Finite Group Databasebeta

← #136943

Subgroups — #136943

Order 512.

370 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 98 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#33626C211yesmaximal, characteristic
2#22688C211yesmaximal, characteristic
3#13285C211yesmaximal, characteristic, complemented
4#11591C211yesmaximal, characteristic
5#9749C211yesmaximal, characteristic, complemented
6#7276C211yesmaximal, characteristic, complemented
7#7167C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443—21no—
10C4×#135C2211yescharacteristic
11#2906—21no—
12#2229—21no—
13#2228—21no—
14#2227C411yescharacteristic, complemented
15#2226C411yescharacteristic, complemented
16#2225C411yescharacteristic, complemented
17#2224C411yescharacteristic, complemented
18#2158—21no—
19C2×C4⋊C16C2211yescharacteristic
20#2138—21no—
21C2×C4×C16C2211yescharacteristic
22#1945—21no—
23#1866—22no—
24#1848C2211yescharacteristic
25#1806—21no—
26C8×C4⋊C4C2211yescharacteristic
27#1564C2211yescharacteristic
28#1503—21no—
29#1481—21no—
30#1382—22no—
31#1332—21no—
32#1331—21no—
33C2×#135D411yescharacteristic
34C2×#131—22no—
35#533—21no—
36C4×#41—21no—
37C22×C16—21no—
38C22×C16—21no—
39#500C2×C411yescharacteristic
40#500C2×C411yescharacteristic
41#499—21no—
42#499—21no—
43#498—21no—
44#443—42no—
45#443—42no—
46C4×Q16C2×C411yescharacteristic
47C4×Q16D411yescharacteristic
48C4×SD16—21no—
49C4×SD16—21no—
50C4×D8C2×C411yescharacteristic
51C4×D8D411yescharacteristic
52#431—21no—
53C2×#108—21no—
54C2×#108D412yes—
55#427—21no—
56#427—21no—
57#427D411yescharacteristic
58#424—22no—
59#424—42no—
60#423—21no—
61#423—41no—
62C2×C4⋊C8—41no—
63C2×C4⋊C8D411yescharacteristic
64C2×#104—22no—
65#416—21no—
66C2×C4×C8—21no—
67C2×C4×C8C2311yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69#367—21no—
70#367—21no—
71#366—21no—
72#366—21no—
73C4⋊C16—21no—
74C4⋊C16C2×C411yescharacteristic
75C4⋊C16C2×C411yescharacteristic
76#358—21no—
77#358—21no—
78#357—21no—
79#357—21no—
80C4×C16—21no—
81C4×C16—21no—
82#339—22no—
83#336—21no—
84#335—21no—
85#334—21no—
86#328—42no—
87Q8⋊C8—41no—
88Q8⋊C8—41no—
89C2×#41—21no—
90#135—21no—
91#135—21no—
92#135SD1612yes—
93#135Q1612yes—
94C2×Q16C22⋊C411yescharacteristic
95C2×SD16—21no—
96C2×D8C22⋊C411yescharacteristic
97#131—44no—
98#131—82no—
99C2×M16—22no—
100C2×M16—42no—
101C22×C8—21no—
102C22×C8—42no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107#125—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—41no—
117C4×D4—41no—
118#117—21no—
119#117—41no—
120#117—41no—
121C2×C4⋊C4—21no—
122C2×C42—21no—
123C2×C42—41no—
124C2×C42C2×D411yescharacteristic
125C2×C16—22no—
126C2×C16—21no—
127C2×C16—21no—
128C2×C16—41no—
129C2×C16—41no—
130C2×C16—41no—
131C2×C16—41no—
132#108—42no—
133#108—41no—
134#108—41no—
135#107—21no—
136#107—21no—
137#107—21no—
138#107—21no—
139#107C22⋊C411yescharacteristic
140#107C22⋊C411yescharacteristic
141#106—21no—
142#106—41no—
143#106—41no—
144C4⋊C8—21no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8D811yescharacteristic
151C4⋊C8D811yescharacteristic
152#104—44no—
153#103—21no—
154#103—21no—
155#102—21no—
156#102—21no—
157C22⋊C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8C22×C411yescharacteristic
162C4×C8C2×D411yescharacteristic
163C4×C8#4111yescharacteristic
164C4×C8#4111yescharacteristic
165#95—21no—
166#41—22no—
167#41—22no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×D4—21no—
174C2×D4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4#12011yescharacteristic
180C22×C4#13511yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182Q16—21no—
183Q16#10312yes—
184SD16—22no—
185SD16—22no—
186D8—21no—
187D8#10312yes—
188M16—42no—
189M16—42no—
190M16—84no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—84no—
199C2×C8—81no—
200C2×C8C2×C22⋊C411yescharacteristic
201C2×C8C4×D411yescharacteristic
202C2×C8#12111yescharacteristic
203C2×C8#12311yescharacteristic
204C2×C8C2×SD1611yescharacteristic
205C2×C8C2×Q1611yescharacteristic
206C2×C8AGL(1,ℤ/8)11yescharacteristic
207C2×C8#13711yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—81no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—81no—
230C42C4×D411yescharacteristic
231C42#12111yescharacteristic
232C42C2×D811yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C16—42no—
235C16—41no—
236C16—41no—
237C23—41no—
238C23#44911yescharacteristic
239Q8—22no—
240Q8—41no—
241Q8—81no—
242Q8—81no—
243Q8—161no—
244D4—22no—
245D4—42no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—161no—
264C2×C4#38611yescharacteristic
265C2×C4C4×D811yescharacteristic
266C2×C4#44211yescharacteristic
267C2×C4#44711yescharacteristic
268C2×C4#45011yescharacteristic
269C2×C4#45111yescharacteristic
270C2×C4#45511yescharacteristic
271C2×C4#45611yescharacteristic
272C2×C4#45911yescharacteristic
273C2×C4#46311yescharacteristic
274C8—21no—
275C8—21no—
276C8—41no—
277C8—41no—
278C8—82no—
279C8—161no—
280C8C2×#10311yescharacteristic
281C8#41711yescharacteristic
282C8#48311yescharacteristic
283C8#48411yescharacteristic
284C22—21no—
285C22—21no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22#164411yescharacteristic
290C22#165411yescharacteristic
291C22#187911yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—42no—
296C4—82no—
297C4—81no—
298C4—81no—
299C4—161no—
300C4—161no—
301C4#163911yescharacteristic
302C4#164911yescharacteristic
303C4#187211yescharacteristic
304C4#188211yescharacteristic
305C4#217811yescharacteristic
306C4#217911yescharacteristic
307C2—21no—
308C2—21no—
309C2—82no—
310C2#898111yescharacteristic
311C2#1301111yescharacteristic
312C2#1302111yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1879
prime radicalC2×#135D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13285C2semidirect
5. #9749C2semidirect
6. #7276C2semidirect
7. #7167C2semidirect
14. #2227C4semidirect
15. #2226C4semidirect
16. #2225C4semidirect
17. #2224C4semidirect

Maximal subgroups

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

subgroupindex
#336262
#226882
#132852
#115912
#97492
#72762
#71672