
3. #12679C2semidirect
4. C2×#1347C2semidirect
5. #7258C2semidirect
8. C2×#466C4semidirect
15. #1808C4semidirect
23. C22×D8C2×C4semidirect
44. C2×#107C2×C4semidirect
45. C2×#107C2×C4semidirect
59. #340D4semidirect
326. C22#1347semidirect

Maximal subgroups

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

subgroupindex
#329302
C22×#3402
#126792
C2×#13472
#72582