Finite Group Databasebeta

← #25082

Subgroups — #25082

Order 256.

541 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×#467C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic
4C22×#103C211yesmaximal, characteristic
5#2434C211yesmaximal, characteristic
6#2360C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#2043C211yesmaximal, characteristic, complemented
10#2016C211yesmaximal, characteristic, complemented
11#1981C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic
15#1850C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#467—24no—
28#462—24no—
29#451—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45#395—21no—
46#395—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#389—21no—
51#389C2211yescharacteristic
52#387—21no—
53#385—21no—
54#385—21no—
55#385—21no—
56C4×C4⋊C4C2211yescharacteristic
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—41no—
84C2×Q16—41no—
85C2×Q16—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8D414yes—
101C4×Q8—22no—
102C4×Q8—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#107—22no—
127C4⋊C8—22no—
128C4⋊C8—41no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#103D414yes—
139C22⋊C8—22no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24C2×D411yescharacteristic
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—28no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—42no—
179C2×Q8C2×D412yes—
180C2×Q8C2×D412yes—
181C2×Q8#4112yes—
182C22×C4—21no—
183C22×C4—21no—
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—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—81no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D412yes—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208Q16—44no—
209Q16—44no—
210Q16—44no—
211C2×C8—21no—
212C2×C8—21no—
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—44no—
223C2×C8C2×D412yes—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—81no—
239C4⋊C4C2×D412yes—
240C4⋊C4C2×D412yes—
241C22⋊C4—24no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4Q1614yescomplemented
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—42no—
250C42—81no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#12112yes—
256C23C2×Q1612yes—
257C23#13712yes—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—48no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275Q8—41no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—48no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—84no—
318C2×C4#12112yes—
319C2×C4C2×Q1612yes—
320C2×C4C2×Q1612yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22#46214yescomplemented
338C22C2×#12111yescharacteristic
339C22D4211yescharacteristic
340C22#54611yescharacteristic
341C22C22×Q1611yescharacteristic
342C22C2×#13711yescharacteristic
343C22#57711yescharacteristic
344C22#57811yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4#52511yescharacteristic
359C4#53811yescharacteristic
360C4D4211yescharacteristic
361C4#54611yescharacteristic
362C2—24no—
363C2#242411yescharacteristic
364C2C2×#46211yescharacteristic
365C2#304611yescharacteristic
366C2#326611yescharacteristic
367C2D4×Q1611yescharacteristic
368C2#329911yescharacteristic
369C2#331211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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×#462C2semidirect
7. #2068C2semidirect
8. #2056C2semidirect
9. #2043C2semidirect
10. #2016C2semidirect
11. #1981C2semidirect
12. #1928C2semidirect
13. #1878C2semidirect
244. C22⋊C4Q16semidirect
337. C22#462semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4622
C2×#4512
C22×#1032
#24342
#23602
#20682
#20562
#20432
#20162
#19812
#19282
#18782
#18722
#18502