Finite Group Databasebeta

← #22862

Subgroups — #22862

Order 256.

567 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×#452C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic
6#1969C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1878C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#452C418yescomplemented
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29#417—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#387—21no—
47#386—21no—
48#386—21no—
49#385—21no—
50#381—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41C2×C414yes—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16C2×C414yescomplemented
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16C2×C414yescomplemented
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8C2×C414yescomplemented
92C4×Q8—22no—
93C4×Q8—41no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—41no—
97C4×D4—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#107—41no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103C2×C414yescomplemented
126#102—22no—
127#102C2×C414yescomplemented
128#102D414yes—
129C22⋊C8C2×C414yescomplemented
130#97—22no—
131C4×C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95D411yescharacteristic
142#95D411yescharacteristic
143C24—21no—
144C24C2×D411yescharacteristic
145#41—28no—
146#41—44no—
147#41—44no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8C22×C412yes—
160C2×Q8C22×C412yes—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4C22×C412yes—
168C2×D4C2×D412yes—
169C2×D4#4112yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C22×C412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201Q16—44no—
202SD16—28no—
203M16—44no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C22×C412yes—
214C2×C8C22×C412yes—
215C2×C8C2×D412yes—
216C4⋊C4—24no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4C22×C412yes—
229C4⋊C4C2×D412yes—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—81no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C4×D412yes—
256C23C22×D411yescharacteristic
257Q8—22no—
258Q8—22no—
259Q8—22no—
260Q8—22no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265D4—22no—
266D4—22no—
267D4—22no—
268D4—48no—
269D4C4×D414yes—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—48no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4C4×D412yes—
313C2×C4#13712yes—
314C2×C4C23×C411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C8—22no—
321C8—22no—
322C8—22no—
323C8—41no—
324C22—22no—
325C22—24no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22C2×C4×D411yescharacteristic
339C22D4211yescharacteristic
340C22#54611yescharacteristic
341C22C2×#13711yescharacteristic
342C22#57511yescharacteristic
343C22#57611yescharacteristic
344C22#57811yescharacteristic
345C4—22no—
346C4—22no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4C2×C4×D411yescharacteristic
358C4#51811yescharacteristic
359C4#54611yescharacteristic
360C4#55111yescharacteristic
361C2—42no—
362C2—44no—
363C2#235811yescharacteristic
364C2C4×#13711yescharacteristic
365C2#295111yescharacteristic
366C2#327311yescharacteristic
367C2#327611yescharacteristic
368C2#331411yescharacteristic
369C2#331511yescharacteristic
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. C2×#440C2semidirect
4. #2360C2semidirect
6. #1969C2semidirect
7. #1935C2semidirect
8. #1928C2semidirect
9. #1894C2semidirect
10. #1878C2semidirect
11. #1877C2semidirect
12. #1862C2semidirect
13. #1852C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
25. #452C4semidirect
79. C2×Q16C2×C4semidirect
82. C2×SD16C2×C4semidirect
91. C22⋊Q8C2×C4semidirect
125. #103C2×C4semidirect
127. #102C2×C4semidirect
129. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4402
C2×#4172
#23602
#23392
#19692
#19352
#19282
#18942
#18782
#18772
#18622
#18522
C4×#1022
#17532