Finite Group Databasebeta

← #23039

Subgroups — #23039

Order 256.

541 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2361C211yesmaximal, characteristic, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1944C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14C4×#106C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#469C418yescomplemented
26#442—24no—
27#428—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#390—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#386—21no—
50#386—21no—
51#385—21no—
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C2×D8—22no—
71C2×D8—22no—
72C2×D8C2×C414yescomplemented
73C2×M16—22no—
74C2×M16—22no—
75C2×M16C2×C414yescomplemented
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#121—22no—
81#121—22no—
82#121C2×C414yescomplemented
83#121C2×C414yescomplemented
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—44no—
89#117—22no—
90#117—41no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#107—22no—
119#107—22no—
120#106—22no—
121#106C2×C414yescomplemented
122#106D414yescomplemented
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102C2×C414yescomplemented
128#102C2×C414yescomplemented
129C22⋊C8—22no—
130#97—22no—
131C4×C8—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141C2×D4—24no—
142C2×D4—24no—
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—21no—
152C2×D4—21no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4C22×C412yes—
158C2×D4C22×C412yes—
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—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C22×C412yes—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186D8—28no—
187M16—28no—
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×C8C22×C412yes—
197C2×C8C22×C412yes—
198C2×C8C2×D412yes—
199C2×C8#4112yes—
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—81no—
211C4⋊C4C22×C412yes—
212C4⋊C4C22×C412yes—
213C4⋊C4C2×D412yes—
214C4⋊C4C2×D412yes—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—22no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—42no—
231C42—42no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23C4×D412yes—
254C23C22×D411yescharacteristic
255D4—22no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—48no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—84no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
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—41no—
299C2×C4—41no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4C4×D412yes—
304C2×C4AGL(1,ℤ/8)12yes—
305C2×C4C23×C411yescharacteristic
306C2×C4C22×D411yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C42−1+411yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8C4×D414yescomplemented
315C22—22no—
316C22—22no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×C4×D411yescharacteristic
337C22D4211yescharacteristic
338C22#54711yescharacteristic
339C22C2×AGL(1,ℤ/8)11yescharacteristic
340C22#57511yescharacteristic
341C22#57611yescharacteristic
342C22#57711yescharacteristic
343C4—22no—
344C4—22no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4C2×C4×D411yescharacteristic
355C4#52011yescharacteristic
356C4#54811yescharacteristic
357C4#55311yescharacteristic
358C2—42no—
359C2—82no—
360C2—82no—
361C2#236111yescharacteristic
362C2C4×AGL(1,ℤ/8)11yescharacteristic
363C2#294611yescharacteristic
364C2#327411yescharacteristic
365C2#327511yescharacteristic
366C2#334511yescharacteristic
367C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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. C2×#442C2semidirect
3. C2×#428C2semidirect
4. #2363C2semidirect
5. #2361C2semidirect
6. #1970C2semidirect
7. #1944C2semidirect
8. #1926C2semidirect
9. #1925C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1864C2semidirect
13. #1862C2semidirect
14. C4×#106C2semidirect
15. #1753C2semidirect
25. #469C4semidirect
72. C2×D8C2×C4semidirect
75. C2×M16C2×C4semidirect
82. #121C2×C4semidirect
83. #121C2×C4semidirect
121. #106C2×C4semidirect
122. #106D4semidirect
127. #102C2×C4semidirect
128. #102C2×C4semidirect
314. C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4422
C2×#4282
#23632
#23612
#19702
#19442
#19262
#19252
#18752
#18742
#18642
#18622
C4×#1062
#17532