Finite Group Databasebeta

← #136641

Subgroups — #136641

Order 512.

508 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C4×#466C211yesmaximal, characteristic
2C2×#1566C211yesmaximal, characteristic, complemented
3#12599C211yesmaximal, characteristic, complemented
4C2×#1333C212yesmaximal, complemented
5#7178C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C4×D8C2211yescharacteristic
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10C4×#121—21no—
11C2×C22⋊C16—21no—
12C2×#346C2212yes—
13#1970C411yescharacteristic
14#1942C411yescharacteristic
15#1934—21no—
16#1893—21no—
17#1808C411yescharacteristic
18C4×#107C2211yescharacteristic
19C4×#102—21no—
20#1566—24no—
21C2×#334—21no—
22C2×#334C2212yes—
23#1374—21no—
24#1333—24no—
25C22×D8C2×C411yescharacteristic
26C22×M16—41no—
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#121—21no—
30C2×C4×D4—21no—
31C2×C4×D4—41no—
32C22×C42D411yescharacteristic
33C22×C16—22no—
34#466—22no—
35#466—22no—
36C4×D8—22no—
37C4×D8—22no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41#423—24no—
42#423—42no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—41no—
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×C4×C8—22no—
49C2×C4×C8—22no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#390—21no—
55#389—21no—
56#388—41no—
57#386—41no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—41no—
61C22⋊C16—24no—
62#346—22no—
63#346—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#334—24no—
68#334—22no—
69#334D414yescomplemented
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C2×D8—22no—
77C2×D8—21no—
78C2×D8—22no—
79C2×D8M1612yescomplemented
80C2×M16—44no—
81C2×M16—82no—
82C2×M16—82no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—41no—
90C22×C8—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22×C411yescharacteristic
94C22×C8C2×D411yescharacteristic
95#121—24no—
96#121—42no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—84no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42C22⋊C411yescharacteristic
117C2×C42D811yescharacteristic
118C2×C42SD1611yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C16—24no—
121C2×C16—42no—
122C2×C16—42no—
123#107—21no—
124#107—22no—
125#107—22no—
126#107M1612yescomplemented
127C4⋊C8—24no—
128C4⋊C8—44no—
129C4⋊C8—42no—
130#102—24no—
131#102—42no—
132#102—42no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8C2×D412yes—
138C4×C8#4112yes—
139#95—41no—
140#95—41no—
141#95—41no—
142C24—41no—
143C24#10211yescharacteristic
144C2×D4—22no—
145C2×D4—44no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—82no—
152C2×D4—82no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4#10211yescharacteristic
175C22×C4#10211yescharacteristic
176C22×C4#10211yescharacteristic
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×SD1611yescharacteristic
180D8—24no—
181D8—42no—
182M16—88no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8#10412yes—
197C2×C8C4×D412yes—
198C2×C8#12112yes—
199C2×C8C2×M1611yescharacteristic
200C2×C8C2×M1611yescharacteristic
201C2×C8#13112yes—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—82no—
209C22⋊C4—44no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—82no—
222C42#12011yescharacteristic
223C42#12311yescharacteristic
224C42AGL(1,ℤ/8)11yescharacteristic
225C42#13711yescharacteristic
226C16—44no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—42no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23#36012yes—
241C23C2×#10211yescharacteristic
242D4—42no—
243D4—42no—
244D4—88no—
245D4—81no—
246D4—81no—
247C2×C4—24no—
248C2×C4—24no—
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—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—162no—
276C2×C4—162no—
277C2×C4#36012yes—
278C2×C4#36112yes—
279C2×C4#38611yescharacteristic
280C2×C4#40711yescharacteristic
281C2×C4#40811yescharacteristic
282C2×C4#41711yescharacteristic
283C2×C4C2×#10411yescharacteristic
284C2×C4#42111yescharacteristic
285C2×C4#44711yescharacteristic
286C2×C4#44811yescharacteristic
287C2×C4#48011yescharacteristic
288C2×C4#48111yescharacteristic
289C8—22no—
290C8—22no—
291C8—41no—
292C8—84no—
293C8#43512yes—
294C8#43612yes—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—161no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22#148611yescharacteristic
310C22#149211yescharacteristic
311C22#187411yescharacteristic
312C22C2×#36011yescharacteristic
313C22C2×#36011yescharacteristic
314C22C2×#36111yescharacteristic
315C22#214811yescharacteristic
316C4—21no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—41no—
323C4—82no—
324C4—161no—
325C4—161no—
326C4#148911yescharacteristic
327C4#149511yescharacteristic
328C4#187011yescharacteristic
329C4#187111yescharacteristic
330C2—42no—
331C2—162no—
332C2#894411yescharacteristic
333C2#1166311yescharacteristic
334C2#1166611yescharacteristic
335C2#1167711yescharacteristic
336C2#1168011yescharacteristic
337C2#1300111yescharacteristic
338C2#1300411yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1566C2semidirect
3. #12599C2semidirect
4. C2×#1333C2semidirect
5. #7178C2semidirect
69. #334D4semidirect
79. C2×D8M16semidirect
126. #107M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
C2×#15662
#125992
C2×#13332
#71782