Finite Group Databasebeta

← #31020

Subgroups — #31020

Order 256.

600 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2414C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1999C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic, complemented
23C2×#120—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic, complemented
26#482—24no—
27#447—24no—
28C2×#107C2211yescharacteristic, complemented
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×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic, complemented
42C2×C22⋊C8C2211yescharacteristic, complemented
43C2×C4×C8C2211yescharacteristic, complemented
44#394—21no—
45#394—21no—
46#392—21no—
47#392—21no—
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic, complemented
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×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—22no—
90#121—22no—
91#121—22no—
92#121—41no—
93#121—41no—
94#121—41no—
95#121—41no—
96#121—41no—
97#120—44no—
98C4×D4—41no—
99C4×D4—41no—
100#117—22no—
101#117—41no—
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⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126#107—22no—
127#106—22no—
128C4⋊C8—41no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102D414yescomplemented
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C4×C8—41no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—81no—
154C24—81no—
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157C2×D4—22no—
158C2×D4—22no—
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—41no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4C2×D412yes—
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—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2411yescharacteristic
219D8—44no—
220D8—44no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8C2×D412yes—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—81no—
244C4⋊C4C2×D412yes—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C42—21no—
261C42—21no—
262C42—42no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—48no—
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—84no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23#13512yes—
305C23C22×D411yescharacteristic
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—44no—
311D4—48no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—84no—
321D4—84no—
322D4—84no—
323D4—84no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
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×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×C4AGL(1,ℤ/8)12yes—
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C2×C4H(5,𝔽2)11yescharacteristic
365C8—22no—
366C8—41no—
367C8—41no—
368C8—41no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—42no—
389C22—816no—
390C22—81no—
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#53411yescharacteristic
406C22D4211yescharacteristic
407C22#54611yescharacteristic
408C22C2×#13511yescharacteristic
409C22C2×AGL(1,ℤ/8)11yescharacteristic
410C22#57611yescharacteristic
411C22#57611yescharacteristic
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4#53511yescharacteristic
422C4#53811yescharacteristic
423C4#54611yescharacteristic
424C4#56011yescharacteristic
425C2—42no—
426C2—81no—
427C2—82no—
428C2—82no—
429C2—161no—
430C2#267811yescharacteristic
431C2#318811yescharacteristic
432C2#319911yescharacteristic
433C2#327411yescharacteristic
434C2#328011yescharacteristic
435C2#329711yescharacteristic
436C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#447C2semidirect
3. #2414C2semidirect
4. #2408C2semidirect
5. #2076C2semidirect
6. #2065C2semidirect
7. #2054C2semidirect
8. #2034C2semidirect
9. #1999C2semidirect
10. #1999C2semidirect
11. #1970C2semidirect
12. #1954C2semidirect
13. #1934C2semidirect
14. #1875C2semidirect
15. #1786C2semidirect
22. C2×#121C22semidirect
25. C2×#117C22semidirect
28. C2×#107C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
41. C2×C22⋊C8C22semidirect
42. C2×C22⋊C8C22semidirect
43. C2×C4×C8C22semidirect
51. #390C22semidirect
52. #389C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
136. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4472
#24142
#24082
#20762
#20652
#20542
#20342
#19992
#19702
#19542
#19342
#18752
#17862