Finite Group Databasebeta

← #31294

Subgroups — #31294

Order 256.

475 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2495C211yesmaximal, characteristic, complemented
2#2435C211yesmaximal, characteristic
3#2092C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1926C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#107C2211yescharacteristic, complemented
21C2×#106C2211yescharacteristic, complemented
22C2×C4⋊C8C2211yescharacteristic, complemented
23C2×#103C2211yescharacteristic, complemented
24C2×#103C2211yescharacteristic, complemented
25C2×#103C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic, complemented
38#400—21no—
39#397—21no—
40#394—21no—
41#394—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic, complemented
45#389C2211yescharacteristic, complemented
46#388—21no—
47#383—21no—
48#382C2211yescharacteristic, complemented
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#379—21no—
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#336C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59C25—21no—
60C22×Q8C2311yescharacteristic
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—21no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#124—22no—
76#124—22no—
77#124—41no—
78#124—41no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—41no—
106#106—41no—
107C4⋊C8—41no—
108#103—41no—
109#103—41no—
110#103—41no—
111#103—41no—
112#102—22no—
113#102—22no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C4×C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C24—42no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24—42no—
139C24—81no—
140C24—81no—
141C24C2×D411yescharacteristic
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×D4—21no—
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—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4#4112yes—
166C2×D4#4112yes—
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—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2411yescharacteristic
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—42no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C22⋊C4—22no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—48no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—84no—
250C23—84no—
251C23—84no—
252C23—84no—
253C23—84no—
254C23—84no—
255C23#13512yes—
256C23C22×D411yescharacteristic
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261D4—22no—
262D4—22no—
263D4—22no—
264D4—22no—
265D4—48no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
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×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—816no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#53511yescharacteristic
340C22#54611yescharacteristic
341C22#54611yescharacteristic
342C22C2×#13511yescharacteristic
343C22#57511yescharacteristic
344C22#57611yescharacteristic
345C22#57711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4#53811yescharacteristic
358C4#53811yescharacteristic
359C4#55111yescharacteristic
360C4#56011yescharacteristic
361C2—42no—
362C2—82no—
363C2—82no—
364C2#268511yescharacteristic
365C2#321111yescharacteristic
366C2#322411yescharacteristic
367C2#329611yescharacteristic
368C2#329711yescharacteristic
369C2#331911yescharacteristic
370C2#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. #2495C2semidirect
3. #2092C2semidirect
4. #2067C2semidirect
5. #2042C2semidirect
6. #2042C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1983C2semidirect
10. #1958C2semidirect
11. #1934C2semidirect
12. #1926C2semidirect
13. #1923C2semidirect
20. C2×#107C22semidirect
21. C2×#106C22semidirect
22. C2×C4⋊C8C22semidirect
23. C2×#103C22semidirect
24. C2×#103C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
37. C2×C4×C8C22semidirect
44. #389C22semidirect
45. #389C22semidirect
48. #382C22semidirect
53. C4×C4⋊C4C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #336C22semidirect
58. #336C22semidirect

Maximal subgroups

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

subgroupindex
#24952
#24352
#20922
#20672
#20422
#20362
#20352
#19832
#19582
#19342
#19262
#19232
#17862