Finite Group Databasebeta

← #22582

Subgroups — #22582

Order 256.

519 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#417C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic, complemented
3#2339C211yesmaximal, characteristic
4#1935C211yesmaximal, characteristic, complemented
5#1933C211yesmaximal, characteristic, complemented
6#1928C211yesmaximal, characteristic, complemented
7#1927C211yesmaximal, characteristic, complemented
8#1864C211yesmaximal, characteristic, complemented
9#1862C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic
13#1762C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117—21no—
21C22×C4⋊C4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24#417C418yescomplemented
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#387—21no—
42#386—21no—
43#386—21no—
44#381—21no—
45#381—21no—
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51C2×#95—21no—
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41C2×C414yes—
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×M16—41no—
75C2×M16—41no—
76C2×M16C2×C414yescomplemented
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—41no—
85#117—44no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2×C414yescomplemented
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—41no—
108#106—41no—
109#103—41no—
110#103—41no—
111#103C2×C414yescomplemented
112#103C2×C414yescomplemented
113#102—22no—
114#102—22no—
115#102C2×C414yes—
116#102C2×C414yes—
117C22⋊C8—22no—
118#97—41no—
119C4×C8—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24C2×D411yescharacteristic
134#41—28no—
135#41—44no—
136#41—44no—
137#41—44no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C22×C412yes—
144C2×D4—24no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4C22×C412yes—
151C2×D4#4112yes—
152C2×D4#4112yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4C22×C412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4#4112yes—
183C22×C4C2411yescharacteristic
184M16—44no—
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×C8C22×C412yes—
196C2×C8C22×C412yes—
197C4⋊C4—24no—
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⋊C4C22×C412yes—
211C4⋊C4C22×C412yes—
212C22⋊C4—22no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—42no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C4×D412yes—
238C23C22×D411yescharacteristic
239Q8—22no—
240Q8—22no—
241Q8—41no—
242Q8—41no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—48no—
247D4#11714yescomplemented
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
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—48no—
269C2×C4—48no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
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—84no—
289C2×C4—84no—
290C2×C4C4×D412yes—
291C2×C4AGL(1,ℤ/8)12yes—
292C2×C4C23×C411yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C8—22no—
299C8—22no—
300C8—41no—
301C8—41no—
302C22—22no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22C2×C4×D411yescharacteristic
317C22#54611yescharacteristic
318C22#54611yescharacteristic
319C22C2×AGL(1,ℤ/8)11yescharacteristic
320C22#57511yescharacteristic
321C22#57711yescharacteristic
322C22#57811yescharacteristic
323C4—22no—
324C4—22no—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4C2×#11711yescharacteristic
336C4#51811yescharacteristic
337C4#55111yescharacteristic
338C4#55111yescharacteristic
339C2—42no—
340C2—44no—
341C2#234211yescharacteristic
342C2C4×AGL(1,ℤ/8)11yescharacteristic
343C2#294711yescharacteristic
344C2#330611yescharacteristic
345C2#331511yescharacteristic
346C2#332311yescharacteristic
347C2#333011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×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
2. #2344C2semidirect
4. #1935C2semidirect
5. #1933C2semidirect
6. #1928C2semidirect
7. #1927C2semidirect
8. #1864C2semidirect
9. #1862C2semidirect
10. #1824C2semidirect
11. #1809C2semidirect
13. #1762C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
24. #417C4semidirect
76. C2×M16C2×C4semidirect
90. C2×C4⋊C4C2×C4semidirect
111. #103C2×C4semidirect
112. #103C2×C4semidirect
247. D4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#4172
#23442
#23392
#19352
#19332
#19282
#19272
#18642
#18622
#18242
#18092
#17882
#17622
C4×#1022
#17532