Finite Group Databasebeta

← #30543

Subgroups — #30543

Order 256.

540 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic
5#2055C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1973C211yesmaximal, characteristic, complemented
13#1925C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C22×Q16C2211yescharacteristic, complemented
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25#489—24no—
26#450—24no—
27C2×#107C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic, complemented
42#400—21no—
43#394—21no—
44#394—21no—
45#393—21no—
46#391—21no—
47#391C2211yescharacteristic, complemented
48#391C2211yescharacteristic, complemented
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#388—21no—
52#380C2211yescharacteristic
53#379—21no—
54#379—21no—
55C4×C4⋊C4C2211yescharacteristic, complemented
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—41no—
82C2×SD16—41no—
83C2×SD16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—41no—
90#124—22no—
91#124—22no—
92#124—41no—
93#124—41no—
94C22⋊Q8—22no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—41no—
122C4⋊C8—41no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127#102—22no—
128#102—22no—
129#102—41no—
130#102—41no—
131#102—41no—
132#102D414yes—
133C22⋊C8—22no—
134C22⋊C8—41no—
135#97—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—81no—
152C24—81no—
153C24C2×D411yescharacteristic
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—81no—
169C2×D4—21no—
170C2×D4—21no—
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—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4C2×D412yes—
187C2×D4#4112yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
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×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2411yescharacteristic
206Q16—44no—
207SD16—44no—
208SD16—44no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8C2×D412yes—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4C2×D412yes—
232C22⋊C4—22no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—81no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—48no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23#13712yes—
278C23C22×D411yescharacteristic
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—84no—
288D4—22no—
289D4—22no—
290D4—22no—
291D4—22no—
292D4—48no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
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×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—816no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22#53511yescharacteristic
367C22D4211yescharacteristic
368C22#54611yescharacteristic
369C22C2×#13711yescharacteristic
370C22#57511yescharacteristic
371C22#57611yescharacteristic
372C22#57711yescharacteristic
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4#53511yescharacteristic
385C4#53811yescharacteristic
386C4#54611yescharacteristic
387C4H(3,𝔽4)11yescharacteristic
388C2—42no—
389C2—82no—
390C2—82no—
391C2#267011yescharacteristic
392C2#321511yescharacteristic
393C2#322411yescharacteristic
394C2#326611yescharacteristic
395C2#327311yescharacteristic
396C2#331111yescharacteristic
397C2#331411yescharacteristic
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×#489C2semidirect
3. #2492C2semidirect
5. #2055C2semidirect
6. #2036C2semidirect
7. #2023C2semidirect
8. #2015C2semidirect
9. #2015C2semidirect
10. #2001C2semidirect
11. #1983C2semidirect
12. #1973C2semidirect
13. #1925C2semidirect
14. #1924C2semidirect
17. C22×Q16C22semidirect
20. C2×C4⋊Q8C22semidirect
27. C2×#107C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
41. C2×#97C22semidirect
47. #391C22semidirect
48. #391C22semidirect
55. C4×C4⋊C4C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4502
#24922
#24352
#20552
#20362
#20232
#20152
#20012
#19832
#19732
#19252
#19242
#17862