Finite Group Databasebeta

← #23708

Subgroups — #23708

Order 256.

521 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#442C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3C4×#121C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic, complemented
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2054C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic, complemented
11#2033C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic
13#1776C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#442—24no—
25#432—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#386—21no—
48#386—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#121—22no—
78#121—22no—
79#121—22no—
80#121—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—44no—
86C4×D4—44no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
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×C42—22no—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#106—22no—
115#106—22no—
116C4⋊C8—22no—
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#97D414yescomplemented
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134C24—21no—
135C24—21no—
136C24C2×D411yescharacteristic
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—21no—
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—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4#4112yes—
154C2×D4#4112yes—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
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—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2411yescharacteristic
185D8—28no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8C2×D412yes—
197C2×C8C2×D412yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4#4112yes—
211C4⋊C4#4112yes—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—22no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—42no—
228C42C2×D412yes—
229C42#4112yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23AGL(1,ℤ/8)12yes—
251C23C22×D411yescharacteristic
252D4—22no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—48no—
257D4—48no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—44no—
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—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4#12112yes—
301C2×C4#12112yes—
302C2×C4AGL(1,ℤ/8)12yes—
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C8—22no—
310C8—22no—
311C8—41no—
312C8—41no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#12111yescharacteristic
335C22#52511yescharacteristic
336C22#52511yescharacteristic
337C22C2×AGL(1,ℤ/8)11yescharacteristic
338C22C2×AGL(1,ℤ/8)11yescharacteristic
339C22#57511yescharacteristic
340C22#57511yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4#52511yescharacteristic
353C4#52911yescharacteristic
354C4#52911yescharacteristic
355C4#53211yescharacteristic
356C2—42no—
357C2—82no—
358C2—82no—
359C2#239211yescharacteristic
360C2#305511yescharacteristic
361C2#305511yescharacteristic
362C2#310911yescharacteristic
363C2#311111yescharacteristic
364C2#311411yescharacteristic
365C2#311511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#442C2semidirect
2. C2×#432C2semidirect
3. C4×#121C2semidirect
4. C4×#121C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #2054C2semidirect
10. #2034C2semidirect
11. #2033C2semidirect
13. #1776C2semidirect
14. #1762C2semidirect
15. #1762C2semidirect
127. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4422
C2×#4322
C4×#1212
#20882
#20772
#20672
#20652
#20542
#20342
#20332
#19992
#17762
#17622