Finite Group Databasebeta

← #128776

Subgroups — #128776

Order 512.

588 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 134 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
2#12599C211yesmaximal, characteristic, complemented
3C2×#1347C212yesmaximal
4C4×#340C211yesmaximal, characteristic, complemented
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×C4×D8C411yescharacteristic, complemented
8C22×C4×C8C2211yescharacteristic
9C4×#121—21no—
10C2×C4×C4⋊C4—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1970C411yescharacteristic, complemented
14#1942C411yescharacteristic, complemented
15#1934—21no—
16#1893—21no—
17#1808C2211yescharacteristic
18C4×#107C411yescharacteristic, complemented
19C4×#102—21no—
20C2×#340—21no—
21C2×#340C2212yes—
22#1374—21no—
23#1347C418yescomplemented
24C22×D8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×#121—21no—
28C2×C4×D4—21no—
29C2×C4×D4—41no—
30C22×C4⋊C4—22no—
31C22×C42—41no—
32C22×C42D411yescharacteristic
33C22×C16—22no—
34#466—22no—
35#466C2×C414yes—
36C4×D8—42no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#390—21no—
49#389—21no—
50#388—41no—
51#386—41no—
52#380—21no—
53C4×C4⋊C4—24no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—42no—
56C4×C22⋊C4—41no—
57C2×#95—41no—
58C22⋊C16—22no—
59C22⋊C16C2×C414yescomplemented
60#346—42no—
61#340—24no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340C2×C414yescomplemented
67C22×D4—21no—
68C22×D4—41no—
69C23×C4—42no—
70C23×C4—41no—
71C23×C4D811yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×D8—21no—
75C2×D8—41no—
76C2×D8—41no—
77C2×D8C4212yescomplemented
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22×C411yescharacteristic
86C22×C8C22×C412yes—
87C22×C8C2×D411yescharacteristic
88C22×C8#4112yes—
89#121—24no—
90#121—42no—
91C4×D4—42no—
92C4×D4—42no—
93C4×D4—84no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—44no—
112C2×C42—41no—
113C2×C42—82no—
114C2×C42—82no—
115C2×C42C22⋊C411yescharacteristic
116C2×C42C22⋊C411yescharacteristic
117C2×C42SD1611yescharacteristic
118C2×C42SD1611yescharacteristic
119C2×C16—24no—
120C2×C16—42no—
121C2×C16—42no—
122#107—21no—
123#107—41no—
124#107—41no—
125#107C4212yescomplemented
126#102—24no—
127#102—42no—
128#102—42no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—41no—
132C4×C8—41no—
133#95—44no—
134#95—41no—
135#95—41no—
136#95—41no—
137C24—41no—
138C24C2×D811yescharacteristic
139C2×D4—22no—
140C2×D4—44no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—82no—
147C2×D4—81no—
148C2×D4—81no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—41no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—41no—
163C22×C4—44no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4#10211yescharacteristic
174C22×C4#10211yescharacteristic
175C22×C4#10211yescharacteristic
176C22×C4#10211yescharacteristic
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D412yes—
179C22×C4C2×SD1611yescharacteristic
180C22×C4#13512yes—
181D8—42no—
182D8—42no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8#10412yes—
196C2×C8C2×C4211yescharacteristic
197C2×C8#11711yescharacteristic
198C2×C8C4×D412yes—
199C4⋊C4—28no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—82no—
210C22⋊C4—44no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C42—22no—
216C42—22no—
217C42—21no—
218C42—21no—
219C42—88no—
220C42—82no—
221C42#9912yes—
222C42#10412yes—
223C16—44no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—42no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23C2×#10211yescharacteristic
236C23C4×D812yes—
237C23#50911yescharacteristic
238C23#50912yes—
239C23#51011yescharacteristic
240D4—42no—
241D4—42no—
242D4—84no—
243D4—84no—
244D4—81no—
245D4—81no—
246C2×C4—24no—
247C2×C4—24no—
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—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—164no—
281C2×C4#32712yes—
282C2×C4#32712yes—
283C2×C4#36012yes—
284C2×C4C4×C22⋊C411yescharacteristic
285C2×C4C2×#9911yescharacteristic
286C2×C4#41011yescharacteristic
287C2×C4#41611yescharacteristic
288C2×C4C2×#10411yescharacteristic
289C2×C4C2×#10411yescharacteristic
290C2×C4C4×SD1612yes—
291C8—22no—
292C8—21no—
293C8—21no—
294C8—22no—
295C8—41no—
296C22—22no—
297C22—22no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22—161no—
311C22C2×#32711yescharacteristic
312C22#150211yescharacteristic
313C22C4×#10211yescharacteristic
314C22#213911yescharacteristic
315C22#214011yescharacteristic
316C22C2×#36011yescharacteristic
317C22#214511yescharacteristic
318C22#217512yes—
319C22#217712yes—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—88no—
326C4—82no—
327C4—161no—
328C4—161no—
329C4#150011yescharacteristic
330C4#150111yescharacteristic
331C4C4×#9911yescharacteristic
332C4C4×#10411yescharacteristic
333C2—22no—
334C2—22no—
335C2—162no—
336C2C4×#32711yescharacteristic
337C2#1170411yescharacteristic
338C2#1170511yescharacteristic
339C2#1174411yescharacteristic
340C2#1174611yescharacteristic
341C2#1263711yescharacteristic
342C2C4×#36011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C2×C42
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12599C2semidirect
4. C4×#340C2semidirect
5. #7258C2semidirect
7. C2×C4×D8C4semidirect
13. #1970C4semidirect
14. #1942C4semidirect
18. C4×#107C4semidirect
23. #1347C4semidirect
59. C22⋊C16C2×C4semidirect
66. #340C2×C4semidirect
77. C2×D8C42semidirect
125. #107C42semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
#125992
C2×#13472
C4×#3402
#72582