Finite Group Databasebeta

← #26246

Subgroups — #26246

Order 256.

503 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic
4#2352C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1961C211yesmaximal, characteristic, complemented
11#1955C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1874C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×#127—21no—
17C2×#127C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#482—24no—
26#481—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#392—21no—
46#392—21no—
47#390—21no—
48#388—21no—
49#387—21no—
50#386—21no—
51#382C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C22⋊C4—21no—
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×C4—21no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#127—22no—
78#127—22no—
79#127—44no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121—22no—
84#121—41no—
85#121—41no—
86#121—41no—
87#121—41no—
88C4×D4—22no—
89C4×D4—22no—
90#117—22no—
91#117—41no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115#107—22no—
116#107—22no—
117#106—22no—
118#106D414yescomplemented
119C4⋊C8—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—41no—
125#102—41no—
126#102—41no—
127#102—41no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132#97—22no—
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×C4—42no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
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—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C2×D412yes—
214C4⋊C4—22no—
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⋊C4C2×D412yes—
226C4⋊C4C2×D412yes—
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—
237C42—42no—
238C42—42no—
239C42—81no—
240C42#4112yes—
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—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
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—84no—
323C2×C4#12312yes—
324C2×C4#12312yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
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—
362C22—81no—
363C22C2×#12311yescharacteristic
364C22D4211yescharacteristic
365C22#54611yescharacteristic
366C22C2×AGL(1,ℤ/8)11yescharacteristic
367C22#57511yescharacteristic
368C22#57611yescharacteristic
369C22#57711yescharacteristic
370C4—22no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4#53211yescharacteristic
381C4#54011yescharacteristic
382C4#55011yescharacteristic
383C4#55911yescharacteristic
384C2—42no—
385C2—81no—
386C2—161no—
387C2—161no—
388C2#244511yescharacteristic
389C2#309411yescharacteristic
390C2#309611yescharacteristic
391C2#326511yescharacteristic
392C2#326811yescharacteristic
393C2#331711yescharacteristic
394C2#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×#481C2semidirect
2. C2×#482C2semidirect
4. #2352C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #2033C2semidirect
8. #2033C2semidirect
9. #1982C2semidirect
10. #1961C2semidirect
11. #1955C2semidirect
12. #1949C2semidirect
13. #1875C2semidirect
14. #1874C2semidirect
15. #1776C2semidirect
118. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4822
#25412
#23522
#20342
#20332
#19822
#19612
#19552
#19492
#18752
#18742
#17762