Finite Group Databasebeta

← #28609

Subgroups — #28609

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic, complemented
3#2562C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic
5#2034C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#127—21no—
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#490—24no—
27#435—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#400—21no—
43#394—21no—
44#392—21no—
45#392—21no—
46#390—21no—
47#388—21no—
48#388—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#127—22no—
83#127—22no—
84#127—44no—
85#125—22no—
86#125—22no—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—41no—
92#121—41no—
93#121—41no—
94#121—41no—
95C4×D4—22no—
96C4×D4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4D411yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
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—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119#107—22no—
120#106—22no—
121C4⋊C8—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129#102—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132#97D414yescomplemented
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
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—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199M16—44no—
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—21no—
210C2×C8—44no—
211C2×C8C2×D412yes—
212C2×C8C2×D412yes—
213C4⋊C4—22no—
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—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4#4112yes—
227C4⋊C4#4112yes—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—42no—
239C42—81no—
240C42C2×D412yes—
241C42#4112yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23AGL(1,ℤ/8)12yes—
270C23C22×D411yescharacteristic
271D4—44no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—84no—
281D4—84no—
282D4—84no—
283D4—84no—
284D4—84no—
285D4—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4#12112yes—
323C2×C4#12112yes—
324C2×C4C22×D411yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C8—22no—
331C8—22no—
332C8—41no—
333C8—41no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C2×#12111yescharacteristic
363C22#54611yescharacteristic
364C22#54611yescharacteristic
365C22C2×AGL(1,ℤ/8)11yescharacteristic
366C22#57511yescharacteristic
367C22#57611yescharacteristic
368C22#57711yescharacteristic
369C4—22no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4#53211yescharacteristic
380C4#53511yescharacteristic
381C4#55911yescharacteristic
382C4#55911yescharacteristic
383C2—42no—
384C2—81no—
385C2—161no—
386C2—161no—
387C2#254411yescharacteristic
388C2#305511yescharacteristic
389C2#306211yescharacteristic
390C2#330911yescharacteristic
391C2#331011yescharacteristic
392C2#331711yescharacteristic
393C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#490C2semidirect
2. C2×#435C2semidirect
3. #2562C2semidirect
5. #2034C2semidirect
6. #2034C2semidirect
7. #2033C2semidirect
8. #2033C2semidirect
9. #1989C2semidirect
10. #1982C2semidirect
11. #1895C2semidirect
12. #1895C2semidirect
13. #1893C2semidirect
14. #1893C2semidirect
15. #1776C2semidirect
132. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4352
#25622
#25412
#20342
#20332
#19892
#19822
#18952
#18932
#17762