Finite Group Databasebeta

← #22038

Subgroups — #22038

Order 256.

578 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic, complemented
3C4×#120C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1954C211yesmaximal, characteristic, complemented
6#1924C211yesmaximal, characteristic, complemented
7#1923C211yesmaximal, characteristic, complemented
8#1923C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1870C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1851C211yesmaximal, characteristic, complemented
14#1779C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic, complemented
20C2×#120C411yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#436—24no—
25C2×C4⋊C8C2211yescharacteristic, complemented
26C2×C4⋊C8C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28#407—24no—
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic, complemented
41C2×C22⋊C8C2211yescharacteristic, complemented
42C2×#97C2211yescharacteristic, complemented
43C2×C4×C8C2211yescharacteristic, complemented
44C2×C4×C8C2211yescharacteristic, complemented
45#386C411yescharacteristic
46#386C411yescharacteristic
47#386C411yescharacteristic
48#385C411yescharacteristic
49#385C411yescharacteristic
50#385C411yescharacteristic
51#379C411yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53C4×C22⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#336C2211yescharacteristic, complemented
56#336C2211yescharacteristic, complemented
57#336C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25C2×C411yescharacteristic
62C22×D4C2×C411yescharacteristic
63C22×D4C2×C411yescharacteristic
64C22×D4C2×C411yescharacteristic
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C2×C411yescharacteristic
72C23×C4C2×C411yescharacteristic
73C23×C4C2×C411yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16D414yescomplemented
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95#120—24no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C4⋊C4C2×C411yescharacteristic
104C2×C4⋊C4C2×C411yescharacteristic
105C2×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C22⋊C4C2×C411yescharacteristic
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C22⋊C4C2×C411yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C4⋊C8—22no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136#97—41no—
137C4×C8—22no—
138C4×C8—41no—
139#95C2×C411yescharacteristic
140#95C2×C411yescharacteristic
141#95C2×C411yescharacteristic
142#95C2×C411yescharacteristic
143#95C2×C411yescharacteristic
144#95C2×C411yescharacteristic
145C24—22no—
146C24—22no—
147C24—22no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24C22×C411yescharacteristic
153C24C22×C411yescharacteristic
154C24C22×C411yescharacteristic
155C24C22×C411yescharacteristic
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4C22×C411yescharacteristic
201C22×C4C22×C411yescharacteristic
202C22×C4C22×C411yescharacteristic
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C22×C411yescharacteristic
206C22×C4C22×C411yescharacteristic
207C22×C4C22×C411yescharacteristic
208C22×C4C22×C411yescharacteristic
209C22×C4C22×C411yescharacteristic
210C22×C4C2×D412yes—
211C22×C4#4112yes—
212C22×C4C2411yescharacteristic
213M16—44no—
214M16—44no—
215M16—44no—
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—21no—
227C2×C8—21no—
228C2×C8—21no—
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×C8C2×D412yes—
239C2×C8C2×D412yes—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4M1614yescomplemented
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23C4×D412yes—
294C23C2×M1612yes—
295C23#13112yes—
296C23C23×C411yescharacteristic
297D4—44no—
298D4—44no—
299D4—44no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
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×C4C4×D412yes—
347C2×C4C2×M1612yes—
348C2×C4C2×M1612yes—
349C2×C4C22×D411yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C2×C4H(5,𝔽2)11yescharacteristic
354C2×C4H(5,𝔽2)11yescharacteristic
355C8—22no—
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—416no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22C2×C4×D411yescharacteristic
393C22#51811yescharacteristic
394C22#51811yescharacteristic
395C22C22×M1611yescharacteristic
396C22C2×#13111yescharacteristic
397C22#56811yescharacteristic
398C22#56811yescharacteristic
399C4—22no—
400C4—22no—
401C4—22no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4#53411yescharacteristic
410C4#53511yescharacteristic
411C4#53811yescharacteristic
412C4#53811yescharacteristic
413C2—42no—
414C2—42no—
415C2—42no—
416C2—81no—
417C2#232111yescharacteristic
418C2#293311yescharacteristic
419C2D4×M1611yescharacteristic
420C2#297411yescharacteristic
421C2#297511yescharacteristic
422C2#297611yescharacteristic
423C2#297811yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#120C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#436C2semidirect
2. C2×#407C2semidirect
4. #1955C2semidirect
5. #1954C2semidirect
6. #1924C2semidirect
7. #1923C2semidirect
8. #1923C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1870C2semidirect
12. #1870C2semidirect
13. #1851C2semidirect
14. #1779C2semidirect
15. #1753C2semidirect
19. C22×M16C22semidirect
25. C2×C4⋊C8C22semidirect
26. C2×C4⋊C8C22semidirect
27. C2×C4⋊C8C22semidirect
39. C2×C22⋊C8C22semidirect
40. C2×C22⋊C8C22semidirect
41. C2×C22⋊C8C22semidirect
42. C2×#97C22semidirect
43. C2×C4×C8C22semidirect
44. C2×C4×C8C22semidirect
55. #336C22semidirect
56. #336C22semidirect
57. #336C22semidirect
58. #336C22semidirect
59. #336C22semidirect
60. #336C22semidirect
86. C2×M16D4semidirect
254. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4072
C4×#1202
#19552
#19542
#19242
#19232
#18702
#18512
#17792
#17532