Finite Group Databasebeta

← #27231

Subgroups — #27231

Order 256.

512 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1923C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124—21no—
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24#477—24no—
25#449—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#394—21no—
41#393—21no—
42#390—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#388—21no—
49#380—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—41no—
87#124—44no—
88#121—22no—
89#121—41no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—41no—
93C4×Q8—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
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⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—22no—
121#107—22no—
122C4⋊C8—22no—
123#103—22no—
124#103—22no—
125#103D414yes—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130#102—41no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142#41—44no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—81no—
152C2×Q8C2×D412yes—
153C2×Q8#4112yes—
154C2×Q8#4112yes—
155C2×D4—24no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—81no—
172C2×D4#4112yes—
173C22×C4—21no—
174C22×C4—24no—
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—42no—
187C22×C4—42no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2411yescharacteristic
194SD16—44no—
195D8—44no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8C2×D412yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4C2×D412yes—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C23—21no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C22×D411yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—22no—
266Q8—84no—
267Q8#12414yescomplemented
268D4—22no—
269D4—22no—
270D4—48no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—84no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—48no—
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—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#13512yes—
315C2×C4AGL(1,ℤ/8)12yes—
316C2×C4C22×D411yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C8—22no—
323C8—41no—
324C8—41no—
325C8—41no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#52511yescharacteristic
348C22D4211yescharacteristic
349C22#54611yescharacteristic
350C22C2×#13511yescharacteristic
351C22C2×AGL(1,ℤ/8)11yescharacteristic
352C22#57611yescharacteristic
353C22#57811yescharacteristic
354C4—24no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4C2×#12411yescharacteristic
366C4#53811yescharacteristic
367C4#54611yescharacteristic
368C4#55111yescharacteristic
369C2—81no—
370C2—82no—
371C2—161no—
372C2#247211yescharacteristic
373C2#310611yescharacteristic
374C2#312411yescharacteristic
375C2#327411yescharacteristic
376C2#328511yescharacteristic
377C2#330511yescharacteristic
378C2#330811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C22×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
1. C2×#477C2semidirect
3. #2491C2semidirect
4. #2341C2semidirect
5. #2076C2semidirect
6. #2054C2semidirect
7. #2033C2semidirect
8. #2023C2semidirect
9. #1970C2semidirect
10. #1925C2semidirect
11. #1923C2semidirect
12. #1880C2semidirect
13. #1880C2semidirect
14. #1788C2semidirect
15. C4×#103C2semidirect
267. Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4492
#24912
#23412
#20762
#20542
#20332
#20232
#19702
#19252
#19232
#18802
#17882
C4×#1032