Finite Group Databasebeta

← #499690

Subgroups — #499690

Order 512.

526 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 130 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#21736C211yesmaximal, characteristic, complemented
2#21693C211yesmaximal, characteristic
3#13542C212yesmaximal, complemented
4#13488C211yesmaximal, characteristic, complemented
5#13487C211yesmaximal, characteristic
6#13414C212yesmaximal
7#13368C211yesmaximal, characteristic, complemented
8#13367C211yesmaximal, characteristic
9#12648C212yesmaximal, complemented
10#12647C212yesmaximal
11#12595C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13C2×#495C2211yescharacteristic
14#2961—21no—
15C2×#442C2211yescharacteristic
16C2×#441C2211yescharacteristic
17C2×#440—21no—
18C4×#124—21no—
19C2×#362C2212yes—
20C2×#361C2212yes—
21#2141C2212yes—
22#2140C2212yes—
23C2×#347C2212yes—
24#1994—21no—
25#1973—21no—
26#1972C2211yescharacteristic
27#1971C2211yescharacteristic
28#1958—21no—
29#1946—21no—
30#1945C2212yes—
31#1849C2211yescharacteristic
32#1846C2211yescharacteristic
33#1773C2212yes—
34#1772—21no—
35C4×#97C2211yescharacteristic
36#1386C2212yes—
37#1386C2212yes—
38#1383C2212yes—
39#1383C2212yes—
40#1377C2211yescharacteristic
41#1377C2211yescharacteristic
42#1376C2212yes—
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46#533—21no—
47C2×#124—21no—
48C2×C4×Q8—21no—
49C2×C4×D4—21no—
50C2×M32C2312yes—
51C2×M32C2312yes—
52#501—24no—
53#495—22no—
54#495—22no—
55#446—44no—
56#442—24no—
57#441—24no—
58#440—44no—
59C2×#108—21no—
60C2×#108C2312yes—
61#427—21no—
62#427C2312yes—
63C2×#107—21no—
64C2×#106C2311yescharacteristic
65C2×#106C2311yescharacteristic
66#424—22no—
67#423—21no—
68C2×#103—21no—
69C2×#103—21no—
70C2×#102—21no—
71C2×#102—21no—
72#405—22no—
73C2×#97D411yescharacteristic
74C2×#97D411yescharacteristic
75C2×#97C2311yescharacteristic
76C2×#97C2311yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391—21no—
80#390—21no—
81#388—42no—
82#382—21no—
83C4×C22⋊C4—42no—
84C43D411yescharacteristic
85#362—24no—
86#361—24no—
87#358—24no—
88#357—24no—
89#347—24no—
90#339—22no—
91#339—22no—
92C22×Q8—21no—
93C22×D4—21no—
94C23×C4—41no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—21no—
98C2×Q16#4112yes—
99C2×SD16—24no—
100C2×SD16—42no—
101C2×SD16—42no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—21no—
105C2×D8#4112yes—
106C2×M16—22no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2411yescharacteristic
111C4⋊Q8—22no—
112#125—22no—
113#124—24no—
114#124—42no—
115C22⋊Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×D4—42no—
120C4×D4—42no—
121#117—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—42no—
128C2×C42—22no—
129C2×C42—41no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133M32—22no—
134M32—22no—
135M32—22no—
136M32—22no—
137C2×C16—22no—
138C2×C16—22no—
139#108—22no—
140#108—22no—
141#108—42no—
142#107—22no—
143#107—42no—
144#107#4114yes—
145#106—22no—
146#106—22no—
147#106—22no—
148#106—22no—
149C4⋊C8—42no—
150C4⋊C8—44no—
151#103—42no—
152#103—42no—
153#102—42no—
154#102—42no—
155#97—22no—
156#97—22no—
157#97—22no—
158#97—42no—
159#97#4114yes—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—22no—
163C4×C8—42no—
164#95—42no—
165#95—42no—
166C24—41no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×D4—22no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—82no—
186C22×C4—84no—
187C22×C4AGL(1,ℤ/8)12yes—
188C22×C4C22×D411yescharacteristic
189Q16—24no—
190Q16—42no—
191SD16—48no—
192D8—24no—
193D8—42no—
194M16—42no—
195M16—42no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8#12412yes—
204C2×C8AGL(1,ℤ/8)12yes—
205C2×C8AGL(1,ℤ/8)12yes—
206C2×C8C2×#4111yescharacteristic
207C2×C8C2×#4111yescharacteristic
208C2×C8C2×#4112yes—
209C2×C8C2×#4112yes—
210C4⋊C4—24no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C22⋊C4—48no—
219C22⋊C4—42no—
220C22⋊C4—82no—
221C22⋊C4—84no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—82no—
228C42—82no—
229C42#12412yes—
230C42#13712yes—
231C16—42no—
232C16—42no—
233C23—42no—
234C23—82no—
235C23—82no—
236C23—81no—
237C23—81no—
238C23C2×AGL(1,ℤ/8)11yescharacteristic
239Q8—42no—
240Q8—42no—
241Q8—81no—
242Q8—81no—
243Q8—162no—
244D4—42no—
245D4—42no—
246D4—81no—
247D4—81no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—164no—
263C2×C4#52511yescharacteristic
264C2×C4#52511yescharacteristic
265C2×C4C2×#12411yescharacteristic
266C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
268C2×C4C2×#13711yescharacteristic
269C8—22no—
270C8—22no—
271C8—22no—
272C8—82no—
273C8#52912yes—
274C8#52912yes—
275C22—22no—
276C22—84no—
277C22—82no—
278C22—161no—
279C22—161no—
280C22—162no—
281C22#310911yescharacteristic
282C22#311011yescharacteristic
283C22#313811yescharacteristic
284C4—42no—
285C4—42no—
286C4—42no—
287C4—82no—
288C4—84no—
289C4—161no—
290C4—161no—
291C4#239811yescharacteristic
292C4#310911yescharacteristic
293C4#311211yescharacteristic
294C4#314111yescharacteristic
295C2—21no—
296C2—21no—
297C2—162no—
298C2#2382211yescharacteristic
299C2#3369012yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21736C2semidirect
3. #13542C2semidirect
4. #13488C2semidirect
7. #13368C2semidirect
9. #12648C2semidirect
11. #12595C2semidirect

Maximal subgroups

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

subgroupindex
#217362
#216932
#135422
#134882
#134872
#134142
#133682
#133672
#126482
#126472
#125952