Finite Group Databasebeta

← #22062

Subgroups — #22062

Order 256.

450 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3002C211yesmaximal, characteristic
2#2930C211yesmaximal, characteristic, complemented
3#2879C211yesmaximal, characteristic, complemented
4#1975C211yesmaximal, characteristic, complemented
5#1974C211yesmaximal, characteristic, complemented
6#1938C211yesmaximal, characteristic, complemented
7#1937C211yesmaximal, characteristic, complemented
8#1929C211yesmaximal, characteristic, complemented
9#1929C211yesmaximal, characteristic, complemented
10#1876C211yesmaximal, characteristic, complemented
11#1876C211yesmaximal, characteristic, complemented
12#1871C211yesmaximal, characteristic, complemented
13#1855C211yesmaximal, characteristic, complemented
14#1789C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#135C2211yescharacteristic
19C2×#131C2211yescharacteristic, complemented
20C22×M16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#525C2211yescharacteristic
23#452C411yescharacteristic, complemented
24#452C411yescharacteristic, complemented
25#451C411yescharacteristic, complemented
26#450C411yescharacteristic, complemented
27#449C411yescharacteristic, complemented
28#449C411yescharacteristic, complemented
29#448C411yescharacteristic, complemented
30#447C411yescharacteristic, complemented
31#436—21no—
32#436—21no—
33#435—21no—
34#435—21no—
35C2×#108C2211yescharacteristic, complemented
36C2×#108C2211yescharacteristic, complemented
37#424C2211yescharacteristic, complemented
38C2×#104C2211yescharacteristic
39C2×#104C2211yescharacteristic
40C2×#104C2211yescharacteristic
41C2×#104C2211yescharacteristic
42C2×#104C2211yescharacteristic, complemented
43C2×#104C2211yescharacteristic, complemented
44#416C2211yescharacteristic
45#416C2211yescharacteristic
46#408—21no—
47#408—21no—
48#408C2211yescharacteristic
49#408C2211yescharacteristic
50#408C2211yescharacteristic
51#408C2211yescharacteristic
52#408C2211yescharacteristic, complemented
53#407—21no—
54#407—21no—
55#407C2211yescharacteristic
56C2×C22⋊C8C2211yescharacteristic
57#405C2211yescharacteristic, complemented
58C2×#97C2211yescharacteristic, complemented
59C2×C4×C8C2211yescharacteristic, complemented
60C4×C22⋊C4C2211yescharacteristic
61#341C2211yescharacteristic, complemented
62#341C2211yescharacteristic, complemented
63#339C2211yescharacteristic, complemented
64#339C2211yescharacteristic, complemented
65#336C2211yescharacteristic, complemented
66#336C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—41no—
70C2×#41—41no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41C2×C411yescharacteristic
74C2×#41C2×C411yescharacteristic
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C2×#41C2311yescharacteristic
78C22×Q8C2×C411yescharacteristic
79C22×D4—21no—
80C22×D4C2×C411yescharacteristic
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83#135—41no—
84#135—41no—
85#135—41no—
86#135—41no—
87C2×Q16C2×C411yescharacteristic
88C2×Q16C2×C411yescharacteristic
89C2×SD16C2×C411yescharacteristic
90C2×SD16C2×C411yescharacteristic
91C2×SD16C2×C411yescharacteristic
92C2×SD16C2×C411yescharacteristic
93C2×D8C2×C411yescharacteristic
94C2×D8C2×C411yescharacteristic
95#131—41no—
96#131—41no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16D411yescharacteristic
104C2×M16D411yescharacteristic
105C2×M16D411yescharacteristic
106C2×M16D411yescharacteristic
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117#123—21no—
118C22⋊Q8C2×C411yescharacteristic
119C22⋊Q8C2×C411yescharacteristic
120#121C2×C411yescharacteristic
121#121C2×C411yescharacteristic
122#120—21no—
123C4×D4—21no—
124C4×D4—21no—
125#117C2311yescharacteristic
126C2×C22⋊C4—21no—
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#108—41no—
130#108—41no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133#104—41no—
134#104—41no—
135#104—41no—
136#104—41no—
137#104—41no—
138#104—41no—
139#103C2×C411yescharacteristic
140#103C2×C411yescharacteristic
141#103C2×C411yescharacteristic
142#103C2×C411yescharacteristic
143#102C2×C411yescharacteristic
144#102C2×C411yescharacteristic
145#102C2×C411yescharacteristic
146#102C2×C411yescharacteristic
147C22⋊C8—21no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8C2×C411yescharacteristic
160C22⋊C8C2×C411yescharacteristic
161C22⋊C8C2×C411yescharacteristic
162C22⋊C8C2×C411yescharacteristic
163#97—21no—
164#97—41no—
165C4×C8—21no—
166C4×C8—41no—
167#95—21no—
168C24—21no—
169C24#4111yescharacteristic
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—41no—
183#41—41no—
184#41—41no—
185#41—41no—
186#41—42no—
187#41—42no—
188#41—42no—
189#41—42no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8C22×C411yescharacteristic
194C2×Q8C22×C411yescharacteristic
195C2×Q8C22×C411yescharacteristic
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4C22×C411yescharacteristic
209C2×D4C22×C411yescharacteristic
210C2×D4C22×C411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C22×C411yescharacteristic
226C22×C4C22×C411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4#4111yescharacteristic
230C22×C4C2411yescharacteristic
231Q16—41no—
232Q16—41no—
233SD16—41no—
234SD16—41no—
235SD16—41no—
236SD16—41no—
237D8—41no—
238D8—41no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—21no—
243M16—21no—
244M16—21no—
245M16—21no—
246M16—21no—
247M16—41no—
248M16—41no—
249M16—41no—
250M16—41no—
251M16—42no—
252M16—42no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8C22×C411yescharacteristic
268C2×C8C22×C411yescharacteristic
269C2×C8C22×C411yescharacteristic
270C2×C8C22×C411yescharacteristic
271C2×C8C2×D411yescharacteristic
272C2×C8C2×D411yescharacteristic
273C2×C8C2×D411yescharacteristic
274C2×C8C2×D411yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4C22×C411yescharacteristic
278C4⋊C4C22×C411yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—42no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23C4×D411yescharacteristic
301C23C4×D411yescharacteristic
302C23C2×#4111yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—42no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
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×C4C4×D411yescharacteristic
358C2×C4C4×D411yescharacteristic
359C2×C4C23×C411yescharacteristic
360C2×C4C22×D411yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C2×C4H(5,𝔽2)11yescharacteristic
365C8—21no—
366C8—21no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—41no—
374C8—41no—
375C8—41no—
376C8—41no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—81no—
395C22—81no—
396C22—84no—
397C22C2×C4×D411yescharacteristic
398C22#53811yescharacteristic
399C22#53811yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4#51811yescharacteristic
410C4#51811yescharacteristic
411C4#53411yescharacteristic
412C4#53511yescharacteristic
413C2—21no—
414C2—21no—
415C2—41no—
416C2—41no—
417C2—81no—
418C2—81no—
419C2—81no—
420C2#232111yescharacteristic
421C2#295611yescharacteristic
422C2#295711yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#3002C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2930C2semidirect
3. #2879C2semidirect
4. #1975C2semidirect
5. #1974C2semidirect
6. #1938C2semidirect
7. #1937C2semidirect
8. #1929C2semidirect
9. #1929C2semidirect
10. #1876C2semidirect
11. #1876C2semidirect
12. #1871C2semidirect
13. #1855C2semidirect
14. #1789C2semidirect
15. #1753C2semidirect
19. C2×#131C22semidirect
23. #452C4semidirect
24. #452C4semidirect
25. #451C4semidirect
26. #450C4semidirect
27. #449C4semidirect
28. #449C4semidirect
29. #448C4semidirect
30. #447C4semidirect
35. C2×#108C22semidirect
36. C2×#108C22semidirect
37. #424C22semidirect
42. C2×#104C22semidirect
43. C2×#104C22semidirect
52. #408C22semidirect
57. #405C22semidirect
58. C2×#97C22semidirect
59. C2×C4×C8C22semidirect
61. #341C22semidirect
62. #341C22semidirect
63. #339C22semidirect
64. #339C22semidirect
65. #336C22semidirect
66. #336C22semidirect

Maximal subgroups

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

subgroupindex
#30022
#29302
#28792
#19752
#19742
#19382
#19372
#19292
#18762
#18712
#18552
#17892
#17532