Finite Group Databasebeta

← #24909

Subgroups — #24909

Order 256.

599 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3#2420C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1936C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic, complemented
23C22×C22⋊C4—21no—
24#447—24no—
25#442—24no—
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic, complemented
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic, complemented
40#397—21no—
41#394—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic, complemented
45#389C2211yescharacteristic, complemented
46#386—21no—
47#386—21no—
48#386—21no—
49#382C2211yescharacteristic, complemented
50#381—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#379—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8D414yes—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#121—22no—
88#121—41no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
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—44no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#107—41no—
119#106—22no—
120C4⋊C8—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131#97—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—24no—
141C24—21no—
142C24—21no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24—81no—
149C24C2×D411yescharacteristic
150C2×D4—24no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4C2×D412yes—
174C2×D4C2×D412yes—
175C2×D4#4112yes—
176C2×D4#4112yes—
177C2×D4#4112yes—
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—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2411yescharacteristic
204D8—28no—
205D8—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8C2×D412yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—81no—
229C22⋊C4—22no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—48no—
259C23—48no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23—84no—
281C23AGL(1,ℤ/8)12yes—
282C23C22×D411yescharacteristic
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—22no—
290D4—48no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—84no—
296D4#12314yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—24no—
346C22—22no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—816no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#52511yescharacteristic
374C22D4211yescharacteristic
375C22#54611yescharacteristic
376C22C2×AGL(1,ℤ/8)11yescharacteristic
377C22#57511yescharacteristic
378C22#57611yescharacteristic
379C22#57611yescharacteristic
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4C2×#12311yescharacteristic
391C4#53811yescharacteristic
392C4#54611yescharacteristic
393C4#55111yescharacteristic
394C2—42no—
395C2—44no—
396C2—82no—
397C2—82no—
398C2#241811yescharacteristic
399C2#311411yescharacteristic
400C2#312211yescharacteristic
401C2#326411yescharacteristic
402C2#327311yescharacteristic
403C2#331811yescharacteristic
404C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×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. #2420C2semidirect
4. D4×C22⋊C4C2semidirect
5. #2092C2semidirect
6. #2077C2semidirect
7. #2054C2semidirect
8. #2033C2semidirect
9. #1936C2semidirect
10. #1924C2semidirect
11. #1879C2semidirect
12. #1879C2semidirect
13. #1862C2semidirect
14. #1804C2semidirect
15. #1786C2semidirect
22. C2×C4×D4C22semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×C4⋊C8C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
39. C2×#97C22semidirect
44. #390C22semidirect
45. #389C22semidirect
49. #382C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C2×#4422
#24202
D4×C22⋊C42
#20922
#20772
#20542
#20332
#19362
#19242
#18792
#18622
#18042
#17862