Finite Group Databasebeta

← #33006

Subgroups — #33006

Order 256.

704 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic
4#2854C211yesmaximal, characteristic, complemented
5#1880C211yesmaximal, characteristic, complemented
6#1880C211yesmaximal, characteristic, complemented
7#1879C211yesmaximal, characteristic, complemented
8#1879C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic, complemented
11#1863C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1817C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#123—21no—
20C2×#121C2211yescharacteristic
21C4×#41—24no—
22C4×#41—24no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30C2×#117C2211yescharacteristic
31C22×C42—21no—
32C22×C42C2211yescharacteristic
33#482—24no—
34#449—24no—
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#388—21no—
53#386—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41D414yes—
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4D411yescharacteristic
78C23×C4D411yescharacteristic
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#123—44no—
91#121—22no—
92#121—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×D4—28no—
98C4×D4—28no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103#117—28no—
104#117—22no—
105#117D414yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—28no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42D411yescharacteristic
131C2×C42D411yescharacteristic
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152#41—28no—
153#41—28no—
154#41—28no—
155#41—28no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8C2×D412yes—
160C2×Q8#4112yes—
161C2×Q8#4112yes—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4C2×D412yes—
178C2×D4#4112yes—
179C2×D4#4112yes—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2411yescharacteristic
213SD16—44no—
214D8—44no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4C2×D412yes—
240C4⋊C4C2×D412yes—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—24no—
251C42—24no—
252C42—22no—
253C42—24no—
254C42—22no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#12012yes—
280C23C22×D411yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—22no—
284Q8#12314yescomplemented
285D4—216no—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294D4#12314yescomplemented
295C2×C4—216no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4#12012yes—
335C2×C4#13512yes—
336C2×C4AGL(1,ℤ/8)12yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4C2×#4111yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C22—22no—
348C22—24no—
349C22—22no—
350C22—22no—
351C22—41no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#12011yescharacteristic
369C22#52511yescharacteristic
370C22#52511yescharacteristic
371C22C2×#13511yescharacteristic
372C22C2×AGL(1,ℤ/8)11yescharacteristic
373C22#57511yescharacteristic
374C22#57511yescharacteristic
375C4—22no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—42no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4C2×#12311yescharacteristic
387C4C2×#12311yescharacteristic
388C4#52511yescharacteristic
389C4#52511yescharacteristic
390C2—42no—
391C2—44no—
392C2—161no—
393C2#284911yescharacteristic
394C2#300211yescharacteristic
395C2#300611yescharacteristic
396C2#310211yescharacteristic
397C2#310611yescharacteristic
398C2#311111yescharacteristic
399C2#311511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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×C4×#41C2semidirect
2. C2×#482C2semidirect
4. #2854C2semidirect
5. #1880C2semidirect
6. #1880C2semidirect
7. #1879C2semidirect
8. #1879C2semidirect
9. #1865C2semidirect
10. #1864C2semidirect
11. #1863C2semidirect
12. #1862C2semidirect
13. #1825C2semidirect
14. #1817C2semidirect
15. #1788C2semidirect
284. Q8#123semidirect
294. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4822
C2×#4492
#28542
#18802
#18792
#18652
#18642
#18632
#18622
#18252
#18172
#17882