Finite Group Databasebeta

← #28139

Subgroups — #28139

Order 256.

446 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3092C211yesmaximal, characteristic, complemented
2#3005C211yesmaximal, characteristic
3#2930C211yesmaximal, characteristic, complemented
4C2×#417C211yesmaximal, characteristic, complemented
5#2064C211yesmaximal, characteristic, complemented
6#2063C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic, complemented
8#2051C211yesmaximal, characteristic, complemented
9#1931C211yesmaximal, characteristic, complemented
10#1930C211yesmaximal, characteristic, complemented
11#1908C211yesmaximal, characteristic, complemented
12#1907C211yesmaximal, characteristic, complemented
13#1821C211yesmaximal, characteristic, complemented
14#1811C211yesmaximal, characteristic, complemented
15#1785C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C22×M16C2211yescharacteristic
21#525C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#484—21no—
24#483—21no—
25#481—21no—
26#480—21no—
27#452—21no—
28#452—21no—
29#449—21no—
30#449—21no—
31#436—21no—
32#436—21no—
33#435—21no—
34#435—21no—
35C2×#108C2211yescharacteristic
36C2×#108C2211yescharacteristic
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#424C2211yescharacteristic
40#419C2211yescharacteristic
41#418C2211yescharacteristic
42#417—21no—
43#417—21no—
44#417—21no—
45#417—21no—
46#417C2211yescharacteristic
47#417C2211yescharacteristic
48C2×#103C2211yescharacteristic
49C2×#103C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×#101C2211yescharacteristic
53C2×#101C2211yescharacteristic
54C2×#100C2211yescharacteristic
55C2×#100C2211yescharacteristic
56#408C2211yescharacteristic
57#407C2211yescharacteristic
58#407C2211yescharacteristic
59#407C2211yescharacteristic
60#405C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#343C2211yescharacteristic
65#341C2211yescharacteristic
66#341C2211yescharacteristic
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—21no—
70C2×#41—21no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41C2311yescharacteristic
76C22×Q8C2311yescharacteristic
77C22×D4—21no—
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2311yescharacteristic
82#137—22no—
83#137—41no—
84AGL(1,ℤ/8)—22no—
85AGL(1,ℤ/8)—41no—
86C2×Q16—21no—
87C2×SD16—21no—
88C2×SD16—21no—
89C2×D8—21no—
90#131—41no—
91#131—41no—
92C2×M16—21no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16D411yescharacteristic
96C2×M16D411yescharacteristic
97C2×M16D411yescharacteristic
98C2×M16D411yescharacteristic
99C2×M16C2311yescharacteristic
100C2×M16C2311yescharacteristic
101C2×M16C2311yescharacteristic
102C2×M16C2311yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2311yescharacteristic
105C22×C8—21no—
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108#123—21no—
109C22⋊Q8D411yescharacteristic
110C22⋊Q8D411yescharacteristic
111#121D411yescharacteristic
112#121D411yescharacteristic
113#120—21no—
114C4×D4—21no—
115C4×D4—21no—
116#117C2311yescharacteristic
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124#108—41no—
125#108—41no—
126#107—21no—
127#107—21no—
128#106—21no—
129#106—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#103—21no—
138#103—21no—
139#103—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#102—21no—
148#101—22no—
149#101—41no—
150#100—22no—
151#100—41no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160#97—21no—
161C4×C8—21no—
162C24—21no—
163C24#4111yescharacteristic
164#41—24no—
165#41—24no—
166#41—21no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—42no—
171#41—42no—
172#41—42no—
173#41—42no—
174#41—42no—
175#41—42no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8C2×D411yescharacteristic
182C2×Q8#4112yes—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C2×D411yescharacteristic
196C2×D4#4112yes—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—82no—
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4#4111yescharacteristic
217C22×C4C2411yescharacteristic
218Q16—42no—
219SD16—42no—
220SD16—42no—
221D8—42no—
222M16—21no—
223M16—21no—
224M16—21no—
225M16—21no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—21no—
230M16—21no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16—41no—
235M16—41no—
236M16—41no—
237M16—41no—
238M16—42no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—42no—
250C2×C8C2×D411yescharacteristic
251C2×C8C2×D411yescharacteristic
252C2×C8C2×D411yescharacteristic
253C2×C8C2×D411yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4C2×D411yescharacteristic
263C4⋊C4C2×D411yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C42—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—81no—
279C23—81no—
280C23#12111yescharacteristic
281C23#12111yescharacteristic
282C23C2×#4111yescharacteristic
283Q8—22no—
284Q8—21no—
285Q8—22no—
286Q8—21no—
287Q8—41no—
288Q8—41no—
289D4—22no—
290D4—21no—
291D4—22no—
292D4—21no—
293D4—44no—
294D4—44no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—81no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4#12111yescharacteristic
328C2×C4#12111yescharacteristic
329C2×C4C22×D411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C8—21no—
336C8—21no—
337C8—21no—
338C8—21no—
339C8—41no—
340C8—41no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—41no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22C2×#12111yescharacteristic
364C22#55111yescharacteristic
365C22#55111yescharacteristic
366C4—21no—
367C4—21no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4#52511yescharacteristic
375C4#53411yescharacteristic
376C4#54611yescharacteristic
377C4#54611yescharacteristic
378C2—21no—
379C2—21no—
380C2—41no—
381C2—41no—
382C2—81no—
383C2—82no—
384C2#253411yescharacteristic
385C2#306811yescharacteristic
386C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3092C2semidirect
3. #2930C2semidirect
4. C2×#417C2semidirect
5. #2064C2semidirect
6. #2063C2semidirect
7. #2052C2semidirect
8. #2051C2semidirect
9. #1931C2semidirect
10. #1930C2semidirect
11. #1908C2semidirect
12. #1907C2semidirect
13. #1821C2semidirect
14. #1811C2semidirect
15. #1785C2semidirect

Maximal subgroups

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

subgroupindex
#30922
#30052
#29302
C2×#4172
#20642
#20632
#20522
#20512
#19312
#19302
#19082
#19072
#18212
#18112
#17852