Finite Group Databasebeta

← #141119

Subgroups — #141119

Order 512.

579 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 114 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#33173C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3C2×#1348C212yesmaximal, complemented
4C2×#1347C212yesmaximal
5C4×#340C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic, complemented
7C2×#467C2211yescharacteristic
8C2×#466C2211yescharacteristic
9C2×#465—21no—
10C22×C4×C8C2211yescharacteristic
11#2850—21no—
12C2×C4×C4⋊C4—21no—
13C2×C4⋊C16—21no—
14C2×C22⋊C16C2212yes—
15#1970C411yescharacteristic, complemented
16#1969C411yescharacteristic, complemented
17#1968—21no—
18#1958—21no—
19#1846C411yescharacteristic, complemented
20C2×#340—21no—
21C2×#340C2212yes—
22#1374—21no—
23#1348—24no—
24#1347—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#124—21no—
31C2×C22⋊Q8—21no—
32C2×#121—21no—
33C22×C4⋊C4—22no—
34C22×C42—41no—
35C22×C42D411yescharacteristic
36C22×C16—22no—
37#495—42no—
38#467—22no—
39#467—22no—
40#466—22no—
41#466—22no—
42#465—44no—
43C2×#107C2×C411yescharacteristic
44C2×#107C2×C411yescharacteristic
45C2×#106—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C4×C8—42no—
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#391—21no—
56#390—21no—
57#388—41no—
58#388—41no—
59#386—41no—
60#386—41no—
61#382—21no—
62C4×C4⋊C4—24no—
63C4×C4⋊C4—42no—
64C2×#95—41no—
65C4⋊C16—42no—
66C22⋊C16—22no—
67C22⋊C16—22no—
68#340—24no—
69#340—22no—
70#340D414yes—
71C22×Q8—21no—
72C22×D4—21no—
73C22×D4—41no—
74C23×C4—42no—
75C23×C4D811yescharacteristic
76C23×C4D811yescharacteristic
77C23×C4C2×D411yescharacteristic
78C2×Q16—22no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×SD16—42no—
82C2×SD16—42no—
83C2×SD16—42no—
84C2×D8—22no—
85C2×D8—41no—
86C2×D8—41no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D412yes—
97C22×C8#4112yes—
98#124—42no—
99#124—42no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102#121—42no—
103#121—42no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—44no—
122C2×C42—82no—
123C2×C42—82no—
124C2×C42C22⋊C411yescharacteristic
125C2×C42C22⋊C411yescharacteristic
126C2×C42SD1611yescharacteristic
127C2×C42SD1611yescharacteristic
128C2×C16—42no—
129C2×C16—42no—
130C2×C16—42no—
131#107—22no—
132#107—22no—
133#106—42no—
134#103—42no—
135#103—41no—
136#103—41no—
137#102—42no—
138#102—41no—
139#102—41no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—41no—
143C4×C8—41no—
144#95—44no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149C24—41no—
150C24C2×D811yescharacteristic
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—82no—
164C2×D4—82no—
165C2×D4—81no—
166C2×D4—81no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4#10211yescharacteristic
189C22×C4#10211yescharacteristic
190C22×C4#10211yescharacteristic
191C22×C4#10211yescharacteristic
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4#12112yes—
194C22×C4C2×SD1611yescharacteristic
195C22×C4AGL(1,ℤ/8)12yes—
196Q16—82no—
197SD16—84no—
198D8—82no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—44no—
211C2×C8#10412yes—
212C2×C8C4×D412yes—
213C2×C8#12411yescharacteristic
214C2×C8#12711yescharacteristic
215C4⋊C4—28no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—88no—
235C42#9912yes—
236C42#10412yes—
237C16—82no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23C2×#10211yescharacteristic
250C23#45912yes—
251C23#50812yes—
252C23#50911yescharacteristic
253C23#51011yescharacteristic
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—81no—
258D4—81no—
259D4—81no—
260D4—81no—
261D4—81no—
262D4—162no—
263D4—162no—
264C2×C4—24no—
265C2×C4—24no—
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—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#32712yes—
299C2×C4#32712yes—
300C2×C4#36012yes—
301C2×C4#39011yescharacteristic
302C2×C4C2×#9911yescharacteristic
303C2×C4#41011yescharacteristic
304C2×C4#41811yescharacteristic
305C2×C4C2×#10411yescharacteristic
306C2×C4C2×#10411yescharacteristic
307C2×C4#46012yes—
308C8—21no—
309C8—22no—
310C8—21no—
311C8—22no—
312C8—41no—
313C22—22no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—161no—
322C22—161no—
323C22—161no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22C2×#32711yescharacteristic
329C22#150211yescharacteristic
330C22#198211yescharacteristic
331C22#213811yescharacteristic
332C22#213911yescharacteristic
333C22C2×#36011yescharacteristic
334C22#214511yescharacteristic
335C22#221012yes—
336C22#221112yes—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—88no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4#150511yescharacteristic
347C4#150711yescharacteristic
348C4#196211yescharacteristic
349C4#196411yescharacteristic
350C2—22no—
351C2—22no—
352C2—321no—
353C2#971211yescharacteristic
354C2#1170311yescharacteristic
355C2#1170411yescharacteristic
356C2#1174111yescharacteristic
357C2#1174611yescharacteristic
358C2#1325411yescharacteristic
359C2#1325511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. C2×#1348C2semidirect
5. C4×#340C2semidirect
6. C2×#495C4semidirect
15. #1970C4semidirect
16. #1969C4semidirect
19. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331732
#127832
C2×#13482
C2×#13472
C4×#3402