Finite Group Databasebeta

← #22868

Subgroups — #22868

Order 256.

665 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1936C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic, complemented
24C2×#117C2211yescharacteristic
25C22×C22⋊C4—21no—
26#447C418yescomplemented
27#442—24no—
28C2×#107C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30#418—24no—
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic, complemented
41C2×C4×C8C2211yescharacteristic, complemented
42#390—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
47#388—21no—
48#386—21no—
49#386—21no—
50#386—21no—
51#380C2211yescharacteristic
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62C25—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2×C414yescomplemented
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—41no—
81C2×D8—41no—
82C2×D8C2×C414yes—
83C2×D8C2×C414yescomplemented
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—41no—
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91#121—22no—
92#121C2×C414yescomplemented
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99#117—22no—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
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—44no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#107—41no—
127#106—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—41no—
133#102C2×C414yescomplemented
134#102C2×C414yescomplemented
135#102D414yescomplemented
136C22⋊C8C2×C414yescomplemented
137#97—22no—
138C4×C8—41no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—21no—
147C24—24no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—81no—
153C24—81no—
154C24C2×D411yescharacteristic
155C2×D4—28no—
156C2×D4—24no—
157C2×D4—24no—
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—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4C22×C412yes—
179C2×D4C22×C412yes—
180C2×D4C22×C412yes—
181C2×D4C2×D412yes—
182C2×D4#4112yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C22×C412yes—
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4#4112yes—
208C22×C4C2411yescharacteristic
209D8—28no—
210D8—44no—
211M16—44no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8C22×C412yes—
222C2×C8C22×C412yes—
223C2×C8C2×D412yes—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—81no—
233C4⋊C4C22×C412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—22no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C42—81no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—48no—
267C23—48no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23C4×D412yes—
290C23C22×D411yescharacteristic
291D4—22no—
292D4—22no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—84no—
304D4C4×D414yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4C4×D412yes—
341C2×C4AGL(1,ℤ/8)12yes—
342C2×C4C23×C411yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C2×#4111yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C8—22no—
349C8—22no—
350C8—22no—
351C8—41no—
352C22—22no—
353C22—24no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—816no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22C2×C4×D411yescharacteristic
382C22D4211yescharacteristic
383C22#54611yescharacteristic
384C22C2×AGL(1,ℤ/8)11yescharacteristic
385C22#57511yescharacteristic
386C22#57611yescharacteristic
387C22#57611yescharacteristic
388C4—22no—
389C4—22no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4C2×C4×D411yescharacteristic
399C4#51811yescharacteristic
400C4#54611yescharacteristic
401C4#55111yescharacteristic
402C2—42no—
403C2—44no—
404C2—82no—
405C2—82no—
406C2#235811yescharacteristic
407C2C4×AGL(1,ℤ/8)11yescharacteristic
408C2#294811yescharacteristic
409C2#327311yescharacteristic
410C2#327411yescharacteristic
411C2#331811yescharacteristic
412C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C2×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
2. C2×#442C2semidirect
3. C2×#418C2semidirect
4. #2363C2semidirect
5. D4×C22⋊C4C2semidirect
6. #1970C2semidirect
7. #1936C2semidirect
8. #1926C2semidirect
9. #1895C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1862C2semidirect
13. #1854C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
23. C2×C4×D4C22semidirect
26. #447C4semidirect
28. C2×#107C22semidirect
29. C2×#106C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
40. C2×#97C22semidirect
41. C2×C4×C8C22semidirect
45. #390C22semidirect
46. #389C22semidirect
53. C4×C4⋊C4C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #336C22semidirect
61. #336C22semidirect
70. C22×D4C2×C4semidirect
83. C2×D8C2×C4semidirect
92. #121C2×C4semidirect
133. #102C2×C4semidirect
134. #102C2×C4semidirect
135. #102D4semidirect
136. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C2×#4422
C2×#4182
#23632
D4×C22⋊C42
#19702
#19362
#19262
#18952
#18792
#18622
#18542
C4×#1022
#17532