Finite Group Databasebeta

← #105142

Subgroups — #105142

Order 512.

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

normal subgroups only

This group has 385 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#28753C211yesmaximal, characteristic
2#9698C211yesmaximal, characteristic
3#9604C211yesmaximal, characteristic, complemented
4#8591C211yesmaximal, characteristic, complemented
5#7659C211yesmaximal, characteristic
6#7571C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#489—21no—
9#2858C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2043—21no—
12#2041—21no—
13#1983—21no—
14#1983—21no—
15#1959—21no—
16#1842C2211yescharacteristic
17#1831C2211yescharacteristic
18#1767C2211yescharacteristic
19C4×C4⋊C8C2211yescharacteristic
20C4×C4⋊C8C2211yescharacteristic
21C2×#335—21no—
22C2×#332—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×#124—21no—
35C2×#124—21no—
36C2×C4×Q8—21no—
37#489—42no—
38#489—42no—
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⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×#103—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×#102—21no—
54C2×#102—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#390—21no—
60#390—21no—
61#390—21no—
62#390—21no—
63#389—21no—
64#389—21no—
65#384D411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C43C2311yescharacteristic
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336D411yescharacteristic
74#336D411yescharacteristic
75#335—42no—
76#332—42no—
77#332—42no—
78#331—42no—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8D812yes—
97#127—41no—
98#127—41no—
99#124—41no—
100#124—41no—
101#124—41no—
102#124—41no—
103#124—41no—
104#124—41no—
105#124—41no—
106#124—41no—
107C4×Q8—82no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C42—41no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—42no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143#103—42no—
144#103—42no—
145#103—81no—
146#103—81no—
147#102—42no—
148#102—42no—
149#102—81no—
150#102—81no—
151#97—41no—
152#97—41no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—81no—
156#95—41no—
157#95—41no—
158C24—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12411yescharacteristic
192C2×C8—22no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—82no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—82no—
240C42C2×D811yescharacteristic
241C42#13511yescharacteristic
242C42#13511yescharacteristic
243C42#13511yescharacteristic
244C42AGL(1,ℤ/8)11yescharacteristic
245C42AGL(1,ℤ/8)11yescharacteristic
246C42#13711yescharacteristic
247C42#13711yescharacteristic
248C23—81no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23#39411yescharacteristic
256Q8—82no—
257Q8—82no—
258Q8—81no—
259Q8—81no—
260Q8—81no—
261Q8—81no—
262D4—81no—
263D4—81no—
264D4—81no—
265D4—81no—
266D4—161no—
267D4—161no—
268D4—162no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4#44711yescharacteristic
292C2×C4#44911yescharacteristic
293C2×C4#45211yescharacteristic
294C2×C4#45211yescharacteristic
295C2×C4#45311yescharacteristic
296C2×C4#45511yescharacteristic
297C2×C4#46311yescharacteristic
298C2×C4#46411yescharacteristic
299C2×C4#46611yescharacteristic
300C2×C4#46811yescharacteristic
301C2×C4#47111yescharacteristic
302C2×C4#47311yescharacteristic
303C2×C4#48011yescharacteristic
304C2×C4#48211yescharacteristic
305C2×C4#48811yescharacteristic
306C2×C4#48911yescharacteristic
307C8—42no—
308C8—81no—
309C8—82no—
310C8—81no—
311C8—81no—
312C8—161no—
313C8—161no—
314C22—161no—
315C22—161no—
316C22—161no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22—161no—
321C22#203311yescharacteristic
322C22#203711yescharacteristic
323C22#204211yescharacteristic
324C22#204811yescharacteristic
325C22#205011yescharacteristic
326C22#205211yescharacteristic
327C22#205311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—82no—
340C4—161no—
341C4—161no—
342C4#162111yescharacteristic
343C4#163011yescharacteristic
344C4#164011yescharacteristic
345C4#164111yescharacteristic
346C2—321no—
347C2#1041111yescharacteristic
348C2#1045811yescharacteristic
349C2#1046411yescharacteristic
350C2#1049611yescharacteristic
351C2#1051011yescharacteristic
352C2#1054711yescharacteristic
353C2#1055111yescharacteristic
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
3. #9604C2semidirect
4. #8591C2semidirect
6. #7571C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#287532
#96982
#96042
#85912
#76592
#75712
#70222