Finite Group Databasebeta

← #24919

Subgroups — #24919

Order 256.

494 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#452C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3#2423C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#1935C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#452—24no—
24#440—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#399—21no—
40#398—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#387—21no—
47#386—21no—
48#386—21no—
49#384C2211yescharacteristic
50#381—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C22⋊C4C2211yescharacteristic
54C2×#95—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16D414yes—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22⋊Q8—22no—
86C22⋊Q8—41no—
87C4×Q8—22no—
88C4×Q8—41no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#107—41no—
115C4⋊C8—41no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—41no—
120#103—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124C22⋊C8—22no—
125C22⋊C8—41no—
126#97—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140#41—44no—
141#41—44no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×D412yes—
155C2×Q8#4112yes—
156C2×D4—24no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4C2×D412yes—
163C2×D4#4112yes—
164C2×D4#4112yes—
165C22×C4—24no—
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—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—44no—
194SD16—28no—
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—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C22⋊C4—22no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—42no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23C22×D411yescharacteristic
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—22no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257D4—22no—
258D4—22no—
259D4—22no—
260D4—48no—
261D4#12314yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
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—48no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
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—84no—
303C2×C4—84no—
304C2×C4#13712yes—
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—24no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#52511yescharacteristic
330C22D4211yescharacteristic
331C22#54611yescharacteristic
332C22C2×#13711yescharacteristic
333C22#57511yescharacteristic
334C22#57611yescharacteristic
335C22#57811yescharacteristic
336C4—22no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4C2×#12311yescharacteristic
349C4#53811yescharacteristic
350C4#54611yescharacteristic
351C4#55111yescharacteristic
352C2—44no—
353C2—81no—
354C2#241811yescharacteristic
355C2#311611yescharacteristic
356C2#312411yescharacteristic
357C2#326811yescharacteristic
358C2#327611yescharacteristic
359C2#331411yescharacteristic
360C2#331511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
2. C2×#440C2semidirect
3. #2423C2semidirect
5. #2095C2semidirect
6. #2075C2semidirect
7. #2056C2semidirect
8. #2036C2semidirect
9. #1935C2semidirect
10. #1924C2semidirect
11. #1878C2semidirect
12. #1877C2semidirect
13. #1862C2semidirect
14. #1802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4402
#24232
#23392
#20952
#20752
#20562
#20362
#19352
#19242
#18782
#18772
#18622
#18022
#17882