Finite Group Databasebeta

← #141167

Subgroups — #141167

Order 512.

507 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 106 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
5#10820C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic, complemented
7C2×#467C2211yescharacteristic
8C2×#466C2211yescharacteristic
9C2×#465—21no—
10C22×C4×C8C2211yescharacteristic
11#2850—21no—
12C2×C4⋊C16—21no—
13C2×C22⋊C16C2212yes—
14#1970C411yescharacteristic, complemented
15#1969C411yescharacteristic, complemented
16#1968—21no—
17#1958—21no—
18#1846C411yescharacteristic, complemented
19C2×#340—21no—
20C2×#340C2212yes—
21C4⋊#95—21no—
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×C42D411yescharacteristic
35C22×C16—22no—
36#495—42no—
37#467—22no—
38#467—22no—
39#466—22no—
40#466—22no—
41#465—44no—
42C2×#107C2×C411yescharacteristic
43C2×#107C2×C411yescharacteristic
44C2×#106—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×C4×C8—42no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#391—21no—
55#390—21no—
56#388—41no—
57#388—41no—
58#386—41no—
59#386—41no—
60#382—21no—
61C2×#95—41no—
62C2×#95—41no—
63C4⋊C16—42no—
64C22⋊C16—22no—
65C22⋊C16—22no—
66#340—22no—
67#340—42no—
68#340D414yes—
69C22×Q8—21no—
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—42no—
73C23×C4D811yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4C2×D411yescharacteristic
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×SD16—42no—
80C2×SD16—42no—
81C2×SD16—42no—
82C2×D8—22no—
83C2×D8—41no—
84C2×D8—41no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C22×C8#4112yes—
96#124—42no—
97#124—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100#121—42no—
101#121—42no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42C22⋊C411yescharacteristic
120C2×C42C22⋊C411yescharacteristic
121C2×C42SD1611yescharacteristic
122C2×C42SD1611yescharacteristic
123C2×C16—42no—
124C2×C16—42no—
125C2×C16—42no—
126#107—22no—
127#107—22no—
128#106—42no—
129#103—42no—
130#103—41no—
131#103—41no—
132#102—42no—
133#102—41no—
134#102—41no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—41no—
138C4×C8—41no—
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—82no—
144#95—82no—
145C24—41no—
146C24C2×D811yescharacteristic
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—82no—
160C2×D4—82no—
161C2×D4—81no—
162C2×D4—81no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—44no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4#10211yescharacteristic
185C22×C4#10211yescharacteristic
186C22×C4#10211yescharacteristic
187C22×C4#10211yescharacteristic
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12112yes—
190C22×C4C2×SD1611yescharacteristic
191C22×C4AGL(1,ℤ/8)12yes—
192Q16—82no—
193SD16—84no—
194D8—82no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—44no—
207C2×C8#9912yes—
208C2×C8C4×D412yes—
209C2×C8#12411yescharacteristic
210C2×C8#12711yescharacteristic
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—41no—
232C16—82no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23C2×#10211yescharacteristic
245C23#45912yes—
246C23#50812yes—
247C23#50911yescharacteristic
248C23#51011yescharacteristic
249Q8—81no—
250Q8—81no—
251Q8—81no—
252Q8—81no—
253D4—81no—
254D4—81no—
255D4—81no—
256D4—81no—
257D4—162no—
258D4—162no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—162no—
292C2×C4—162no—
293C2×C4—162no—
294C2×C4—162no—
295C2×C4#36012yes—
296C2×C4#39011yescharacteristic
297C2×C4C2×#9911yescharacteristic
298C2×C4#41011yescharacteristic
299C2×C4#41811yescharacteristic
300C2×C4#42111yescharacteristic
301C2×C4#42111yescharacteristic
302C2×C4#46012yes—
303C8—21no—
304C8—22no—
305C8—21no—
306C8—22no—
307C8—41no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—161no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22—161no—
321C22—161no—
322C22—161no—
323C22#151711yescharacteristic
324C22#151911yescharacteristic
325C22#198211yescharacteristic
326C22#213811yescharacteristic
327C22#213911yescharacteristic
328C22C2×#36011yescharacteristic
329C22#214511yescharacteristic
330C22#221012yes—
331C22#221112yes—
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—164no—
338C4—161no—
339C4—161no—
340C4—161no—
341C4#151011yescharacteristic
342C4#151411yescharacteristic
343C4#196311yescharacteristic
344C4#196411yescharacteristic
345C2—22no—
346C2—22no—
347C2—321no—
348C2#971511yescharacteristic
349C2#1185511yescharacteristic
350C2#1186011yescharacteristic
351C2#1188511yescharacteristic
352C2#1189011yescharacteristic
353C2#1325411yescharacteristic
354C2#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. #10820C2semidirect
6. C2×#495C4semidirect
14. #1970C4semidirect
15. #1969C4semidirect
18. #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
#108202