Finite Group Databasebeta

← #22141

Subgroups — #22141

Order 256.

470 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3C4×#121C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1955C211yesmaximal, characteristic, complemented
6#1954C211yesmaximal, characteristic, complemented
7#1940C211yesmaximal, characteristic, complemented
8#1924C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1797C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121C411yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#435—24no—
24#432—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#390C411yescharacteristic
46#389C411yescharacteristic
47#388C411yescharacteristic
48#388C411yescharacteristic
49#386C411yescharacteristic
50#386C411yescharacteristic
51#380C411yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4C2×C411yescharacteristic
62C22×D4C2×C411yescharacteristic
63C22×D4C2×C411yescharacteristic
64C23×C4C2×C411yescharacteristic
65C23×C4C2×C411yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×M16—22no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#121—24no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4C2×C411yescharacteristic
91C2×C4⋊C4C2×C411yescharacteristic
92C2×C4⋊C4C2×C411yescharacteristic
93C2×C4⋊C4C2×C411yescharacteristic
94C2×C4⋊C4C2×C411yescharacteristic
95C2×C22⋊C4C2×C411yescharacteristic
96C2×C22⋊C4C2×C411yescharacteristic
97C2×C22⋊C4C2×C411yescharacteristic
98C2×C22⋊C4C2×C411yescharacteristic
99C2×C22⋊C4C2×C411yescharacteristic
100C2×C22⋊C4C2×C411yescharacteristic
101C2×C22⋊C4C2×C411yescharacteristic
102C2×C22⋊C4C2×C411yescharacteristic
103C2×C22⋊C4C2×C411yescharacteristic
104C2×C22⋊C4C2×C411yescharacteristic
105C2×C42—22no—
106C2×C42—22no—
107C2×C42C2×C411yescharacteristic
108C2×C42C2×C411yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129#97—22no—
130#97—41no—
131#97D414yescomplemented
132C4×C8—41no—
133#95C2×C411yescharacteristic
134#95C2×C411yescharacteristic
135#95C2×C411yescharacteristic
136#95C2×C411yescharacteristic
137#95C2×C411yescharacteristic
138#95C2×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141C24C22×C411yescharacteristic
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
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—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C22×C411yescharacteristic
184C22×C4C22×C411yescharacteristic
185C22×C4C22×C411yescharacteristic
186C22×C4C2411yescharacteristic
187M16—44no—
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×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
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—22no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4M1614yescomplemented
228C42—22no—
229C42—22no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42C2×D412yes—
237C42#4112yes—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
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—21no—
256C23—21no—
257C23C2×M1612yes—
258C23#13112yes—
259C23C23×C411yescharacteristic
260D4—44no—
261D4—44no—
262D4—44no—
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—21no—
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—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—44no—
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—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4C4×D412yes—
307C2×C4C4×D412yes—
308C2×C4C2×M1612yes—
309C2×C4C2×M1612yes—
310C2×C4C22×D411yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C42−1+411yescharacteristic
316C8—22no—
317C8—22no—
318C8—41no—
319C8—41no—
320C8—41no—
321C8—41no—
322C8—41no—
323C8—41no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22C2×C4×D411yescharacteristic
346C22#51811yescharacteristic
347C22#52011yescharacteristic
348C22C22×M1611yescharacteristic
349C22C2×#13111yescharacteristic
350C22#56811yescharacteristic
351C22#56811yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4#53511yescharacteristic
364C4#53711yescharacteristic
365C4#54011yescharacteristic
366C4#54211yescharacteristic
367C2—42no—
368C2—42no—
369C2—81no—
370C2#232411yescharacteristic
371C2#293311yescharacteristic
372C2D4×M1611yescharacteristic
373C2#297411yescharacteristic
374C2#298011yescharacteristic
375C2#298911yescharacteristic
376C2#299211yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#121C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#435C2semidirect
2. C2×#432C2semidirect
4. #1955C2semidirect
5. #1955C2semidirect
6. #1954C2semidirect
7. #1940C2semidirect
8. #1924C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1797C2semidirect
15. #1771C2semidirect
131. #97D4semidirect
227. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×#4322
C4×#1212
#19552
#19542
#19402
#19242
#19232
#18702
#18692
#18512
#18502
#17972
#17712