Finite Group Databasebeta

← #27262

Subgroups — #27262

Order 256.

565 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×#489C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic
5#2071C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1959C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic
15#1752C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic, complemented
19C2×#124C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#489—24no—
25C2×#103C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#103C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#102—24no—
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic, complemented
40C2×#97C2211yescharacteristic, complemented
41#398—21no—
42#395—21no—
43#394—21no—
44#394—21no—
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic, complemented
48#389C2211yescharacteristic, complemented
49#385—21no—
50#384C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#379—21no—
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×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—41no—
84#124—22no—
85#124—22no—
86#124—41no—
87#124—41no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
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—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#103—22no—
120#103—22no—
121#103—41no—
122#103—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130#102D414yes—
131C22⋊C8—22no—
132#97—22no—
133#97—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—42no—
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—28no—
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—21no—
168C2×D4—21no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4C2×D412yes—
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—42no—
188C22×C4—42no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4#4112yes—
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
198C2×C8—21no—
199C2×C8—21no—
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—44no—
210C2×C8C2×D412yes—
211C4⋊C4—21no—
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—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4C2×D412yes—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—48no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—84no—
264C23—84no—
265C23—84no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23#12412yes—
270C23AGL(1,ℤ/8)12yes—
271C23#13712yes—
272C23C22×D411yescharacteristic
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—84no—
278D4—22no—
279D4—22no—
280D4—22no—
281D4—22no—
282D4—48no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
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—41no—
322C2×C4—41no—
323C2×C4—84no—
324C2×C4#12412yes—
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—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—816no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#48914yescomplemented
357C22C2×#12411yescharacteristic
358C22D4211yescharacteristic
359C22#54611yescharacteristic
360C22C2×AGL(1,ℤ/8)11yescharacteristic
361C22C2×#13711yescharacteristic
362C22#57611yescharacteristic
363C22#57711yescharacteristic
364C4—22no—
365C4—41no—
366C4—42no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4#52511yescharacteristic
376C4#53811yescharacteristic
377C4#54611yescharacteristic
378C4#55111yescharacteristic
379C2—24no—
380C2—82no—
381C2—82no—
382C2#247211yescharacteristic
383C2C2×#48911yescharacteristic
384C2#313611yescharacteristic
385C2#326411yescharacteristic
386C2#326611yescharacteristic
387C2#330611yescharacteristic
388C2#331411yescharacteristic
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×#489C2semidirect
3. #2440C2semidirect
5. #2071C2semidirect
6. #2042C2semidirect
7. #2042C2semidirect
8. #2023C2semidirect
9. #1959C2semidirect
10. #1936C2semidirect
11. #1936C2semidirect
12. #1892C2semidirect
13. #1872C2semidirect
15. #1752C2semidirect
18. C2×C4⋊Q8C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
39. C2×#97C22semidirect
40. C2×#97C22semidirect
47. #389C22semidirect
48. #389C22semidirect
50. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
356. C22#489semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C22×#1022
#24402
#24352
#20712
#20422
#20232
#19592
#19362
#18922
#18722
#17862
#17522