Finite Group Databasebeta

← #24915

Subgroups — #24915

Order 256.

518 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3#2420C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic, complemented
5#2097C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1935C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#449—24no—
24#442—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#397—21no—
40#394—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#386—21no—
46#386—21no—
47#386—21no—
48#382C2211yescharacteristic
49#381—21no—
50#381—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C22⋊C4C2211yescharacteristic
54C2×#95—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×D8—22no—
81C2×D8—22no—
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⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—41no—
114#106—22no—
115C4⋊C8—41no—
116#103—22no—
117#103—22no—
118#103—41no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—41no—
124C22⋊C8—22no—
125C22⋊C8—41no—
126#97—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141#41—44no—
142#41—44no—
143#41—44no—
144#41—44no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8#4112yes—
151C2×D4—24no—
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—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—81no—
165C2×D4C2×D412yes—
166C2×D4C2×D412yes—
167C2×D4#4112yes—
168C2×D4#4112yes—
169C22×C4—24no—
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×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199SD16—44no—
200D8—28no—
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—
211C2×C8—21no—
212C2×C8C2×D412yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—81no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C22×D411yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—41no—
263Q8—41no—
264D4—22no—
265D4—22no—
266D4—22no—
267D4—22no—
268D4—22no—
269D4—48no—
270D4—41no—
271D4—41no—
272D4—84no—
273D4#12314yes—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—48no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C8—22no—
322C8—41no—
323C8—41no—
324C8—41no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#52511yescharacteristic
347C22D4211yescharacteristic
348C22#54611yescharacteristic
349C22#57511yescharacteristic
350C22#57511yescharacteristic
351C22#57611yescharacteristic
352C22#57711yescharacteristic
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4C2×#12311yescharacteristic
365C4#53811yescharacteristic
366C4#54611yescharacteristic
367C4#55111yescharacteristic
368C2—44no—
369C2—81no—
370C2—82no—
371C2#241811yescharacteristic
372C2#311111yescharacteristic
373C2#311911yescharacteristic
374C2#326711yescharacteristic
375C2#327311yescharacteristic
376C2#330711yescharacteristic
377C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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. #2339C2semidirect
5. #2097C2semidirect
6. #2077C2semidirect
7. #2055C2semidirect
8. #2033C2semidirect
9. #1935C2semidirect
10. #1924C2semidirect
11. #1879C2semidirect
12. #1877C2semidirect
13. #1862C2semidirect
14. #1804C2semidirect
15. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4422
#24202
#23392
#20972
#20772
#20552
#20332
#19352
#19242
#18792
#18772
#18622
#18042
#17882