Finite Group Databasebeta

← #24757

Subgroups — #24757

Order 256.

493 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic, complemented
4C2×#423C211yesmaximal, characteristic, complemented
5#2850C211yesmaximal, characteristic
6C4×C22⋊Q8C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2056C211yesmaximal, characteristic, complemented
10#2055C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic
14#1894C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#470—24no—
27#468—24no—
28#464—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31#423—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#388—21no—
50#387—21no—
51#386—21no—
52#386—21no—
53#385—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2211yescharacteristic
58C4×C22⋊C4—21no—
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C2×M16—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#124—22no—
87#124—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8D414yes—
92#121—22no—
93#121—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
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×C42—22no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42D411yescharacteristic
118C2×C42D411yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8D414yescomplemented
127#103—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#102—22no—
133#102—22no—
134#102—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8C2×D412yes—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4C2×D411yescharacteristic
190C22×C4#4112yes—
191C22×C4C2411yescharacteristic
192Q16—44no—
193SD16—44no—
194M16—44no—
195M16—44no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D412yes—
207C2×C8C2×D412yes—
208C4⋊C4—24no—
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—42no—
220C4⋊C4—42no—
221C4⋊C4C2×D412yes—
222C22⋊C4—24no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—42no—
235C42—42no—
236C42C2×D412yes—
237C42#4112yes—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#12112yes—
252C23C22×D411yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—44no—
284C2×C4—44no—
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#12112yes—
305C2×C4#12112yes—
306C2×C4#12112yes—
307C2×C4#13712yes—
308C2×C4C22×D411yescharacteristic
309C2×C4C22×D411yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C42−1+411yescharacteristic
314C8—22no—
315C8—22no—
316C8—41no—
317C8—41no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#12111yescharacteristic
333C22C2×#12111yescharacteristic
334C22#53711yescharacteristic
335C22C2×#13711yescharacteristic
336C22#57511yescharacteristic
337C22#57511yescharacteristic
338C22#57511yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4D4211yescharacteristic
352C4#54711yescharacteristic
353C4#54811yescharacteristic
354C4#55311yescharacteristic
355C2—42no—
356C2—161no—
357C2#241611yescharacteristic
358C2#303711yescharacteristic
359C2#304011yescharacteristic
360C2#305411yescharacteristic
361C2#305611yescharacteristic
362C2#325211yescharacteristic
363C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#470C2semidirect
3. C2×#464C2semidirect
4. C2×#423C2semidirect
6. C4×C22⋊Q8C2semidirect
7. #2068C2semidirect
8. #2065C2semidirect
9. #2056C2semidirect
10. #2055C2semidirect
11. #1983C2semidirect
12. #1949C2semidirect
14. #1894C2semidirect
15. #1805C2semidirect
126. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4682
C2×#4642
C2×#4232
#28502
C4×C22⋊Q82
#20682
#20652
#20562
#20552
#19832
#19492
#18952
#18942
#18052