Finite Group Databasebeta

← #28476

Subgroups — #28476

Order 256.

550 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic, complemented
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#495C211yesmaximal, characteristic
4C2×#462C211yesmaximal, characteristic, complemented
5C2×#463C211yesmaximal, characteristic
6C2×#440C211yesmaximal, characteristic, complemented
7#2547C211yesmaximal, characteristic, complemented
8#2545C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#2015C211yesmaximal, characteristic, complemented
12#2012C211yesmaximal, characteristic
13#1973C211yesmaximal, characteristic, complemented
14#1969C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×D8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C4⋊Q8C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×#124C2211yescharacteristic
29C2×#124C2211yescharacteristic
30C2×#124C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33#497—24no—
34#497—24no—
35#495—24no—
36#463—24no—
37#462—24no—
38#440—24no—
39C2×#107C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#97C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#397—22no—
51#396—22no—
52#391C2211yescharacteristic
53#391C2211yescharacteristic
54#391C2211yescharacteristic
55#390C2211yescharacteristic
56#388—22no—
57#380—22no—
58C4×C4⋊C4C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—41no—
89C2×SD16—41no—
90C2×SD16—41no—
91C2×SD16D414yes—
92C2×D8—22no—
93C2×D8—22no—
94C2×D8—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—41no—
103#124—22no—
104#124—22no—
105#124—22no—
106#124—22no—
107#124—22no—
108#124—41no—
109#124—41no—
110#124—41no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×D4—22no—
114C4×D4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4D412yes—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C22⋊C4—21no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132C2×C42C2311yescharacteristic
133#107—22no—
134C4⋊C8D414yes—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#102—22no—
140#102—22no—
141#97—22no—
142#97D414yes—
143C4×C8—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95D412yes—
148C24—21no—
149C24—21no—
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×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8C2×D412yes—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4C2×D412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4C2×D412yes—
195C22×C4C2×D412yes—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199Q16—44no—
200Q16—44no—
201Q16—44no—
202Q16—44no—
203SD16—28no—
204SD16—44no—
205SD16—44no—
206SD16—44no—
207D8—44no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8C2×D412yes—
215C2×C8C2×D412yes—
216C2×C8C2×D412yes—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C2×C8#4112yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42C2×D412yes—
242C42C2×D412yes—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23C22×D411yescharacteristic
254Q8—22no—
255Q8—22no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270D4—22no—
271D4—22no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
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—42no—
301C2×C4—42no—
302C2×C4—84no—
303C2×C4#12712yes—
304C2×C4#12712yes—
305C2×C4C22×D411yescharacteristic
306C2×C4C22×D411yescharacteristic
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C8—22no—
312C8—22no—
313C8—22no—
314C8#12114yes—
315C22—44no—
316C22—42no—
317C22—82no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22—82no—
324C22—81no—
325C22C2×#12711yescharacteristic
326C22D4211yescharacteristic
327C22#55011yescharacteristic
328C22#57512yes—
329C22#57812yes—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4C2×#12111yescharacteristic
341C4#53511yescharacteristic
342C4D4211yescharacteristic
343C4#55011yescharacteristic
344C2—82no—
345C2—161no—
346C2#254111yescharacteristic
347C2#315311yescharacteristic
348C2#315611yescharacteristic
349C2#326912yes—
350C2#340212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#497C2semidirect
2. C2×#497C2semidirect
4. C2×#462C2semidirect
6. C2×#440C2semidirect
7. #2547C2semidirect
8. #2545C2semidirect
9. #2016C2semidirect
10. #2015C2semidirect
11. #2015C2semidirect
13. #1973C2semidirect
14. #1969C2semidirect
15. #1771C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4952
C2×#4622
C2×#4632
C2×#4402
#25472
#25452
#20162
#20152
#20122
#19732
#19692
#17712