Finite Group Databasebeta

← #24901

Subgroups — #24901

Order 256.

578 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3#2423C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic, complemented
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C22⋊C4—21no—
24#450—24no—
25C4×SD16—24no—
26C2×#106C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic, complemented
40#399—21no—
41#398—21no—
42#391C2211yescharacteristic, complemented
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic, complemented
46#388—21no—
47#387—21no—
48#386—21no—
49#386—21no—
50#384C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#379—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16D414yes—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22⋊Q8—22no—
86C22⋊Q8—41no—
87C4×Q8—22no—
88C4×Q8—41no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#106—22no—
119#106—41no—
120C4⋊C8—41no—
121#103—22no—
122#103—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C4×C8—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—24no—
141C24—21no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×D412yes—
155C2×D4—24no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4C2×D412yes—
171C2×D4#4112yes—
172C2×D4#4112yes—
173C2×D4#4112yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2411yescharacteristic
198SD16—28no—
199SD16—44no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8C2×D412yes—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4SD1614yescomplemented
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—42no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—48no—
254C23—48no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—84no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23—84no—
270C23C2×SD1612yes—
271C23C22×D411yescharacteristic
272Q8—22no—
273Q8—22no—
274Q8—41no—
275Q8—41no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—22no—
281D4—48no—
282D4—41no—
283D4—41no—
284D4#12314yes—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
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—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4C2×SD1612yes—
323C2×C4C2×SD1612yes—
324C2×C4C22×D411yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C8—22no—
331C8—41no—
332C8—41no—
333C8—41no—
334C22—22no—
335C22—24no—
336C22—22no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—816no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#52511yescharacteristic
357C22D4211yescharacteristic
358C22#54611yescharacteristic
359C22C22×SD1611yescharacteristic
360C22#57511yescharacteristic
361C22#57611yescharacteristic
362C22#57711yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—42no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4C2×#12311yescharacteristic
375C4#53811yescharacteristic
376C4#54611yescharacteristic
377C4#55111yescharacteristic
378C2—42no—
379C2—44no—
380C2—82no—
381C2#241811yescharacteristic
382C2#310111yescharacteristic
383C2#311811yescharacteristic
384C2#326811yescharacteristic
385C2D4×SD1611yescharacteristic
386C2#329511yescharacteristic
387C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×SD16C2semidirect
3. #2423C2semidirect
5. #2094C2semidirect
6. #2077C2semidirect
7. #2055C2semidirect
8. #2035C2semidirect
9. #1934C2semidirect
10. #1923C2semidirect
11. #1880C2semidirect
12. #1877C2semidirect
13. #1864C2semidirect
14. #1802C2semidirect
20. C2×C4×Q8C22semidirect
26. C2×#106C22semidirect
27. C2×#106C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
39. C2×C4×C8C22semidirect
42. #391C22semidirect
45. #389C22semidirect
50. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
237. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×C4×SD162
#24232
D4×C22⋊C42
#20942
#20772
#20552
#20352
#19342
#19232
#18802
#18772
#18642
#18022
#17862