Finite Group Databasebeta

← #31013

Subgroups — #31013

Order 256.

475 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#462C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic
3#2432C211yesmaximal, characteristic
4#2423C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1959C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#462—24no—
26#452—24no—
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#399—21no—
43#398—21no—
44#394—21no—
45#394—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#385—21no—
53#385—21no—
54#384C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16D414yes—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—41no—
89#124—22no—
90#124—41no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C4×Q8—22no—
96C4×Q8—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#106—41no—
122C4⋊C8—22no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—41no—
127#103—41no—
128#103—41no—
129#103—41no—
130#102—22no—
131#102—41no—
132#102—41no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135#97—41no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24C2×D411yescharacteristic
146#41—44no—
147#41—44no—
148#41—44no—
149#41—44no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8C2×D412yes—
167C2×Q8C2×D412yes—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
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—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199Q16—44no—
200Q16—44no—
201SD16—44no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C2×D412yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C22⋊C4—22no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271D4—22no—
272D4—22no—
273D4—48no—
274D4—41no—
275D4—41no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—48no—
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×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#13712yes—
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—22no—
324C8—41no—
325C8—41no—
326C8—41no—
327C22—41no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#53511yescharacteristic
342C22D4211yescharacteristic
343C22#54611yescharacteristic
344C22C2×#13711yescharacteristic
345C22#57511yescharacteristic
346C22#57711yescharacteristic
347C22#57811yescharacteristic
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4#53411yescharacteristic
361C4#53811yescharacteristic
362C4#54611yescharacteristic
363C4#56011yescharacteristic
364C2—81no—
365C2—82no—
366C2#267811yescharacteristic
367C2#321611yescharacteristic
368C2#322511yescharacteristic
369C2#327011yescharacteristic
370C2#327111yescharacteristic
371C2#331511yescharacteristic
372C2#333111yescharacteristic
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
1. C2×#462C2semidirect
4. #2423C2semidirect
5. #2072C2semidirect
6. #2065C2semidirect
7. #2043C2semidirect
8. #2035C2semidirect
9. #2015C2semidirect
10. #2001C2semidirect
11. #1971C2semidirect
12. #1959C2semidirect
13. #1924C2semidirect
14. #1880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4522
#24322
#24232
#20722
#20652
#20432
#20352
#20152
#20012
#19712
#19592
#19242
#18802
#17882