Finite Group Databasebeta

← #24769

Subgroups — #24769

Order 256.

473 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic, complemented
4C2×#424C211yesmaximal, characteristic, complemented
5#2620C211yesmaximal, characteristic
6#2323C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2056C211yesmaximal, characteristic, complemented
10#2055C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1948C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic
14#1892C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#470—24no—
27#465—24no—
28#464—24no—
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31#424—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#396—21no—
45#396—21no—
46#394—21no—
47#394—21no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic
53#388—21no—
54#388—21no—
55#387—21no—
56#384C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—41no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#124—22no—
85#124—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8D414yes—
90#121—22no—
91#121—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#106—22no—
124#106—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8D414yescomplemented
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—41no—
134#102—22no—
135#102—22no—
136#102—41no—
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—81no—
166C2×D4—81no—
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×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186Q16—44no—
187SD16—44no—
188M16—44no—
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×C8—21no—
198C2×C8—21no—
199C2×C8—44no—
200C2×C8C2×D412yes—
201C2×C8C2×D412yes—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4C2×D412yes—
220C22⋊C4—24no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C42—21no—
231C42—21no—
232C42—42no—
233C42—81no—
234C42C2×D412yes—
235C42#4112yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#12112yes—
250C23C22×D411yescharacteristic
251Q8—22no—
252Q8—22no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—84no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
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#12112yes—
302C2×C4#12112yes—
303C2×C4#12112yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—22no—
311C8—22no—
312C8—41no—
313C8—41no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C2×#12111yescharacteristic
329C22C2×#12111yescharacteristic
330C22#53711yescharacteristic
331C22#57711yescharacteristic
332C22#57711yescharacteristic
333C22#57711yescharacteristic
334C22#57811yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4D4211yescharacteristic
348C4#54711yescharacteristic
349C4#54811yescharacteristic
350C4#55311yescharacteristic
351C2—42no—
352C2—161no—
353C2#241611yescharacteristic
354C2#304311yescharacteristic
355C2#304611yescharacteristic
356C2#306011yescharacteristic
357C2#306311yescharacteristic
358C2#325811yescharacteristic
359C2#326311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ 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×#424C2semidirect
6. #2323C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #2056C2semidirect
10. #2055C2semidirect
11. #1983C2semidirect
12. #1948C2semidirect
14. #1892C2semidirect
15. #1803C2semidirect
128. 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×#4652
C2×#4642
C2×#4242
#26202
#23232
#20662
#20652
#20562
#20552
#19832
#19482
#18952
#18922
#18032