Finite Group Databasebeta

← #27224

Subgroups — #27224

Order 256.

576 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1968C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1923C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic, complemented
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C22⋊C4—21no—
24#476—24no—
25#450—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#400—21no—
41#393—21no—
42#391—21no—
43#391C2211yescharacteristic, complemented
44#391C2211yescharacteristic, complemented
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#388—21no—
48#386—21no—
49#386—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C4⋊Q8—22no—
87C4⋊Q8—41no—
88C22⋊Q8—22no—
89C22⋊Q8—41no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—41no—
93C4×D4—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#106—22no—
117#106—22no—
118C4⋊C8—22no—
119#103—41no—
120#103—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102D414yes—
127C22⋊C8—22no—
128C22⋊C8—41no—
129C4×C8—41no—
130#95—21no—
131#95—21no—
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×Q8—21no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—81no—
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4C2×D412yes—
174C2×D4#4112yes—
175C2×D4#4112yes—
176C2×D4#4112yes—
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—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2411yescharacteristic
201SD16—44no—
202SD16—44no—
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×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8C2×D412yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4C2×D412yes—
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—81no—
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—41no—
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—84no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—22no—
281D4—22no—
282D4—48no—
283D4—41no—
284D4—41no—
285D4#12414yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—41no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4C2×SD1612yes—
324C2×C4C2×SD1612yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—24no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—816no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#52511yescharacteristic
358C22D4211yescharacteristic
359C22#54611yescharacteristic
360C22C22×SD1611yescharacteristic
361C22#57511yescharacteristic
362C22#57611yescharacteristic
363C22#57711yescharacteristic
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—42no—
369C4—41no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4C2×#12411yescharacteristic
376C4#53811yescharacteristic
377C4#54611yescharacteristic
378C4#55111yescharacteristic
379C2—42no—
380C2—44no—
381C2—82no—
382C2#247211yescharacteristic
383C2#310111yescharacteristic
384C2#311811yescharacteristic
385C2#327311yescharacteristic
386C2D4×SD1611yescharacteristic
387C2#329511yescharacteristic
388C2#331111yescharacteristic
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
1. C2×#476C2semidirect
3. #2492C2semidirect
5. #2077C2semidirect
6. #2055C2semidirect
7. #2035C2semidirect
8. #2023C2semidirect
9. #1968C2semidirect
10. #1926C2semidirect
11. #1923C2semidirect
12. #1877C2semidirect
13. #1877C2semidirect
15. C4×#102C2semidirect
19. C2×C4⋊Q8C22semidirect
26. C2×#106C22semidirect
27. C2×#106C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
39. C2×C4×C8C22semidirect
43. #391C22semidirect
44. #391C22semidirect
53. C4×C4⋊C4C22semidirect
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×#4762
C2×#4502
#24922
D4×C22⋊C42
#20772
#20552
#20352
#20232
#19682
#19262
#19232
#18772
#17862
C4×#1022