Finite Group Databasebeta

← #31028

Subgroups — #31028

Order 256.

575 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3#2415C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic, complemented
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic, complemented
26#482—24no—
27#450—24no—
28C2×#107C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic, complemented
42C2×C22⋊C8C2211yescharacteristic, complemented
43C2×C4×C8C2211yescharacteristic, complemented
44#397—21no—
45#394—21no—
46#394—21no—
47#392—21no—
48#391C2211yescharacteristic, complemented
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic, complemented
53#388—21no—
54#380C2211yescharacteristic
55#380C2211yescharacteristic
56#379—21no—
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4D411yescharacteristic
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22⋊Q8—22no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95#120—44no—
96C4×D4—41no—
97C4×D4—41no—
98#117—22no—
99#117—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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⋊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—22no—
126C4⋊C8—41no—
127#103—41no—
128#103—41no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102D414yescomplemented
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C4×C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—81no—
153C24—81no—
154C24C2×D411yescharacteristic
155C24C2×D411yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4C2×D412yes—
189C2×D4#4112yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
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—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2411yescharacteristic
213SD16—44no—
214SD16—44no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8C2×D412yes—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C2×D412yes—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—81no—
253C22⋊C4—81no—
254C42—21no—
255C42—21no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—48no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23—84no—
291C23#13512yes—
292C23C22×D411yescharacteristic
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297D4—22no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—44no—
302D4—48no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—84no—
308D4—84no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
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—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4AGL(1,ℤ/8)12yes—
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C8—22no—
354C8—41no—
355C8—41no—
356C8—41no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—42no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—42no—
377C22—816no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#53411yescharacteristic
387C22D4211yescharacteristic
388C22#54611yescharacteristic
389C22C2×#13511yescharacteristic
390C22C2×AGL(1,ℤ/8)11yescharacteristic
391C22#57711yescharacteristic
392C22#57711yescharacteristic
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4#53511yescharacteristic
404C4#53811yescharacteristic
405C4#54611yescharacteristic
406C4#56011yescharacteristic
407C2—42no—
408C2—81no—
409C2—82no—
410C2—82no—
411C2#267811yescharacteristic
412C2#318811yescharacteristic
413C2#320011yescharacteristic
414C2#327211yescharacteristic
415C2#328211yescharacteristic
416C2#329611yescharacteristic
417C2#332411yescharacteristic
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×#482C2semidirect
2. C2×#450C2semidirect
3. #2415C2semidirect
5. #2077C2semidirect
6. #2067C2semidirect
7. #2055C2semidirect
8. #2036C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1968C2semidirect
12. #1954C2semidirect
13. #1934C2semidirect
14. #1875C2semidirect
15. #1786C2semidirect
20. C2×C22⋊Q8C22semidirect
25. C2×#117C22semidirect
28. C2×#107C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
41. C2×C22⋊C8C22semidirect
42. C2×C22⋊C8C22semidirect
43. C2×C4×C8C22semidirect
48. #391C22semidirect
52. #389C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
134. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4502
#24152
#24082
#20772
#20672
#20552
#20362
#20002
#19682
#19542
#19342
#18752
#17862