Finite Group Databasebeta

← #26245

Subgroups — #26245

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic
4#2354C211yesmaximal, characteristic, complemented
5#2035C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1955C211yesmaximal, characteristic, complemented
12#1948C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#485—24no—
25#482—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44#399—21no—
45#392—21no—
46#390—21no—
47#390—21no—
48#390—21no—
49#388—21no—
50#388—21no—
51#384C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#124—22no—
73#124—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—41no—
77C22⋊Q8—41no—
78#121—22no—
79#121—22no—
80#121—41no—
81#121—41no—
82C4×D4—22no—
83C4×D4—22no—
84#117—22no—
85#117—22no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116#107D414yescomplemented
117#106—22no—
118#106—22no—
119#106—22no—
120C4⋊C8—22no—
121#103—22no—
122#103—22no—
123#103—41no—
124#103—41no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133#97—22no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×D4—22no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
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—21no—
180C22×C4—21no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2411yescharacteristic
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8C2×D412yes—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4C2×D412yes—
213C4⋊C4C2×D412yes—
214C4⋊C4#4112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42—81no—
231C42#4112yes—
232C42#4112yes—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23C22×D411yescharacteristic
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259D4—44no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—84no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
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#12312yes—
306C2×C4#12312yes—
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C8—22no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C2×#12311yescharacteristic
339C22D4211yescharacteristic
340C22#54611yescharacteristic
341C22#57511yescharacteristic
342C22#57511yescharacteristic
343C22#57711yescharacteristic
344C22#57711yescharacteristic
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4#53211yescharacteristic
357C4#54011yescharacteristic
358C4#55011yescharacteristic
359C4#55911yescharacteristic
360C2—81no—
361C2—81no—
362C2—161no—
363C2#244511yescharacteristic
364C2#309311yescharacteristic
365C2#309911yescharacteristic
366C2#326711yescharacteristic
367C2#327111yescharacteristic
368C2#331111yescharacteristic
369C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
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×#485C2semidirect
2. C2×#482C2semidirect
4. #2354C2semidirect
5. #2035C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #1983C2semidirect
10. #1959C2semidirect
11. #1955C2semidirect
12. #1948C2semidirect
13. #1875C2semidirect
14. #1873C2semidirect
15. #1776C2semidirect
116. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4822
#25402
#23542
#20352
#20342
#19832
#19592
#19552
#19482
#18752
#18732
#17762