Finite Group Databasebeta

← #22588

Subgroups — #22588

Order 256.

444 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2924C211yesmaximal, characteristic, complemented
2#2896C211yesmaximal, characteristic
3#2880C211yesmaximal, characteristic
4#1938C211yesmaximal, characteristic, complemented
5#1937C211yesmaximal, characteristic, complemented
6#1929C211yesmaximal, characteristic, complemented
7#1929C211yesmaximal, characteristic, complemented
8#1866C211yesmaximal, characteristic, complemented
9#1866C211yesmaximal, characteristic, complemented
10#1826C211yesmaximal, characteristic, complemented
11#1811C211yesmaximal, characteristic, complemented
12#1789C211yesmaximal, characteristic
13#1765C211yesmaximal, characteristic, complemented
14#1764C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#131C2211yescharacteristic
18C2×#131C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#433—21no—
22#433—21no—
23#432—21no—
24#431—21no—
25C2×#108C2211yescharacteristic, complemented
26C2×#108C2211yescharacteristic, complemented
27#424C2211yescharacteristic, complemented
28#424C2211yescharacteristic, complemented
29C2×#104C2211yescharacteristic
30C2×#104C2211yescharacteristic
31C2×#104C2211yescharacteristic
32C2×#104C2211yescharacteristic
33#419C411yescharacteristic, complemented
34#419C411yescharacteristic, complemented
35#418C411yescharacteristic, complemented
36#418C411yescharacteristic, complemented
37#417C411yescharacteristic, complemented
38#417C411yescharacteristic, complemented
39#417C411yescharacteristic, complemented
40#417C411yescharacteristic, complemented
41#416C2211yescharacteristic
42#416C2211yescharacteristic
43#416C2211yescharacteristic
44#416C2211yescharacteristic
45#408—21no—
46#408—21no—
47#408C2211yescharacteristic
48#408C2211yescharacteristic
49#407—21no—
50#407C2211yescharacteristic
51#407C2211yescharacteristic
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2211yescharacteristic
54#405C2211yescharacteristic, complemented
55#405C2211yescharacteristic, complemented
56C2×#97C2211yescharacteristic, complemented
57C2×C4×C8C2211yescharacteristic, complemented
58C4×C22⋊C4C2211yescharacteristic
59#341C2211yescharacteristic, complemented
60#341C2211yescharacteristic, complemented
61#339C2211yescharacteristic, complemented
62#339C2211yescharacteristic, complemented
63#339C2211yescharacteristic, complemented
64#339C2211yescharacteristic, complemented
65#336C2211yescharacteristic, complemented
66#336C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41C2×C411yescharacteristic
74C2×#41C2×C411yescharacteristic
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C22×Q8C2×C411yescharacteristic
78C22×D4—21no—
79C22×D4C2×C411yescharacteristic
80C23×C4—21no—
81C23×C4C2311yescharacteristic
82#131—22no—
83#131—22no—
84#131—41no—
85#131—41no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16C2×C411yescharacteristic
91C2×M16C2×C411yescharacteristic
92C2×M16C2×C411yescharacteristic
93C2×M16C2×C411yescharacteristic
94C2×M16C2311yescharacteristic
95C2×M16C2311yescharacteristic
96C2×M16C2311yescharacteristic
97C2×M16C2311yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104#117—21no—
105#117—21no—
106#117C2×C411yescharacteristic
107#117C2×C411yescharacteristic
108#117C2311yescharacteristic
109#117C2311yescharacteristic
110C2×C4⋊C4C2×C411yescharacteristic
111C2×C4⋊C4C2×C411yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#108—41no—
118#108—41no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123#104—22no—
124#104—22no—
125#104—41no—
126#104—41no—
127#103C2×C411yescharacteristic
128#103C2×C411yescharacteristic
129#103C2×C411yescharacteristic
130#103C2×C411yescharacteristic
131#103C2×C411yescharacteristic
132#103C2×C411yescharacteristic
133#103C2×C411yescharacteristic
134#103C2×C411yescharacteristic
135#102C2×C411yescharacteristic
136#102C2×C411yescharacteristic
137#102C2×C411yescharacteristic
138#102C2×C411yescharacteristic
139#102C2×C411yescharacteristic
140#102C2×C411yescharacteristic
141#102C2×C411yescharacteristic
142#102C2×C411yescharacteristic
143C22⋊C8—21no—
144C22⋊C8—21no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8D411yescharacteristic
152C22⋊C8D411yescharacteristic
153C22⋊C8D411yescharacteristic
154C22⋊C8D411yescharacteristic
155#97—21no—
156#97—21no—
157#97—41no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—41no—
161#95—21no—
162C24—21no—
163C24#4111yescharacteristic
164#41—21no—
165#41—21no—
166#41—22no—
167#41—22no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—42no—
173#41—42no—
174#41—41no—
175#41—41no—
176#41—42no—
177#41—42no—
178#41#4112yes—
179#41#4112yes—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8C22×C411yescharacteristic
184C2×Q8C22×C411yescharacteristic
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4C22×C411yescharacteristic
195C2×D4C22×C411yescharacteristic
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×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4C22×C411yescharacteristic
212C22×C4C22×C411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4#4111yescharacteristic
216C22×C4C2411yescharacteristic
217M16—21no—
218M16—21no—
219M16—21no—
220M16—21no—
221M16—21no—
222M16—21no—
223M16—21no—
224M16—21no—
225M16—41no—
226M16—41no—
227M16—41no—
228M16—41no—
229M16—42no—
230M16—42no—
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×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C22×C411yescharacteristic
246C2×C8C22×C411yescharacteristic
247C2×C8C22×C411yescharacteristic
248C2×C8C22×C411yescharacteristic
249C2×C8C2×D411yescharacteristic
250C2×C8C2×D411yescharacteristic
251C2×C8C2×D411yescharacteristic
252C2×C8C2×D411yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4C22×C411yescharacteristic
256C4⋊C4C22×C411yescharacteristic
257C4⋊C4C22×C411yescharacteristic
258C4⋊C4C22×C411yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—81no—
281C23C4×D411yescharacteristic
282C23C4×D411yescharacteristic
283C23C2×#4111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—42no—
288Q8#11712yes—
289D4—22no—
290D4—22no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—42no—
295D4—42no—
296D4—42no—
297D4—41no—
298D4—41no—
299D4#11712yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4C4×D411yescharacteristic
332C2×C4C4×D411yescharacteristic
333C2×C4C23×C411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—84no—
368C22C2×C4×D411yescharacteristic
369C22#55111yescharacteristic
370C22#55111yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4C2×#11711yescharacteristic
382C4#51811yescharacteristic
383C4#54611yescharacteristic
384C4#54611yescharacteristic
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—42no—
390C2—81no—
391C2#234211yescharacteristic
392C2#295511yescharacteristic
393C2#295611yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2924C2semidirect
4. #1938C2semidirect
5. #1937C2semidirect
6. #1929C2semidirect
7. #1929C2semidirect
8. #1866C2semidirect
9. #1866C2semidirect
10. #1826C2semidirect
11. #1811C2semidirect
13. #1765C2semidirect
14. #1764C2semidirect
15. #1753C2semidirect
25. C2×#108C22semidirect
26. C2×#108C22semidirect
27. #424C22semidirect
28. #424C22semidirect
33. #419C4semidirect
34. #419C4semidirect
35. #418C4semidirect
36. #418C4semidirect
37. #417C4semidirect
38. #417C4semidirect
39. #417C4semidirect
40. #417C4semidirect
54. #405C22semidirect
55. #405C22semidirect
56. C2×#97C22semidirect
57. C2×C4×C8C22semidirect
59. #341C22semidirect
60. #341C22semidirect
61. #339C22semidirect
62. #339C22semidirect
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
#29242
#28962
#28802
#19382
#19372
#19292
#18662
#18262
#18112
#17892
#17652
#17642
#17532