Finite Group Databasebeta

← #30405

Subgroups — #30405

Order 256.

503 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic, complemented
3#2536C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic
5#2092C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2074C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#467—24no—
27#452—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#398—21no—
42#394—21no—
43#394—21no—
44#392—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#388—21no—
48#388—21no—
49#386—21no—
50#385—21no—
51#382C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16D414yescomplemented
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—41no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95#121—22no—
96#121—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×D4—41no—
100C4×D4—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#107—22no—
125#106—41no—
126#103—22no—
127#103—22no—
128#103—22no—
129#103—41no—
130#103—41no—
131#102—22no—
132#102—41no—
133#102—41no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147#41—44no—
148#41—44no—
149#41—44no—
150#41—44no—
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×Q8—21no—
160C2×Q8—21no—
161C2×Q8C2×D412yes—
162C2×Q8C2×D412yes—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4#4112yes—
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—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205Q16—44no—
206SD16—44no—
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×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—44no—
219C2×C8C2×D412yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C42—42no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#13512yes—
270C23C22×D411yescharacteristic
271Q8—22no—
272Q8—22no—
273Q8—22no—
274Q8—22no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279D4—22no—
280D4—22no—
281D4—44no—
282D4—48no—
283D4—41no—
284D4—41no—
285D4—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—48no—
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—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4C22×D411yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#53411yescharacteristic
360C22D4211yescharacteristic
361C22#54611yescharacteristic
362C22C2×#13511yescharacteristic
363C22#57511yescharacteristic
364C22#57711yescharacteristic
365C22#57811yescharacteristic
366C4—41no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4#53411yescharacteristic
378C4#53811yescharacteristic
379C4#54611yescharacteristic
380C4#56011yescharacteristic
381C2—42no—
382C2—81no—
383C2—82no—
384C2#266811yescharacteristic
385C2#319111yescharacteristic
386C2#320111yescharacteristic
387C2#327111yescharacteristic
388C2#328511yescharacteristic
389C2#329611yescharacteristic
390C2#331511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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×#467C2semidirect
2. C2×#452C2semidirect
3. #2536C2semidirect
5. #2092C2semidirect
6. #2076C2semidirect
7. #2074C2semidirect
8. #2035C2semidirect
9. #2035C2semidirect
10. #2000C2semidirect
11. #1893C2semidirect
12. #1880C2semidirect
13. #1878C2semidirect
14. #1869C2semidirect
15. #1788C2semidirect
78. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4522
#25362
#24072
#20922
#20762
#20742
#20352
#20002
#18932
#18802
#18782
#18692
#17882