Finite Group Databasebeta

← #24136

Subgroups — #24136

Order 256.

619 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic, complemented
4#2537C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1992C211yesmaximal, characteristic, complemented
10#1955C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17#553—24no—
18#553—24no—
19C2×#126—21no—
20C2×#125C2211yescharacteristic
21C2×#123—21no—
22C2×#121—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#478—24no—
35#442—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×#97C2211yescharacteristic
51#394—21no—
52#394—21no—
53#384C2211yescharacteristic
54#380—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#126—28no—
83#125—22no—
84#125D414yescomplemented
85#123—28no—
86#121—28no—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95#121—44no—
96C4×D4—28no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4D414yescomplemented
103#117—22no—
104#117—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#106—22no—
135C4⋊C8—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—41no—
141#102—41no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146#97—22no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
153C2×D4—24no—
154C2×D4—22no—
155C2×D4—24no—
156C2×D4—22no—
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—21no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4C2×D412yes—
179C2×D4#4112yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
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—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2411yescharacteristic
212D8—28no—
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×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8#4112yes—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—81no—
240C4⋊C4C2×D412yes—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—24no—
255C42C2×D412yes—
256C42C2×D412yes—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23AGL(1,ℤ/8)12yes—
285C23C22×D411yescharacteristic
286D4—22no—
287D4—22no—
288D4—22no—
289D4—22no—
290D4—44no—
291D4—48no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—84no—
297D4—84no—
298D4—84no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
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—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#12012yes—
339C2×C4#12012yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C42−1+411yescharacteristic
346C8—22no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—41no—
369C22—42no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C2×#12011yescharacteristic
379C22#54611yescharacteristic
380C22#54711yescharacteristic
381C22C2×AGL(1,ℤ/8)11yescharacteristic
382C22#57511yescharacteristic
383C22#57611yescharacteristic
384C22#57711yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4#52511yescharacteristic
396C4#53711yescharacteristic
397C4#54811yescharacteristic
398C4#55011yescharacteristic
399C2—42no—
400C2—81no—
401C2—82no—
402C2—82no—
403C2#240711yescharacteristic
404C2#300611yescharacteristic
405C2#301211yescharacteristic
406C2#330711yescharacteristic
407C2#331811yescharacteristic
408C2#334511yescharacteristic
409C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#553C2semidirect
2. C2×#478C2semidirect
3. C2×#442C2semidirect
4. #2537C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1992C2semidirect
10. #1955C2semidirect
11. #1875C2semidirect
12. #1874C2semidirect
13. #1854C2semidirect
14. #1853C2semidirect
15. #1802C2semidirect
84. #125D4semidirect
102. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4782
C2×#4422
#25372
#20342
#20332
#19992
#19922
#19552
#18752
#18742
#18542
#18532
#18022