Finite Group Databasebeta

← #25112

Subgroups — #25112

Order 256.

659 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×C8×D4C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2417C211yesmaximal, characteristic, complemented
6#2408C211yesmaximal, characteristic
7#2065C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic, complemented
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic, complemented
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#465—24no—
29#450—24no—
30C8×D4—24no—
31C2×#106C2211yescharacteristic, complemented
32C2×#106C2211yescharacteristic, complemented
33C2×C4⋊C8C2211yescharacteristic, complemented
34C2×#103C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#102—24no—
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic, complemented
45C2×C4×C8C2211yescharacteristic, complemented
46#398—21no—
47#396—21no—
48#394—21no—
49#392—21no—
50#391C2211yescharacteristic, complemented
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic, complemented
54#386—21no—
55#380—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic, complemented
58#379—21no—
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4D411yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×SD16—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—41no—
100#121—22no—
101#121—22no—
102#121—41no—
103#121—41no—
104#120—44no—
105C4×D4—22no—
106C4×D4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42C2311yescharacteristic
132#106—22no—
133#106—41no—
134C4⋊C8—22no—
135#103—22no—
136#103—41no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102D414yescomplemented
143C22⋊C8—22no—
144C22⋊C8D414yescomplemented
145C4×C8—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24—21no—
153C24—42no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—81no—
159C24—81no—
160C24C2×D411yescharacteristic
161C24C2×D411yescharacteristic
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—28no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4SD1614yescomplemented
194C2×D4C2×D412yes—
195C2×D4#4112yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—81no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D412yes—
219C22×C4C2×D411yescharacteristic
220C22×C4#4112yes—
221C22×C4C2411yescharacteristic
222SD16—44no—
223SD16—44no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8C2×D412yes—
236C2×C8C2×D412yes—
237C2×C8C2×D412yes—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4C2×D412yes—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—48no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—84no—
293C23—84no—
294C23—84no—
295C23—84no—
296C23—84no—
297C23—84no—
298C23#12112yes—
299C23C2×SD1612yes—
300C23C2×SD1612yes—
301C23#13512yes—
302C23C22×D411yescharacteristic
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307D4—28no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—48no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—84no—
356C2×C4#12112yes—
357C2×C4C2×SD1612yes—
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4C2×#4111yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C8—22no—
365C8—22no—
366C8—22no—
367C8—41no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—816no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#46514yescomplemented
397C22C2×#12111yescharacteristic
398C22D4211yescharacteristic
399C22#54611yescharacteristic
400C22C22×SD1611yescharacteristic
401C22C2×#13511yescharacteristic
402C22#57611yescharacteristic
403C22#57711yescharacteristic
404C4—22no—
405C4—22no—
406C4—41no—
407C4—42no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4#52511yescharacteristic
415C4#53811yescharacteristic
416C4D4211yescharacteristic
417C4#54611yescharacteristic
418C2—24no—
419C2—42no—
420C2—82no—
421C2—82no—
422C2#242411yescharacteristic
423C2C2×#46511yescharacteristic
424C2#306211yescharacteristic
425C2#328011yescharacteristic
426C2D4×SD1611yescharacteristic
427C2#329511yescharacteristic
428C2#329611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#465C2semidirect
2. C2×#450C2semidirect
3. C2×C8×D4C2semidirect
4. C22×#102C2semidirect
5. #2417C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
9. #2034C2semidirect
10. #2000C2semidirect
11. #1968C2semidirect
12. #1934C2semidirect
13. #1893C2semidirect
14. #1892C2semidirect
15. #1853C2semidirect
20. C23×C8C22semidirect
22. C2×C22⋊Q8C22semidirect
31. C2×#106C22semidirect
32. C2×#106C22semidirect
33. C2×C4⋊C8C22semidirect
34. C2×#103C22semidirect
35. C2×#103C22semidirect
44. C2×C22⋊C8C22semidirect
45. C2×C4×C8C22semidirect
50. #391C22semidirect
53. #389C22semidirect
57. #380C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
142. #102D4semidirect
144. C22⋊C8D4semidirect
193. C2×D4SD16semidirect
396. C22#465semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4502
C2×C8×D42
C22×#1022
#24172
#24082
#20652
#20552
#20342
#20002
#19682
#19342
#18932
#18922
#18532