Finite Group Databasebeta

← #27281

Subgroups — #27281

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic
3#2439C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic
5#2073C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1958C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic
15#1753C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic, complemented
19C2×#124C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#489—24no—
24#418—24no—
25C2×#103C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#103C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic, complemented
39C2×C4×C8C2211yescharacteristic, complemented
40#396—21no—
41#394—21no—
42#394—21no—
43#393—21no—
44#391—21no—
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic, complemented
48#389C2211yescharacteristic, complemented
49#387—21no—
50#382C2211yescharacteristic, 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×C4C2311yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83#124—22no—
84#124—22no—
85#124—41no—
86#124—41no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89#117—22no—
90#117—41no—
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×C22⋊C4—21no—
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—21no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#103—22no—
117#103—22no—
118#103—41no—
119#103—41no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102D414yes—
128C22⋊C8—22no—
129#97—22no—
130C4×C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C24—42no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—81no—
146C24—81no—
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—81no—
157C2×D4—28no—
158C2×D4—21no—
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—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4C2×D412yes—
175C2×D4#4112yes—
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×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4#4112yes—
192C22×C4#4112yes—
193C22×C4C2411yescharacteristic
194M16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
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×C8C2×D412yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4C2×D412yes—
221C22⋊C4—41no—
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—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—48no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—84no—
261C23—84no—
262C23—84no—
263C23—84no—
264C23—84no—
265C23—84no—
266C23#12412yes—
267C23C22×D411yescharacteristic
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—84no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—22no—
277D4—48no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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#12412yes—
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C8—22no—
326C8—41no—
327C8—41no—
328C8—41no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—816no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C2×#12411yescharacteristic
352C22D4211yescharacteristic
353C22#54611yescharacteristic
354C22#57511yescharacteristic
355C22#57511yescharacteristic
356C22#57611yescharacteristic
357C22#57711yescharacteristic
358C4—22no—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4#52511yescharacteristic
370C4#53811yescharacteristic
371C4#54611yescharacteristic
372C4#55111yescharacteristic
373C2—42no—
374C2—82no—
375C2—82no—
376C2#247211yescharacteristic
377C2#313611yescharacteristic
378C2#314011yescharacteristic
379C2#327311yescharacteristic
380C2#327511yescharacteristic
381C2#331111yescharacteristic
382C2#331911yescharacteristic
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. #2439C2semidirect
5. #2073C2semidirect
6. #2042C2semidirect
7. #2042C2semidirect
8. #2023C2semidirect
9. #1958C2semidirect
10. #1936C2semidirect
11. #1934C2semidirect
12. #1894C2semidirect
13. #1873C2semidirect
15. #1753C2semidirect
18. C2×C4⋊Q8C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
38. C2×#97C22semidirect
39. C2×C4×C8C22semidirect
47. #389C22semidirect
48. #389C22semidirect
50. #382C22semidirect
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×#4892
C2×#4182
#24392
#24352
#20732
#20422
#20232
#19582
#19362
#19342
#18942
#18732
#17862
#17532