Finite Group Databasebeta

← #28617

Subgroups — #28617

Order 256.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#495C211yesmaximal, characteristic
2C2×#488C211yesmaximal, characteristic, complemented
3#2861C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic, complemented
5#2056C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic
9#1991C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic
13#1878C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15C4×C4⋊C8C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#495—24no—
26#488—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#388—21no—
46#388—21no—
47#388—21no—
48#388—21no—
49#388—21no—
50#388—21no—
51#382C2211yescharacteristic
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C43C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—41no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#125—22no—
83#125—22no—
84#124—22no—
85#124—22no—
86#124—22no—
87#124D414yes—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×D4—41no—
91C4×D4—41no—
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×C22⋊C4—21no—
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×C42—21no—
111C2×C42D411yescharacteristic
112C2×C42D411yescharacteristic
113C2×C42D411yescharacteristic
114C2×C42D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C4⋊C8—22no—
119C4⋊C8—22no—
120#103—22no—
121#103—22no—
122#103—41no—
123#103—41no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128C4×C8—22no—
129C4×C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D412yes—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4C2×D412yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
180Q16—44no—
181SD16—44no—
182SD16—44no—
183D8—44no—
184C2×C8—21no—
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#4112yes—
195C2×C8#4112yes—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—41no—
225C42—41no—
226C42—42no—
227C42C2×D412yes—
228C42#4112yes—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C22×D411yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252D4—22no—
253D4—22no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
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—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
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—84no—
293C2×C4#12112yes—
294C2×C4#12112yes—
295C2×C4#13512yes—
296C2×C4#13512yes—
297C2×C4C22×D411yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C2×C4H(5,𝔽2)11yescharacteristic
303C8—22no—
304C8—22no—
305C8—41no—
306C8—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×#12111yescharacteristic
322C22#53211yescharacteristic
323C22#53511yescharacteristic
324C22C2×#13511yescharacteristic
325C22C2×#13511yescharacteristic
326C22#57511yescharacteristic
327C22#57511yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—42no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#54611yescharacteristic
341C4#54611yescharacteristic
342C4#55911yescharacteristic
343C4#55911yescharacteristic
344C2—82no—
345C2—161no—
346C2#254411yescharacteristic
347C2#303511yescharacteristic
348C2#303511yescharacteristic
349C2#316811yescharacteristic
350C2#318311yescharacteristic
351C2#321111yescharacteristic
352C2#321211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#488C2semidirect
3. #2861C2semidirect
4. C4×#124C2semidirect
5. #2056C2semidirect
6. #2055C2semidirect
7. #2055C2semidirect
9. #1991C2semidirect
10. #1958C2semidirect
11. #1880C2semidirect
13. #1878C2semidirect
14. #1877C2semidirect
15. C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4952
C2×#4882
#28612
C4×#1242
#20562
#20552
#20542
#19912
#19582
#18802
#18792
#18782
#18772
C4×C4⋊C82