Finite Group Databasebeta

← #28126

Subgroups — #28126

Order 256.

564 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3#2324C211yesmaximal, characteristic, complemented
4#2309C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1798C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#482—24no—
24#447—24no—
25C2×#107C2211yescharacteristic, complemented
26C2×#106C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, 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
39#390—21no—
40#390—21no—
41#390—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic, complemented
47#389C2211yescharacteristic, complemented
48#388—21no—
49#386—21no—
50#385—21no—
51#380C2211yescharacteristic
52#379—21no—
53#379—21no—
54C4×C4⋊C4C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C2×D8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#121—22no—
86#121D414yes—
87C4×D4—41no—
88C4×D4—41no—
89#117—22no—
90#117—22no—
91#117—41no—
92#117—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
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—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—22no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—24no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—42no—
143C24—42no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C2×D4—24no—
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—21no—
165C2×D4—21no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C2×D4#4112yes—
176C2×D4#4112yes—
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—42no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D412yes—
195C22×C4#4112yes—
196C22×C4C2411yescharacteristic
197D8—44no—
198D8—44no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—81no—
223C4⋊C4C2×D412yes—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—81no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—48no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
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—84no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23#12112yes—
281C23C22×D411yescharacteristic
282D4—22no—
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—22no—
288D4—48no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—84no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
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—21no—
313C2×C4—21no—
314C2×C4—21no—
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—84no—
330C2×C4#12112yes—
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C8—41no—
338C8—41no—
339C8—41no—
340C8—41no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—816no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C2×#12111yescharacteristic
371C22#52511yescharacteristic
372C22#53411yescharacteristic
373C22#57611yescharacteristic
374C22#57611yescharacteristic
375C22#57611yescharacteristic
376C22#57711yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4#54611yescharacteristic
388C4#54611yescharacteristic
389C4#55111yescharacteristic
390C4#55111yescharacteristic
391C2—42no—
392C2—82no—
393C2—82no—
394C2—82no—
395C2#253411yescharacteristic
396C2#304111yescharacteristic
397C2#304511yescharacteristic
398C2#311811yescharacteristic
399C2#312211yescharacteristic
400C2#319811yescharacteristic
401C2#319911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
3. #2324C2semidirect
4. #2309C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2054C2semidirect
8. #2054C2semidirect
9. #1926C2semidirect
10. #1925C2semidirect
11. #1879C2semidirect
12. #1879C2semidirect
13. #1825C2semidirect
14. #1798C2semidirect
15. #1786C2semidirect
20. C2×C4×D4C22semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×C4⋊C8C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
46. #390C22semidirect
47. #389C22semidirect
54. C4×C4⋊C4C22semidirect
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×#4822
C2×#4472
#23242
#23092
#20672
#20652
#20542
#19262
#19252
#18792
#18252
#17982
#17862