Finite Group Databasebeta

← #105146

Subgroups — #105146

Order 512.

378 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 84 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#29145C211yesmaximal, characteristic
2#9697C211yesmaximal, characteristic, complemented
3#9604C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7662C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic, complemented
8#2862C2211yescharacteristic
9C4×#127C2211yescharacteristic
10#2023—21no—
11#2023—21no—
12#1961—21no—
13#1842C2211yescharacteristic
14#1836—21no—
15#1831C2211yescharacteristic
16#1804—21no—
17#1804—21no—
18#1767C2211yescharacteristic
19C4×C4⋊C8C2211yescharacteristic
20C4×C4⋊C8C2211yescharacteristic
21C2×#335—21no—
22C2×#333—21no—
23C2×#332—21no—
24C2×#331—21no—
25#1296—21no—
26#1277—21no—
27#1272—21no—
28#1272—21no—
29#1270—21no—
30#1270—21no—
31C2×C4⋊Q8D411yescharacteristic
32C2×C4⋊Q8D411yescharacteristic
33C2×#127D411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×#97—21no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52#391—21no—
53#391—21no—
54#391—21no—
55#391—21no—
56#390—21no—
57#390—21no—
58#390—21no—
59#384D411yescharacteristic
60#382—21no—
61#382—21no—
62C43C2311yescharacteristic
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336D411yescharacteristic
72#336D411yescharacteristic
73#335—42no—
74#333—42no—
75#332—42no—
76#331—42no—
77C22×Q8—21no—
78C22×Q8—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C23×C4—41no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94#127—21no—
95#127—21no—
96#127—21no—
97#127Q1612yes—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—81no—
103C4×D4—81no—
104C4×D4—82no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C4⋊C8—22no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—42no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137#102—81no—
138#102—81no—
139#102—81no—
140#102—81no—
141#97—41no—
142#97—41no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—81no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24—41no—
151C2×Q8—42no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8—81no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12411yescharacteristic
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—82no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42C2×Q1611yescharacteristic
230C42#13511yescharacteristic
231C42#13511yescharacteristic
232C42#13511yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42#13711yescharacteristic
236C42#13711yescharacteristic
237C23—42no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23#39411yescharacteristic
245Q8—161no—
246Q8—161no—
247Q8—162no—
248D4—42no—
249D4—42no—
250D4—82no—
251D4—82no—
252D4—82no—
253D4—82no—
254D4—81no—
255D4—81no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—162no—
279C2×C4—162no—
280C2×C4#44911yescharacteristic
281C2×C4#44911yescharacteristic
282C2×C4#45111yescharacteristic
283C2×C4#45211yescharacteristic
284C2×C4#45311yescharacteristic
285C2×C4#45511yescharacteristic
286C2×C4#46311yescharacteristic
287C2×C4#46411yescharacteristic
288C2×C4#46711yescharacteristic
289C2×C4#46811yescharacteristic
290C2×C4#47111yescharacteristic
291C2×C4#47211yescharacteristic
292C2×C4#48411yescharacteristic
293C2×C4#48511yescharacteristic
294C2×C4#48811yescharacteristic
295C2×C4#48911yescharacteristic
296C8—42no—
297C8—81no—
298C8—82no—
299C8—81no—
300C8—81no—
301C8—161no—
302C8—161no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22#203611yescharacteristic
311C22#203711yescharacteristic
312C22#204211yescharacteristic
313C22#204811yescharacteristic
314C22#205011yescharacteristic
315C22#205111yescharacteristic
316C22#205311yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—161no—
329C4—161no—
330C4—161no—
331C4#162411yescharacteristic
332C4#163211yescharacteristic
333C4#164411yescharacteristic
334C4#164611yescharacteristic
335C2—162no—
336C2#1041311yescharacteristic
337C2#1046111yescharacteristic
338C2#1046911yescharacteristic
339C2#1049611yescharacteristic
340C2#1050811yescharacteristic
341C2#1054711yescharacteristic
342C2#1055311yescharacteristic
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. #9697C2semidirect
3. #9604C2semidirect
5. #7662C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#291452
#96972
#96042
#85882
#76622
#75692
#70222