Finite Group Databasebeta

← #104941

Subgroups — #104941

Order 512.

411 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 86 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#28252C211yesmaximal, characteristic
2#8787C211yesmaximal, characteristic, complemented
3#8589C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7588C211yesmaximal, characteristic, complemented
6#7587C211yesmaximal, characteristic
7#7040C211yesmaximal, characteristic, complemented
8C2×#460—21no—
9C2×#461—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2055—22no—
13#1842C2211yescharacteristic
14#1842C2211yescharacteristic
15#1831C2211yescharacteristic
16#1831C2211yescharacteristic
17#1803—21no—
18#1802—21no—
19#1744C2211yescharacteristic
20C2×#332—21no—
21C2×#331—21no—
22#1296—21no—
23#1296—21no—
24#1277—22no—
25#1271—22no—
26#1270—22no—
27C22×SD16—22no—
28C2×C4⋊Q8D411yescharacteristic
29C2×#127D411yescharacteristic
30C2×C4×Q8—21no—
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34#461—42no—
35#461—42no—
36#460—42no—
37#460—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D412yes—
46C2×#103—22no—
47C2×#102—22no—
48C2×C4×C8—21no—
49#391—21no—
50#390—21no—
51#390—21no—
52#390—21no—
53#389—21no—
54#389—21no—
55#384D411yescharacteristic
56#384D411yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C43C2311yescharacteristic
60#340—22no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336D412yes—
67#332—42no—
68#331—42no—
69C22×Q8—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C23×C4—41no—
74C2×SD16—44no—
75C2×SD16—44no—
76C2×SD16—82no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8D812yes—
86#127—21no—
87#127—21no—
88#127—21no—
89#127SD1612yes—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—82no—
99C4×D4—82no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—41no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134#103—44no—
135#102—44no—
136C4×C8—41no—
137C4×C8—41no—
138#95—41no—
139#95—41no—
140C24—41no—
141C2×Q8—22no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4—81no—
160C2×D4—81no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4#12011yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12311yescharacteristic
174C22×C4#12411yescharacteristic
175SD16—88no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—81no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—81no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—82no—
209C42—21no—
210C42—21no—
211C42—22no—
212C42—41no—
213C42—41no—
214C42—41no—
215C42—81no—
216C42—81no—
217C42C2×D811yescharacteristic
218C42C2×SD1611yescharacteristic
219C42#13511yescharacteristic
220C42#13511yescharacteristic
221C42AGL(1,ℤ/8)11yescharacteristic
222C42AGL(1,ℤ/8)11yescharacteristic
223C42AGL(1,ℤ/8)11yescharacteristic
224C42#13711yescharacteristic
225C23—42no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23#39411yescharacteristic
233Q8—42no—
234Q8—42no—
235Q8—81no—
236Q8—81no—
237D4—42no—
238D4—42no—
239D4—82no—
240D4—82no—
241D4—82no—
242D4—82no—
243D4—81no—
244D4—81no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—162no—
268C2×C4—162no—
269C2×C4#44711yescharacteristic
270C2×C4#44911yescharacteristic
271C2×C4#45011yescharacteristic
272C2×C4#45211yescharacteristic
273C2×C4#45411yescharacteristic
274C2×C4#45511yescharacteristic
275C2×C4#46511yescharacteristic
276C2×C4#46611yescharacteristic
277C2×C4#46811yescharacteristic
278C2×C4#46911yescharacteristic
279C2×C4#47112yes—
280C2×C4#48011yescharacteristic
281C2×C4#48111yescharacteristic
282C2×C4#48211yescharacteristic
283C2×C4#48211yescharacteristic
284C8—82no—
285C8—82no—
286C8—161no—
287C8—161no—
288C8—161no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—161no—
293C22—161no—
294C22—161no—
295C22—161no—
296C22#203311yescharacteristic
297C22#203411yescharacteristic
298C22#204011yescharacteristic
299C22#204611yescharacteristic
300C22#204811yescharacteristic
301C22#204911yescharacteristic
302C22#205311yescharacteristic
303C4—22no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—82no—
314C4—161no—
315C4—161no—
316C4#162511yescharacteristic
317C4#163511yescharacteristic
318C4#163911yescharacteristic
319C4#164611yescharacteristic
320C2—162no—
321C2#1043111yescharacteristic
322C2#1044211yescharacteristic
323C2#1044511yescharacteristic
324C2#1046511yescharacteristic
325C2#1046611yescharacteristic
326C2#1051511yescharacteristic
327C2#1052711yescharacteristic
subgroupG/H
center, socleC23#394
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8787C2semidirect
3. #8589C2semidirect
5. #7588C2semidirect
7. #7040C2semidirect

Maximal subgroups

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

subgroupindex
#282522
#87872
#85892
#85882
#75882
#75872
#70402