Finite Group Databasebeta

← #27780

Subgroups — #27780

Order 256.

485 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×#490C211yesmaximal, characteristic, complemented
2C2×#486C211yesmaximal, characteristic, complemented
3#2550C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1961C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#121—21no—
22C2×#121—21no—
23C2×C4×D4C2211yescharacteristic
24#490—24no—
25#486—24no—
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#398—21no—
43#395—21no—
44#392—21no—
45#392—21no—
46#391—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389—21no—
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52#382C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#127—22no—
79#127—22no—
80#127—22no—
81#127—22no—
82#127—41no—
83#127—41no—
84#125—22no—
85#125—41no—
86#121—44no—
87#121—44no—
88C4×D4—22no—
89C4×D4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
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×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C4⋊C8—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—41no—
122#102—41no—
123#102D414yescomplemented
124#97—22no—
125#97—22no—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—81no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
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—42no—
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—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4C2×D412yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2411yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8C2×D412yes—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C2×D412yes—
219C4⋊C4#4112yes—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—81no—
230C42#4112yes—
231C42#4112yes—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23C22×D411yescharacteristic
254Q8—84no—
255D4—22no—
256D4—22no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—84no—
268D4—84no—
269D4—84no—
270D4—84no—
271D4—84no—
272D4—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
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—41no—
301C2×C4—41no—
302C2×C4—84no—
303C2×C4#12412yes—
304C2×C4#12412yes—
305C2×C4AGL(1,ℤ/8)12yes—
306C2×C4AGL(1,ℤ/8)12yes—
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C42−1+411yescharacteristic
313C8—22no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C2×#12411yescharacteristic
339C22D4211yescharacteristic
340C22#54811yescharacteristic
341C22C2×AGL(1,ℤ/8)11yescharacteristic
342C22C2×AGL(1,ℤ/8)11yescharacteristic
343C22#57611yescharacteristic
344C22#57711yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4#53211yescharacteristic
357C4#54211yescharacteristic
358C4#54711yescharacteristic
359C4#55511yescharacteristic
360C2—82no—
361C2—161no—
362C2—161no—
363C2#249111yescharacteristic
364C2#313611yescharacteristic
365C2#313811yescharacteristic
366C2#326411yescharacteristic
367C2#326511yescharacteristic
368C2#336611yescharacteristic
369C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
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×#490C2semidirect
2. C2×#486C2semidirect
3. #2550C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
9. #1982C2semidirect
10. #1961C2semidirect
11. #1936C2semidirect
12. #1934C2semidirect
13. #1804C2semidirect
14. #1771C2semidirect
15. #1762C2semidirect
123. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4862
#25502
#25412
#20412
#20252
#20232
#19822
#19612
#19362
#19342
#18042
#17712
#17622