Finite Group Databasebeta

← #29897

Subgroups — #29897

Order 256.

489 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2848C211yesmaximal, characteristic
2#2280C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2035C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1779C211yesmaximal, characteristic
16C24×C4C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#107C2211yescharacteristic, complemented
23C2×#107C2211yescharacteristic, complemented
24C2×#106C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#103C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#385—21no—
37#385—21no—
38#385—21no—
39#381—21no—
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#379—21no—
48#379—21no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51#340C2211yescharacteristic, complemented
52#340C2211yescharacteristic, complemented
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59C25D411yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4D411yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83#117—41no—
84#117—41no—
85#117—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42—21no—
111#107—41no—
112#107—41no—
113#106—41no—
114#106—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
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—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4#4112yes—
189C22×C4#4112yes—
190C22×C4C2411yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—42no—
233C42—42no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—44no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23#12312yes—
260C23#13512yes—
261C23#13712yes—
262C23C22×D411yescharacteristic
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#12312yes—
315C2×C4C2×#4111yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4H(5,𝔽2)11yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C8—41no—
322C8—41no—
323C8—41no—
324C8—41no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—416no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22C2×#12311yescharacteristic
348C22#53411yescharacteristic
349C22#53411yescharacteristic
350C22C2×#13511yescharacteristic
351C22C2×#13711yescharacteristic
352C22#57511yescharacteristic
353C22#57511yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4#53811yescharacteristic
366C4#53811yescharacteristic
367C4#53811yescharacteristic
368C4#53811yescharacteristic
369C2—42no—
370C2—42no—
371C2—42no—
372C2#262211yescharacteristic
373C2#309111yescharacteristic
374C2#309511yescharacteristic
375C2#319011yescharacteristic
376C2#319111yescharacteristic
377C2#319611yescharacteristic
378C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2077C2semidirect
4. #2077C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2036C2semidirect
8. #2036C2semidirect
9. #2035C2semidirect
10. #2035C2semidirect
11. #1873C2semidirect
12. #1873C2semidirect
13. #1854C2semidirect
14. #1854C2semidirect
22. C2×#107C22semidirect
23. C2×#107C22semidirect
24. C2×#106C22semidirect
25. C2×#106C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
51. #340C22semidirect
52. #340C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect

Maximal subgroups

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

subgroupindex
#28482
#22802
#20772
#20762
#20752
#20362
#20352
#18732
#18542
#17792