Finite Group Databasebeta

← #238817

Subgroups — #238817

Order 512.

992 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C2×#3245C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic, complemented
3#17750C212yesmaximal, complemented
4#17544C212yesmaximal, complemented
5#10513C218yesmaximal, complemented
6#7988C211yesmaximal, characteristic, complemented
7C2×#532C2212yes—
8#3245—22no—
9#3245—24no—
10#3245C2214yes—
11C2×#474—22no—
12C2×#466—22no—
13C2×#459—22no—
14#2920C2212yes—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C4×#117C2211yescharacteristic
18#1832C2214yescomplemented
19#1831—22no—
20#1729—22no—
21#1560—24no—
22#1551C2218yescomplemented
23#1548—22no—
24#1545—22no—
25#1507—24no—
26#1505C2218yescomplemented
27#1502—24no—
28#1290C2214yescomplemented
29C22×#41—22no—
30C22×D8—22no—
31C22×M16—21no—
32C23×C8D412yes—
33#532—22no—
34#532—22no—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39#532D414yes—
40#532C2314yes—
41C2×C4⋊Q8D412yes—
42C2×#127D412yes—
43C2×#124—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#474—28no—
51#474—28no—
52#466—28no—
53#466—28no—
54#459—28no—
55#459—28no—
56#432—22no—
57#432C2314yes—
58#431—22no—
59#431—22no—
60C2×#107—22no—
61#423—22no—
62#423—22no—
63#423—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×#102—24no—
71#407—22no—
72C4×M16—22no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4D412yes—
75C4×C22⋊C4—21no—
76C4×C22⋊C4D412yes—
77#376C2312yes—
78C43—21no—
79#336—28no—
80#333—28no—
81#331—28no—
82#327—216no—
83C2×#41—24no—
84C2×#41—24no—
85C2×#41—42no—
86C2×#41—42no—
87C2×#41—88no—
88C22×Q8—22no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—42no—
93C23×C4—42no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C2×D8—28no—
98C2×D8—84no—
99C2×M16—44no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—42no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—44no—
107C22×C8C2×D414yes—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8C2×D414yes—
112#127—22no—
113#127—22no—
114#127—22no—
115#127C2×D414yes—
116#124—24no—
117#124—24no—
118#124—42no—
119#124—42no—
120#121—24no—
121#121—24no—
122#121—24no—
123#121—24no—
124#121—42no—
125#121—42no—
126#121—42no—
127#121—42no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136#117—42no—
137#117—42no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151C2×C42#4112yes—
152C2×C42#4112yes—
153C2×C42#4111yescharacteristic
154C2×C42C2411yescharacteristic
155#107—28no—
156C4⋊C8—24no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—24no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162#102—216no—
163C22⋊C8—24no—
164C22⋊C8—42no—
165#97—24no—
166C4×C8—42no—
167#95—22no—
168#95C2×D414yes—
169C24—42no—
170C24#12111yescharacteristic
171#41—88no—
172#41—88no—
173#41—88no—
174#41—88no—
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—84no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—84no—
189C2×D4—84no—
190C2×D4—88no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—88no—
199C22×C4—81no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12112yes—
204C22×C4#12112yes—
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12312yes—
208C22×C4#12312yes—
209C22×C4#12411yescharacteristic
210C22×C4#12411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214D8—88no—
215M16—82no—
216M16—84no—
217M16—82no—
218C2×C8—24no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—82no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—42no—
236C42—42no—
237C42—44no—
238C42—82no—
239C42#12012yes—
240C42#12112yes—
241C42#12112yes—
242C42#12312yes—
243C42#12312yes—
244C42#12412yes—
245C42C22×D412yes—
246C42C2×#4112yes—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—44no—
251C23—84no—
252C23—84no—
253C23—84no—
254C23#39411yescharacteristic
255C23#39411yescharacteristic
256C23C2×#12111yescharacteristic
257Q8—84no—
258Q8—84no—
259D4—44no—
260D4—44no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—88no—
265D4—88no—
266D4—84no—
267D4—84no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—88no—
287C2×C4—84no—
288C2×C4—88no—
289C2×C4#39412yes—
290C2×C4#39411yescharacteristic
291C2×C4#39411yescharacteristic
292C2×C4#39412yes—
293C2×C4#45412yes—
294C2×C4#45512yes—
295C2×C4#47214yes—
296C2×C4C2×#12011yescharacteristic
297C2×C4C2×#12112yes—
298C2×C4C2×#12312yes—
299C2×C4C2×#12411yescharacteristic
300C8—44no—
301C8—84no—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—44no—
311C22—84no—
312C22—84no—
313C22—84no—
314C22—168no—
315C22#204012yes—
316C22C2×#39411yescharacteristic
317C22C2×#47212yes—
318C4—21no—
319C4—21no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—84no—
324C4—84no—
325C4#204012yes—
326C4#204514yes—
327C4C2×#45511yescharacteristic
328C4C2×#45411yescharacteristic
329C2—22no—
330C2—22no—
331C2—41no—
332C2—164no—
333C2C2×#204011yescharacteristic
334C2C2×#204511yescharacteristic
335C2#2353511yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#3245C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20755C2semidirect
3. #17750C2semidirect
4. #17544C2semidirect
5. #10513C2semidirect
6. #7988C2semidirect
18. #1832C22semidirect
22. #1551C22semidirect
26. #1505C22semidirect
28. #1290C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32452
#207552
#177502
#175442
#105132
#79882