Finite Group Databasebeta

← #25156

Subgroups — #25156

Order 256.

634 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#461C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic
3C2×#450C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2429C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic
7#2065C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1864C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic, complemented
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic, complemented
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C22⋊C4—21no—
28#461—24no—
29#450—24no—
30#450—24no—
31C4×SD16—24no—
32C2×#106C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic, complemented
46#398—21no—
47#396—21no—
48#391C2211yescharacteristic, complemented
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic, complemented
53#388—21no—
54#386—21no—
55#386—21no—
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—41no—
88C2×SD16—41no—
89C2×SD16D414yes—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8—41no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C4×Q8—22no—
101C4×Q8—41no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—41no—
105C4×D4—41no—
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⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#106—22no—
132#106—41no—
133C4⋊C8—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102D414yes—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C4×C8—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—24no—
153C24—21no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—81no—
159C24—81no—
160C24C2×D411yescharacteristic
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×Q8C2×D412yes—
173C2×D4—24no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4C2×D412yes—
189C2×D4C2×D412yes—
190C2×D4#4112yes—
191C2×D4#4112yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
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—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2411yescharacteristic
216SD16—28no—
217SD16—44no—
218SD16—44no—
219SD16—44no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8C2×D412yes—
231C2×C8C2×D412yes—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4C2×D412yes—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4SD1614yescomplemented
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—42no—
262C23—21no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—48no—
273C23—48no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23C2×SD1612yes—
290C23C22×D411yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—41no—
299D4—22no—
300D4—22no—
301D4—22no—
302D4—22no—
303D4—22no—
304D4—48no—
305D4—41no—
306D4—41no—
307D4#12114yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
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—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—84no—
344C2×C4—84no—
345C2×C4C2×SD1612yes—
346C2×C4C2×SD1612yes—
347C2×C4C22×D411yescharacteristic
348C2×C4C22×D411yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C8—22no—
354C8—22no—
355C8—41no—
356C8—41no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—816no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#52511yescharacteristic
380C22D4211yescharacteristic
381C22D4211yescharacteristic
382C22C22×SD1611yescharacteristic
383C22#57511yescharacteristic
384C22#57711yescharacteristic
385C22#57711yescharacteristic
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4C2×#12111yescharacteristic
398C4#53811yescharacteristic
399C4#54611yescharacteristic
400C4#54611yescharacteristic
401C2—42no—
402C2—44no—
403C2—82no—
404C2#242411yescharacteristic
405C2#310111yescharacteristic
406C2#312311yescharacteristic
407C2#326711yescharacteristic
408C2#327511yescharacteristic
409C2D4×SD1611yescharacteristic
410C2D4×SD1611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
1. C2×#461C2semidirect
4. C2×C4×SD16C2semidirect
5. #2429C2semidirect
7. #2065C2semidirect
8. #2035C2semidirect
9. #2035C2semidirect
10. #2013C2semidirect
11. #1968C2semidirect
12. #1923C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. #1864C2semidirect
21. C2×C4⋊Q8C22semidirect
24. C2×C4×Q8C22semidirect
32. C2×#106C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
45. C2×C4×C8C22semidirect
48. #391C22semidirect
52. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
256. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4502
C2×C4×SD162
#24292
D4×C22⋊C42
#20652
#20352
#20132
#19682
#19232
#18802
#18772
#18642