Finite Group Databasebeta

← #28358

Subgroups — #28358

Order 256.

576 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic
4C2×#417C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic, complemented
6#2432C211yesmaximal, characteristic, complemented
7#2095C211yesmaximal, characteristic, complemented
8#2093C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1970C211yesmaximal, characteristic, complemented
13#1959C211yesmaximal, characteristic, complemented
14#1956C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#123—22no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#484—24no—
28#468—24no—
29#449—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33#417—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#399—22no—
48#394—22no—
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#385—22no—
54#384C2211yescharacteristic
55#384C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16D414yes—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C4⋊Q8—22no—
91C4⋊Q8—41no—
92#124—22no—
93#124—41no—
94#123—48no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97#121—22no—
98#121—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4D412yes—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#107—22no—
121#107—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132C22⋊C8D414yes—
133#97—41no—
134C4×C8—41no—
135#95—22no—
136#95D412yes—
137C24—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140#41—28no—
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×Q8—21no—
155C2×Q8#4112yes—
156C2×D4—21no—
157C2×D4—21no—
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—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—44no—
171C2×D4—42no—
172C2×D4#4112yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4C2×D412yes—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4C2411yescharacteristic
194SD16—44no—
195SD16—44no—
196D8—44no—
197M16—44no—
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—
207C2×C8C2×D412yes—
208C2×C8C2×D412yes—
209C2×C8C2×D412yes—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C22⋊C4—24no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23#12712yes—
249C23C22×D411yescharacteristic
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258D4—22no—
259D4—22no—
260D4—48no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—84no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—48no—
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—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—84no—
295C2×C4#12712yes—
296C2×C4AGL(1,ℤ/8)14yes—
297C2×C4C22×D411yescharacteristic
298C2×C4C22×D411yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C2×C4H(5,𝔽2)11yescharacteristic
303C8—22no—
304C8—22no—
305C8—22no—
306C8—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—44no—
313C22—42no—
314C22—81no—
315C22—81no—
316C22—82no—
317C22—82no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22C2×#12711yescharacteristic
323C22#54611yescharacteristic
324C22#54611yescharacteristic
325C22C2×AGL(1,ℤ/8)12yes—
326C22#57812yes—
327C4—22no—
328C4—24no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4#52511yescharacteristic
337C4#53411yescharacteristic
338C4#54611yescharacteristic
339C4#54611yescharacteristic
340C2—42no—
341C2—82no—
342C2—161no—
343C2#253911yescharacteristic
344C2#315111yescharacteristic
345C2#315611yescharacteristic
346C2#330812yes—
347C2#333012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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×#484C2semidirect
2. C2×#468C2semidirect
4. C2×#417C2semidirect
5. #2501C2semidirect
6. #2432C2semidirect
7. #2095C2semidirect
8. #2093C2semidirect
9. #2013C2semidirect
10. #2012C2semidirect
11. #1973C2semidirect
12. #1970C2semidirect
13. #1959C2semidirect
14. #1956C2semidirect
15. #1753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4682
C2×#4492
C2×#4172
#25012
#24322
#20952
#20932
#20132
#20122
#19732
#19702
#19592
#19562
#17532