Finite Group Databasebeta

← #22144

Subgroups — #22144

Order 256.

441 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×#435C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2392C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1954C211yesmaximal, characteristic, complemented
6#1954C211yesmaximal, characteristic, complemented
7#1939C211yesmaximal, 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#1798C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#435—24no—
23#433—24no—
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C22⋊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×#97C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#398C411yescharacteristic
45#397C411yescharacteristic
46#396C411yescharacteristic
47#396C411yescharacteristic
48#394C411yescharacteristic
49#394C411yescharacteristic
50#394C411yescharacteristic
51#392C411yescharacteristic
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×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×M16—22no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—41no—
87#117—22no—
88#117—22no—
89C2×C4⋊C4C2×C411yescharacteristic
90C2×C4⋊C4C2×C411yescharacteristic
91C2×C4⋊C4C2×C411yescharacteristic
92C2×C4⋊C4C2×C411yescharacteristic
93C2×C4⋊C4C2×C411yescharacteristic
94C2×C4⋊C4C2×C411yescharacteristic
95C2×C4⋊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×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128#97—41no—
129#97D414yescomplemented
130C4×C8—22no—
131C4×C8—41no—
132#95C2×C411yescharacteristic
133#95C2×C411yescharacteristic
134#95C2×C411yescharacteristic
135#95C2×C411yescharacteristic
136#95C2×C411yescharacteristic
137#95C2×C411yescharacteristic
138C24C22×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
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×C4C22×C411yescharacteristic
169C22×C4C22×C411yescharacteristic
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C411yescharacteristic
172C22×C4C22×C411yescharacteristic
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C2411yescharacteristic
180M16—44no—
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×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—44no—
204C2×C8C2×D412yes—
205C2×C8C2×D412yes—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42C2×D412yes—
232C42#4112yes—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
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#13112yes—
254C23C23×C411yescharacteristic
255D4—44no—
256D4—44no—
257D4—44no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—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—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—41no—
299C2×C4—41no—
300C2×C4C4×D412yes—
301C2×C4C4×D412yes—
302C2×C4#13112yes—
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C42−1+411yescharacteristic
309C8—22no—
310C8—22no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22C2×C4×D411yescharacteristic
339C22#51811yescharacteristic
340C22#52011yescharacteristic
341C22C2×#13111yescharacteristic
342C22C2×#13111yescharacteristic
343C22#56811yescharacteristic
344C22#56811yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4#53511yescharacteristic
357C4#53711yescharacteristic
358C4#54011yescharacteristic
359C4#54211yescharacteristic
360C2—42no—
361C2—81no—
362C2—81no—
363C2#232411yescharacteristic
364C2#293311yescharacteristic
365C2#293511yescharacteristic
366C2#297711yescharacteristic
367C2#297811yescharacteristic
368C2#299211yescharacteristic
369C2#299311yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radical#392C4
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×#433C2semidirect
4. #1955C2semidirect
5. #1954C2semidirect
6. #1954C2semidirect
7. #1939C2semidirect
8. #1924C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1798C2semidirect
15. #1771C2semidirect
129. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×#4332
#23922
#19552
#19542
#19392
#19242
#19232
#18702
#18692
#18512
#18502
#17982
#17712