Finite Group Databasebeta

← #28352

Subgroups — #28352

Order 256.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#416C211yesmaximal, characteristic, complemented
5#2499C211yesmaximal, characteristic, complemented
6#2431C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1971C211yesmaximal, characteristic, complemented
13#1961C211yesmaximal, characteristic, complemented
14#1959C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#482—24no—
27#465—24no—
28#452—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32#416—24no—
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×#97C2211yescharacteristic
46#397—22no—
47#392—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#387—22no—
53#386—22no—
54#384C2211yescharacteristic
55#382C2211yescharacteristic
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×D4—22no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8D414yescomplemented
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#127—22no—
91#127—41no—
92#124—22no—
93#124—41no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96#121—22no—
97#121—22no—
98#117—22no—
99#117—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4D412yes—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116#106—22no—
117#106—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128C22⋊C8D414yescomplemented
129#97—41no—
130#97—41no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95D412yes—
136C24—21no—
137C24—21no—
138C24C2×D411yescharacteristic
139#41—28no—
140#41—44no—
141#41—44no—
142#41—44no—
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×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8#4112yes—
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×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—82no—
173C2×D4—82no—
174C2×D4#4112yes—
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—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4C2411yescharacteristic
194Q16—44no—
195SD16—44no—
196SD16—44no—
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—44no—
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—42no—
220C22⋊C4—42no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—42no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23#12712yes—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255D4—22no—
256D4—22no—
257D4—48no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—88no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
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—48no—
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—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4#12712yes—
294C2×C4C22×D411yescharacteristic
295C2×C4C22×D411yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C8—22no—
301C8—22no—
302C8—22no—
303C8—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—82no—
318C22—82no—
319C22C2×#12711yescharacteristic
320C22#54611yescharacteristic
321C22#54611yescharacteristic
322C22#57512yes—
323C22#57712yes—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4#52511yescharacteristic
334C4#53411yescharacteristic
335C4#54611yescharacteristic
336C4#54611yescharacteristic
337C2—42no—
338C2—82no—
339C2—161no—
340C2#253911yescharacteristic
341C2#315011yescharacteristic
342C2#315411yescharacteristic
343C2#331112yes—
344C2#331712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2431C2
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×#482C2semidirect
2. C2×#465C2semidirect
3. C2×#452C2semidirect
4. C2×#416C2semidirect
5. #2499C2semidirect
7. #2097C2semidirect
8. #2094C2semidirect
9. #2015C2semidirect
10. #2014C2semidirect
11. #1973C2semidirect
12. #1971C2semidirect
13. #1961C2semidirect
14. #1959C2semidirect
15. #1752C2semidirect
85. C22×C8D4semidirect
128. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4652
C2×#4522
C2×#4162
#24992
#24312
#20972
#20942
#20152
#20142
#19732
#19712
#19612
#19592
#17522