Finite Group Databasebeta

← #24090

Subgroups — #24090

Order 256.

561 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#427C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17#553—24no—
18#553—24no—
19C2×#126—21no—
20C2×#125C2211yescharacteristic
21C2×#123—21no—
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#482—24no—
32#427—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#397—21no—
47#394—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4D411yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#126—28no—
79#125—22no—
80#125—22no—
81#123—28no—
82#121—28no—
83#121—22no—
84#121—22no—
85#121—22no—
86#121D414yes—
87C4×D4—28no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92#117—22no—
93#117—22no—
94#117—22no—
95#117D414yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#107—22no—
127#106—22no—
128#106—22no—
129#106—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24C2×D411yescharacteristic
144C24C2×D411yescharacteristic
145C2×D4—24no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—24no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4C2×D412yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2411yescharacteristic
190C2×C8—28no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8#4112yes—
202C2×C8#4112yes—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4C2×D412yes—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—22no—
227C22⋊C4—24no—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C42—24no—
234C42—21no—
235C42—21no—
236C42—42no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#12012yes—
257C23#13512yes—
258C23C22×D411yescharacteristic
259D4—22no—
260D4—22no—
261D4—44no—
262D4—48no—
263D4—41no—
264D4—41no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4#12012yes—
309C2×C4C22×D411yescharacteristic
310C2×C4C22×D411yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C42−1+411yescharacteristic
314C2×C42−1+411yescharacteristic
315C8—22no—
316C8—22no—
317C8—41no—
318C8—41no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C2×#12011yescharacteristic
341C22#54711yescharacteristic
342C22#54711yescharacteristic
343C22C2×#13511yescharacteristic
344C22#57511yescharacteristic
345C22#57611yescharacteristic
346C22#57711yescharacteristic
347C4—22no—
348C4—41no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4#52511yescharacteristic
359C4#53611yescharacteristic
360C4#54711yescharacteristic
361C4#54711yescharacteristic
362C2—42no—
363C2—42no—
364C2—82no—
365C2#240611yescharacteristic
366C2#300211yescharacteristic
367C2#301211yescharacteristic
368C2#333711yescharacteristic
369C2#333911yescharacteristic
370C2#334711yescharacteristic
371C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#553C2
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
2. C2×#482C2semidirect
3. C2×#427C2semidirect
4. #2485C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1893C2semidirect
10. #1869C2semidirect
11. #1864C2semidirect
12. #1862C2semidirect
13. #1853C2semidirect
14. #1852C2semidirect
15. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4822
C2×#4272
#24852
#20592
#20342
#20332
#18932
#18692
#18642
#18622
#18532
#18522
#18252