Finite Group Databasebeta

← #30404

Subgroups — #30404

Order 256.

588 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3#2538C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic, complemented
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic, complemented
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#120—21no—
25C2×C4×Q8C2211yescharacteristic, complemented
26C2×C4×D4C2211yescharacteristic
27#465—24no—
28#450—24no—
29C2×#106C2211yescharacteristic, complemented
30C2×#106C2211yescharacteristic, 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
42#396—21no—
43#396—21no—
44#394—21no—
45#392—21no—
46#390—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389—21no—
50#384C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic, complemented
54#379—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, 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—41no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16D414yescomplemented
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94#121—22no—
95#121—22no—
96#121—41no—
97#121—41no—
98#120—44no—
99C4×Q8—41no—
100C4×Q8—41no—
101C4×D4—41no—
102C4×D4—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129#106—22no—
130#106—41no—
131#103—22no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—41no—
137#102—41no—
138#102—41no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—41no—
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×Q8C2×D412yes—
162C2×D4—21no—
163C2×D4—21no—
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—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—44no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4SD1614yescomplemented
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×C8—44no—
227C2×C8C2×D412yes—
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—21no—
237C4⋊C4—21no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C42—21no—
258C42—21no—
259C42—42no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—48no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23C2×SD1612yes—
294C23C2×SD1612yes—
295C23C22×D411yescharacteristic
296Q8—22no—
297Q8—22no—
298Q8—41no—
299Q8—41no—
300D4—22no—
301D4—22no—
302D4—22no—
303D4—22no—
304D4—44no—
305D4—48no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—84no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
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×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4C2×SD1612yes—
350C2×C4C22×D411yescharacteristic
351C2×C4C2×#4111yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C2×C4H(5,𝔽2)11yescharacteristic
354C2×C4H(5,𝔽2)11yescharacteristic
355C2×C4H(5,𝔽2)11yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—816no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#53411yescharacteristic
390C22D4211yescharacteristic
391C22#54611yescharacteristic
392C22C22×SD1611yescharacteristic
393C22#57511yescharacteristic
394C22#57611yescharacteristic
395C22#57711yescharacteristic
396C4—22no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4#53411yescharacteristic
407C4#53811yescharacteristic
408C4#54611yescharacteristic
409C4#56011yescharacteristic
410C2—42no—
411C2—42no—
412C2—82no—
413C2—82no—
414C2#266811yescharacteristic
415C2#318711yescharacteristic
416C2#319811yescharacteristic
417C2#326811yescharacteristic
418C2D4×SD1611yescharacteristic
419C2#329511yescharacteristic
420C2#331111yescharacteristic
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×#465C2semidirect
2. C2×#450C2semidirect
3. #2538C2semidirect
5. #2094C2semidirect
6. #2077C2semidirect
7. #2071C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2000C2semidirect
11. #1893C2semidirect
12. #1880C2semidirect
13. #1880C2semidirect
14. #1869C2semidirect
15. #1786C2semidirect
19. C23×C8C22semidirect
21. C2×C22⋊Q8C22semidirect
25. C2×C4×Q8C22semidirect
29. C2×#106C22semidirect
30. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
41. C2×C22⋊C8C22semidirect
50. #384C22semidirect
53. #380C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
82. C2×SD16D4semidirect
187. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4502
#25382
#24082
#20942
#20772
#20712
#20352
#20342
#20002
#18932
#18802
#18692
#17862