Finite Group Databasebeta

← #25976

Subgroups — #25976

Order 256.

568 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic
3#2441C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#125C2211yescharacteristic, complemented
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C22⋊C4—21no—
23#478—24no—
24C2×#107C2211yescharacteristic, complemented
25C2×#107C2211yescharacteristic, complemented
26C2×#106C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×C4⋊C8C2211yescharacteristic, complemented
30#418—24no—
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#399—21no—
41#396—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
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×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—22no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4Q814yes—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#125—22no—
82#125—41no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—41no—
86C4×D4—41no—
87#117—22no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
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#107—22no—
117#107—41no—
118#106—22no—
119#106—41no—
120C4⋊C8—22no—
121C4⋊C8—41no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129#102D414yes—
130C22⋊C8—22no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95D411yescharacteristic
136#95D411yescharacteristic
137C24—21no—
138C24—21no—
139C24—24no—
140C24—42no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—81no—
145C24—81no—
146C24C2×D411yescharacteristic
147C2×D4—28no—
148C2×D4—24no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4C2×D412yes—
163C2×D4C2×Q812yes—
164C2×D4C2×Q812yes—
165C2×D4#4112yes—
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—21no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×Q812yes—
188C22×C4#4112yes—
189C22×C4C2411yescharacteristic
190M16—44no—
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×C8C2×D412yes—
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—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4C2×D412yes—
221C22⋊C4—22no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—42no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—48no—
248C23—48no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—84no—
260C23—84no—
261C23—84no—
262C23—84no—
263C23—84no—
264C23C22⋊Q812yes—
265C23C22×D411yescharacteristic
266D4—22no—
267D4—22no—
268D4—22no—
269D4—22no—
270D4—22no—
271D4—48no—
272D4—41no—
273D4—41no—
274D4C22⋊Q814yes—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
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—84no—
311C2×C4—84no—
312C2×C4C22⋊Q812yes—
313C2×C4AGL(1,ℤ/8)12yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C22×Q811yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C8—22no—
321C8—41no—
322C8—41no—
323C8—41no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—24no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—816no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C2×C22⋊Q811yescharacteristic
347C22D4211yescharacteristic
348C22#54611yescharacteristic
349C22C2×AGL(1,ℤ/8)11yescharacteristic
350C22#57511yescharacteristic
351C22#57611yescharacteristic
352C22#57711yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4C2×C22⋊Q811yescharacteristic
365C4#54311yescharacteristic
366C4#54611yescharacteristic
367C4#55111yescharacteristic
368C2—42no—
369C2—44no—
370C2—82no—
371C2#243711yescharacteristic
372C2#307711yescharacteristic
373C2#308111yescharacteristic
374C2#326811yescharacteristic
375C2#327211yescharacteristic
376C2#331811yescharacteristic
377C2#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
1. C2×#478C2semidirect
3. #2441C2semidirect
5. #2094C2semidirect
6. #2067C2semidirect
7. #2067C2semidirect
8. #2025C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1854C2semidirect
12. #1851C2semidirect
13. #1809C2semidirect
14. #1802C2semidirect
18. C2×#125C22semidirect
24. C2×#107C22semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×#106C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×C4⋊C8C22semidirect
45. #389C22semidirect
46. #389C22semidirect
50. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4182
#24412
D4×C22⋊C42
#20942
#20672
#20252
#18642
#18622
#18542
#18512
#18092
#18022
#17862