Finite Group Databasebeta

← #26812

Subgroups — #26812

Order 256.

455 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3#2491C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic
9#1972C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#482—24no—
24#474—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#394—21no—
41#393—21no—
42#390—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#386—21no—
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×D8—22no—
69C2×D8—22no—
70C2×D8—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77#124—44no—
78#121—22no—
79#121—22no—
80#121—22no—
81#121—41no—
82C4×D4—22no—
83C4×D4—41no—
84#117—22no—
85#117—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#107D414yescomplemented
114#106—22no—
115#106—22no—
116#106—41no—
117C4⋊C8—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126#97—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24—21no—
135C24C2×D411yescharacteristic
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×D4—24no—
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—42no—
152C2×D4—42no—
153C2×D4—81no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4#4112yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2411yescharacteristic
178D8—44no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8C2×D412yes—
190C2×C8#4112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4C2×D412yes—
203C4⋊C4C2×D412yes—
204C4⋊C4#4112yes—
205C22⋊C4—24no—
206C22⋊C4—22no—
207C22⋊C4—41no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—42no—
224C42—81no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
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—
246C23C22×D411yescharacteristic
247Q8—84no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—84no—
257D4—84no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
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—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4AGL(1,ℤ/8)12yes—
296C2×C4C22×D411yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C2×C42−1+411yescharacteristic
302C8—22no—
303C8—22no—
304C8—41no—
305C8—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—41no—
311C22—42no—
312C22—41no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#52511yescharacteristic
328C22D4211yescharacteristic
329C22#54711yescharacteristic
330C22C2×AGL(1,ℤ/8)11yescharacteristic
331C22#57511yescharacteristic
332C22#57611yescharacteristic
333C22#57711yescharacteristic
334C4—22no—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—42no—
340C4—42no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#53211yescharacteristic
346C4#54211yescharacteristic
347C4#54811yescharacteristic
348C4#55111yescharacteristic
349C2—81no—
350C2—82no—
351C2—161no—
352C2#246111yescharacteristic
353C2#311511yescharacteristic
354C2#311811yescharacteristic
355C2#326411yescharacteristic
356C2#327111yescharacteristic
357C2#334711yescharacteristic
358C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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×#474C2semidirect
3. #2491C2semidirect
4. #2363C2semidirect
5. #2065C2semidirect
6. #2058C2semidirect
7. #2034C2semidirect
9. #1972C2semidirect
10. #1926C2semidirect
11. #1925C2semidirect
12. #1924C2semidirect
13. #1879C2semidirect
14. #1854C2semidirect
15. #1776C2semidirect
113. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4742
#24912
#23632
#20652
#20582
#20342
#19992
#19722
#19262
#19252
#19242
#18792
#18542
#17762