Finite Group Databasebeta

← #31000

Subgroups — #31000

Order 256.

627 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2433C211yesmaximal, characteristic, complemented
4#2420C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1961C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic, complemented
20C2×#127C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#120—21no—
25C2×#120—21no—
26C2×C4×D4C2211yescharacteristic, complemented
27#459—24no—
28#447—24no—
29C2×#106C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic, complemented
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic, complemented
44#397—21no—
45#394—21no—
46#392—21no—
47#392—21no—
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic, complemented
51#389C2211yescharacteristic, complemented
52#386—21no—
53#382C2211yescharacteristic, complemented
54#381—21no—
55#380C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4C2311yescharacteristic
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—41no—
85C2×D8—41no—
86C2×D8—41no—
87C2×D8—41no—
88C2×D8—41no—
89C2×D8D414yescomplemented
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94#127—22no—
95#127—22no—
96#127—41no—
97#127—41no—
98#121—22no—
99#121—41no—
100#121—41no—
101#121—41no—
102#120—44no—
103#120—44no—
104C4×D4—22no—
105C4×D4—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#106—41no—
126C4⋊C8—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102—41no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139#97—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95D411yescharacteristic
145#95D411yescharacteristic
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—81no—
156C24—81no—
157C24C2×D411yescharacteristic
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—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4C2×D412yes—
205C2×D4C2×D412yes—
206C2×D4#4112yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2411yescharacteristic
226D8—44no—
227D8—44no—
228D8—44no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8C2×D412yes—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—81no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—48no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23AGL(1,ℤ/8)12yes—
308C23C22×D411yescharacteristic
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—48no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—84no—
327D4—84no—
328D4—84no—
329D4—84no—
330D4—84no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—84no—
363C2×C4AGL(1,ℤ/8)12yes—
364C2×C4C22×D411yescharacteristic
365C2×C4C2×#4111yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C8—22no—
371C8—41no—
372C8—41no—
373C8—41no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—816no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#53511yescharacteristic
411C22D4211yescharacteristic
412C22#54611yescharacteristic
413C22C2×AGL(1,ℤ/8)11yescharacteristic
414C22C2×AGL(1,ℤ/8)11yescharacteristic
415C22#57611yescharacteristic
416C22#57611yescharacteristic
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4#53411yescharacteristic
427C4#53811yescharacteristic
428C4#54611yescharacteristic
429C4#56011yescharacteristic
430C2—42no—
431C2—82no—
432C2—82no—
433C2—82no—
434C2—161no—
435C2#267811yescharacteristic
436C2#321711yescharacteristic
437C2#322011yescharacteristic
438C2#326411yescharacteristic
439C2#326411yescharacteristic
440C2#331811yescharacteristic
441C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#459C2semidirect
2. C2×#447C2semidirect
3. #2433C2semidirect
4. #2420C2semidirect
5. #2073C2semidirect
6. #2065C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1972C2semidirect
12. #1961C2semidirect
13. #1924C2semidirect
14. #1879C2semidirect
15. #1786C2semidirect
19. C22×D8C22semidirect
26. C2×C4×D4C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
40. C2×#102C22semidirect
43. C2×#97C22semidirect
50. #390C22semidirect
51. #389C22semidirect
53. #382C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
89. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
C2×#4472
#24332
#24202
#20732
#20652
#20412
#20342
#20112
#19992
#19722
#19612
#19242
#18792
#17862