Finite Group Databasebeta

← #25176

Subgroups — #25176

Order 256.

631 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×#440C211yesmaximal, characteristic, complemented
5#2429C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic
7#2067C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, 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—
31#440—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#107C2211yescharacteristic, 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×#97C2211yescharacteristic, 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#107—22no—
132#107—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—
144#97—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—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—81no—
256C22⋊C4—81no—
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—
289C23#13712yes—
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—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
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—41no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4AGL(1,ℤ/8)12yes—
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—42no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—816no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#52511yescharacteristic
380C22D4211yescharacteristic
381C22D4211yescharacteristic
382C22C2×AGL(1,ℤ/8)11yescharacteristic
383C22C2×#13711yescharacteristic
384C22#57711yescharacteristic
385C22#57711yescharacteristic
386C4—22no—
387C4—41no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—42no—
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#311011yescharacteristic
406C2#312311yescharacteristic
407C2#326511yescharacteristic
408C2#326611yescharacteristic
409C2#326611yescharacteristic
410C2#327211yescharacteristic
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×#440C2semidirect
5. #2429C2semidirect
7. #2067C2semidirect
8. #2036C2semidirect
9. #2036C2semidirect
10. #2013C2semidirect
11. #1973C2semidirect
12. #1924C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. #1862C2semidirect
21. C2×C4⋊Q8C22semidirect
24. C2×C4×Q8C22semidirect
32. C2×#107C22semidirect
33. C2×#107C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
45. C2×#97C22semidirect
48. #391C22semidirect
52. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect

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×#4402
#24292
D4×C22⋊C42
#20672
#20362
#20132
#19732
#19242
#18802
#18772
#18622