Finite Group Databasebeta

← #298924

Subgroups — #298924

Order 512.

1010 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 186 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×#3139C211yesmaximal, characteristic
2#33210C211yesmaximal, characteristic
3#25902C212yesmaximal
4#25674C212yesmaximal
5C2×#1718C212yesmaximal, complemented
6#10319C212yesmaximal, complemented
7#10313C212yesmaximal, complemented
8#9833C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#500C2212yes—
13C2×#498C2211yescharacteristic
14#3139—28no—
15C2×#491—22no—
16C2×#487C2212yes—
17C2×#487C2212yes—
18C2×#487C2212yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#419—22no—
22C2×C4×M16C2211yescharacteristic
23#2857C2212yes—
24#2036—24no—
25#2024—24no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1956C2212yes—
31#1943—22no—
32#1872—24no—
33#1863—24no—
34#1845C2211yescharacteristic
35#1809—22no—
36C4×C22⋊C8C2212yes—
37#1718—28no—
38C2×#322C2212yes—
39#1273C2211yescharacteristic
40#1273C2211yescharacteristic
41C23×Q8—21no—
42C22×M16D412yes—
43#533—24no—
44#533—41no—
45#533—41no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2312yes—
54C2×#125—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C4×Q8—22no—
58C2×#117—21no—
59C22×C4⋊C4—22no—
60C22×C42C2311yescharacteristic
61#500—24no—
62#500—42no—
63#498—24no—
64#491—216no—
65#487—24no—
66#487—24no—
67#487—24no—
68#487—24no—
69#487—24no—
70#487—24no—
71#487—42no—
72#487—42no—
73#487—42no—
74#487D418yes—
75C2×#107—22no—
76C2×#107—24no—
77C2×#106—22no—
78#419—216no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×C22⋊C8D414yes—
86C4×M16—24no—
87C2×#97D412yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2312yes—
90C2×C4×C8C2312yes—
91C2×C4×C8C2311yescharacteristic
92#391—22no—
93#389—22no—
94#389—22no—
95#384C2312yes—
96#380—22no—
97#380—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336D414yes—
103#322—24no—
104C22×Q8—24no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—42no—
109C22×Q8—44no—
110C22×Q8—41no—
111C22×Q8—44no—
112C23×C4—41no—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—42no—
118C22×C8C2×D412yes—
119C22×C8C2×D414yes—
120C22×C8C2×D414yes—
121C22×C8C2×D412yes—
122C4⋊Q8—24no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—42no—
136C4⋊Q8C2×D414yes—
137C4⋊Q8C2×D414yes—
138#125—24no—
139#125—42no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—48no—
143C4×Q8—44no—
144C4×Q8—44no—
145#117—24no—
146#117—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—42no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—41no—
167C2×C42C2×D412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2411yescharacteristic
170#107—48no—
171#107—44no—
172#106—44no—
173#103—216no—
174#103—216no—
175#103—48no—
176#103—48no—
177#103—48no—
178#103—48no—
179C22⋊C8—44no—
180#97—24no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8C2×D414yes—
187#95—42no—
188#95—42no—
189C24#12711yescharacteristic
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—44no—
193C2×Q8—416no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—82no—
205C2×Q8—88no—
206C2×Q8—82no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4#12712yes—
222C22×C4#12711yescharacteristic
223C22×C4#13714yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D412yes—
226M16—48no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—22no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8#12014yes—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—84no—
257C22⋊C4—44no—
258C22⋊C4—84no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—22no—
263C42—44no—
264C42C22×D412yes—
265C42H(5,𝔽2)11yescharacteristic
266C42H(5,𝔽2)11yescharacteristic
267C23—22no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23#49712yes—
272C23C2×#12711yescharacteristic
273Q8—48no—
274Q8—48no—
275Q8—88no—
276Q8—84no—
277Q8—84no—
278Q8—84no—
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—88no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#49712yes—
303C2×C4C2×#12011yescharacteristic
304C2×C4#53511yescharacteristic
305C2×C4D4212yes—
306C2×C4D4212yes—
307C2×C4C2×#13712yes—
308C2×C4#57512yes—
309C2×C4#57514yes—
310C2×C4#57814yes—
311C8—24no—
312C8—42no—
313C8—44no—
314C8—44no—
315C22—22no—
316C22—24no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22#243311yescharacteristic
321C22C2×#49711yescharacteristic
322C22#315311yescharacteristic
323C22#315312yes—
324C22#315612yes—
325C4—22no—
326C4—42no—
327C4—84no—
328C4—84no—
329C4—162no—
330C4—162no—
331C4#300712yes—
332C4#321512yes—
333C4#326914yes—
334C4#327714yes—
335C2—42no—
336C2#1968112yes—
337C2#1973211yescharacteristic
338C2#1975411yescharacteristic
339C2#2573011yescharacteristic
340C2#2579212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#1718C2semidirect
6. #10319C2semidirect
7. #10313C2semidirect
8. #9833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#332102
#259022
#256742
C2×#17182
#103192
#103132
#98332
#81472