Finite Group Databasebeta

← #28110

Subgroups — #28110

Order 256.

576 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic, complemented
3#2485C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic, complemented
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#120—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#436—24no—
25C2×#107C2211yescharacteristic, complemented
26C2×#106C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28#418—24no—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic, complemented
39C2×C4×C8C2211yescharacteristic, complemented
40#397—21no—
41#394—21no—
42#394—21no—
43#392—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic, complemented
47#389C2211yescharacteristic, complemented
48#388—21no—
49#387—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic, complemented
53#380C2211yescharacteristic, complemented
54#379—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—22no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×M16—22no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16D414yescomplemented
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#121—22no—
87#121—22no—
88#121—41no—
89#121—41no—
90#120—44no—
91C4×D4—22no—
92C4×D4—22no—
93#117—22no—
94#117—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119#107—41no—
120#106—41no—
121C4⋊C8—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C4×C8—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4#4112yes—
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×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D412yes—
198C22×C4C2×D411yescharacteristic
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201M16—44no—
202M16—44no—
203C2×C8—21no—
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×C8C2×D412yes—
214C2×C8C2×D412yes—
215C4⋊C4—22no—
216C4⋊C4—21no—
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—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—42no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—48no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23#12112yes—
281C23C22×D411yescharacteristic
282D4—22no—
283D4—22no—
284D4—22no—
285D4—22no—
286D4—44no—
287D4—48no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—84no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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×C4—41no—
329C2×C4—41no—
330C2×C4#12112yes—
331C2×C4AGL(1,ℤ/8)12yes—
332C2×C4C22×D411yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C8—22no—
339C8—22no—
340C8—41no—
341C8—41no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—42no—
362C22—816no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C2×#12111yescharacteristic
372C22#54611yescharacteristic
373C22#54611yescharacteristic
374C22C2×AGL(1,ℤ/8)11yescharacteristic
375C22#57511yescharacteristic
376C22#57611yescharacteristic
377C22#57711yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4#52511yescharacteristic
389C4#53411yescharacteristic
390C4#55111yescharacteristic
391C4#55111yescharacteristic
392C2—42no—
393C2—81no—
394C2—82no—
395C2—82no—
396C2#253411yescharacteristic
397C2#305511yescharacteristic
398C2#306211yescharacteristic
399C2#332211yescharacteristic
400C2#332311yescharacteristic
401C2#332411yescharacteristic
402C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
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×#436C2semidirect
2. C2×#418C2semidirect
3. #2485C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1934C2semidirect
10. #1934C2semidirect
11. #1895C2semidirect
12. #1895C2semidirect
13. #1825C2semidirect
14. #1809C2semidirect
15. #1786C2semidirect
18. C22×M16C22semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×C4⋊C8C22semidirect
38. C2×C22⋊C8C22semidirect
39. C2×C4×C8C22semidirect
46. #389C22semidirect
47. #389C22semidirect
52. #380C22semidirect
53. #380C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
81. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4182
#24852
#24082
#20672
#20652
#20342
#20332
#19342
#18952
#18252
#18092
#17862