Finite Group Databasebeta

← #138512

Subgroups — #138512

Order 512.

469 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 84 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#29733C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic
3#11892C212yesmaximal, complemented
4C2×#1314C211yesmaximal, characteristic
5#7232C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8C2211yescharacteristic
9#2848—21no—
10C2×C22⋊C16C2212yes—
11#2075C411yescharacteristic
12#2075C411yescharacteristic
13#2035—21no—
14#2035—21no—
15#2001C411yescharacteristic
16#2001C411yescharacteristic
17#1892—21no—
18#1855—24no—
19#1810—21no—
20#1810C2212yes—
21#1807C2211yescharacteristic
22#1350—24no—
23#1314—21no—
24#1314—21no—
25C24×C4D411yescharacteristic
26C22×Q16C2×C411yescharacteristic
27C22×M16—22no—
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8—21no—
33C22×C16—22no—
34#467—22no—
35#467—22no—
36C2×#108—22no—
37C2×#108—22no—
38C2×#107C2×C411yescharacteristic
39C2×#106—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43#407—22no—
44#407—22no—
45#407—41no—
46#407—41no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—41no—
50C2×C22⋊C8—41no—
51C2×C22⋊C8C2×C411yescharacteristic
52C2×C22⋊C8C2×C411yescharacteristic
53C2×C22⋊C8D411yescharacteristic
54C2×C22⋊C8D411yescharacteristic
55#385—41no—
56#385—41no—
57#385—41no—
58#380—21no—
59#380—21no—
60#380—21no—
61#379—41no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#341—22no—
65#341D414yes—
66#340—21no—
67#340—21no—
68#340C2×C411yescharacteristic
69#340C2×C411yescharacteristic
70C25SD1611yescharacteristic
71C22×Q8—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4D811yescharacteristic
81C23×C4D811yescharacteristic
82C23×C4SD1611yescharacteristic
83C23×C4C2×D411yescharacteristic
84C2×Q16—22no—
85C2×Q16—81no—
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—82no—
89C2×M16—82no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—42no—
93C22×C8—81no—
94C22×C8—81no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22×C411yescharacteristic
98C22×C8C2×D412yes—
99C22×C8C2×D411yescharacteristic
100C22×C8#4112yes—
101C22⋊Q8—41no—
102C22⋊Q8—41no—
103C22⋊Q8—41no—
104C22⋊Q8—41no—
105C22⋊Q8—81no—
106C22⋊Q8—81no—
107C22⋊Q8—81no—
108C22⋊Q8—81no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C16—42no—
122C2×C16—82no—
123#108—82no—
124#108—82no—
125#107—22no—
126#106—81no—
127#103—42no—
128#103—81no—
129#103—81no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24—21no—
142C24—21no—
143C24—22no—
144C24—41no—
145C24—42no—
146C24—81no—
147C24#10211yescharacteristic
148C24#10211yescharacteristic
149C24C2×SD1611yescharacteristic
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
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—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4#10211yescharacteristic
186C22×C4#10211yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4AGL(1,ℤ/8)11yescharacteristic
192C22×C4#13711yescharacteristic
193Q16—82no—
194M16—44no—
195M16—44no—
196M16—84no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—44no—
207C2×C8—81no—
208C2×C8—84no—
209C2×C8C4×D412yes—
210C2×C8#12112yes—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C16—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23—81no—
247C23—81no—
248C23—84no—
249C23—84no—
250C23#35912yes—
251C23C2×#10211yescharacteristic
252C23#44811yescharacteristic
253C23#48111yescharacteristic
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—81no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4#38611yescharacteristic
296C2×C4#41711yescharacteristic
297C2×C4#44711yescharacteristic
298C2×C4#48011yescharacteristic
299C8—22no—
300C8—22no—
301C8—41no—
302C8—41no—
303C8—84no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—81no—
322C22—84no—
323C22—81no—
324C22—84no—
325C22#187411yescharacteristic
326C22C2×#35911yescharacteristic
327C22#214311yescharacteristic
328C22#223912yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—161no—
335C4—161no—
336C4—161no—
337C4—161no—
338C4#191111yescharacteristic
339C4#191311yescharacteristic
340C2—21no—
341C2—21no—
342C2—22no—
343C2—41no—
344C2—42no—
345C2—81no—
346C2#920311yescharacteristic
347C2#1299011yescharacteristic
348C2#1300911yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11892C2semidirect
5. #7232C2semidirect

Maximal subgroups

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

subgroupindex
#297332
#219482
#118922
C2×#13142
#72322