Finite Group Databasebeta

← #25509

Subgroups — #25509

Order 256.

546 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic
4#2625C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic
10#1892C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#542—24no—
21#542—24no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#126—21no—
24C2×#124—21no—
25C2×#124C2211yescharacteristic
26C2×#123—21no—
27C2×C22⋊Q8—21no—
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×#117C2211yescharacteristic
34#470—24no—
35#465—24no—
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46#400—21no—
47#398—21no—
48#397—21no—
49#397—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88#126—28no—
89#124—28no—
90#124—22no—
91#124—22no—
92#123—28no—
93C22⋊Q8—28no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8D414yes—
98#121—22no—
99#121D414yes—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×D4—22no—
103C4×D4—22no—
104#117—22no—
105#117—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#106—22no—
132#106—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—41no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
150C2×Q8—24no—
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—81no—
163C2×Q8—81no—
164C2×Q8C2×D412yes—
165C2×D4—24no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4C2×D412yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D412yes—
194C22×C4C2×D412yes—
195C22×C4C2411yescharacteristic
196Q16—44no—
197SD16—44no—
198M16—44no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8#4112yes—
211C2×C8#4112yes—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4C2×D412yes—
229C4⋊C4C2×D412yes—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—24no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C42—24no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#12012yes—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—84no—
268D4—22no—
269D4—22no—
270D4—41no—
271D4—41no—
272D4—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
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—84no—
313C2×C4#12012yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C42−1+411yescharacteristic
320C8—22no—
321C8—22no—
322C8—41no—
323C8—41no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C2×#12011yescharacteristic
339C22#53211yescharacteristic
340C22#53711yescharacteristic
341C22#57711yescharacteristic
342C22#57711yescharacteristic
343C22#57711yescharacteristic
344C22#57811yescharacteristic
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4#54611yescharacteristic
358C4#54611yescharacteristic
359C4#54711yescharacteristic
360C4#54711yescharacteristic
361C2—42no—
362C2—82no—
363C2#243011yescharacteristic
364C2#300911yescharacteristic
365C2#301311yescharacteristic
366C2#317311yescharacteristic
367C2#317811yescharacteristic
368C2#325811yescharacteristic
369C2#326311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#470C2semidirect
4. #2625C2semidirect
5. #2058C2semidirect
6. #2042C2semidirect
7. #2024C2semidirect
8. #2015C2semidirect
10. #1892C2semidirect
11. #1878C2semidirect
12. #1877C2semidirect
13. #1865C2semidirect
14. #1864C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4702
C2×#4652
#26252
#20582
#20422
#20242
#20152
#18952
#18922
#18782
#18772
#18652
#18642
#17292