Finite Group Databasebeta

← #30727

Subgroups — #30727

Order 256.

471 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#470C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic
4#2417C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2015C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1927C211yesmaximal, characteristic, complemented
12#1919C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1870C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#481—24no—
29#470—24no—
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43#399—21no—
44#398—21no—
45#396—21no—
46#392—21no—
47#391C2211yescharacteristic
48#390—21no—
49#390—21no—
50#389C2211yescharacteristic
51#386—21no—
52#380—21no—
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4D411yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×M16—22no—
77C2×M16—41no—
78C2×M16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#124—41no—
84#124—41no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93#121—22no—
94#121—22no—
95#121—41no—
96#121—41no—
97C4×D4—41no—
98C4×D4—41no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126#106—22no—
127#106D414yescomplemented
128#103—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—41no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147C2×Q8—21no—
148C2×Q8—21no—
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×D4—22no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2411yescharacteristic
201Q16—44no—
202SD16—44no—
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×C8—21no—
214C2×C8—21no—
215C2×C8C2×D412yes—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C4⋊C4C2×D412yes—
232C4⋊C4#4112yes—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—81no—
244C42—21no—
245C42—21no—
246C42—81no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#13712yes—
268C23C22×D411yescharacteristic
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277D4—44no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—84no—
283D4—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
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—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4C22×D411yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C8—22no—
333C8—41no—
334C8—41no—
335C8—41no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#53411yescharacteristic
358C22D4211yescharacteristic
359C22#54611yescharacteristic
360C22C2×#13711yescharacteristic
361C22#57511yescharacteristic
362C22#57711yescharacteristic
363C22#57711yescharacteristic
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4#53511yescharacteristic
376C4#54011yescharacteristic
377C4#55011yescharacteristic
378C4#56011yescharacteristic
379C2—42no—
380C2—81no—
381C2—161no—
382C2#267311yescharacteristic
383C2#319611yescharacteristic
384C2#320411yescharacteristic
385C2#326611yescharacteristic
386C2#327511yescharacteristic
387C2#331611yescharacteristic
388C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#481C2semidirect
2. C2×#470C2semidirect
4. #2417C2semidirect
5. #2042C2semidirect
6. #2035C2semidirect
7. #2015C2semidirect
8. #2001C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1927C2semidirect
12. #1919C2semidirect
13. #1895C2semidirect
14. #1872C2semidirect
15. #1870C2semidirect
127. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4702
#25402
#24172
#20422
#20352
#20152
#20012
#20002
#19272
#19192
#18952
#18722
#18702