Finite Group Databasebeta

← #30523

Subgroups — #30523

Order 256.

556 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3#2490C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1970C211yesmaximal, characteristic, complemented
13#1925C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic, complemented
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#125C2211yescharacteristic, complemented
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25#488—24no—
26#447—24no—
27C2×#107C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic, complemented
42#397—21no—
43#394—21no—
44#394—21no—
45#394—21no—
46#390—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic, complemented
50#390C2211yescharacteristic, complemented
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×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×SD16—41no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8—41no—
83C2×D8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88#125—22no—
89#125—41no—
90#124—22no—
91#124—22no—
92#124—41no—
93#124—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96#121—22no—
97#121—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⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#107—41no—
126C4⋊C8—41no—
127#103—22no—
128#103—22no—
129#103—41no—
130#102—22no—
131#102—22no—
132#102—41no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102D414yes—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C4×C8—22no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—21no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—81no—
153C24—81no—
154C24C2×D411yescharacteristic
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
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—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4C2×D412yes—
188C2×D4#4112yes—
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
206SD16—44no—
207D8—44no—
208D8—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—42no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23#13512yes—
289C23C22×D411yescharacteristic
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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×C4C22×D411yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C8—22no—
349C8—41no—
350C8—41no—
351C8—41no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—816no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22#53511yescharacteristic
382C22D4211yescharacteristic
383C22#54611yescharacteristic
384C22C2×#13511yescharacteristic
385C22#57511yescharacteristic
386C22#57611yescharacteristic
387C22#57611yescharacteristic
388C4—41no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4#53511yescharacteristic
399C4#53811yescharacteristic
400C4#54611yescharacteristic
401C4H(3,𝔽4)11yescharacteristic
402C2—42no—
403C2—82no—
404C2—82no—
405C2—161no—
406C2#267011yescharacteristic
407C2#321111yescharacteristic
408C2#322011yescharacteristic
409C2#326811yescharacteristic
410C2#328011yescharacteristic
411C2#329711yescharacteristic
412C2#332211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#488C2semidirect
3. #2490C2semidirect
4. #2435C2semidirect
5. #2054C2semidirect
6. #2036C2semidirect
7. #2025C2semidirect
8. #2012C2semidirect
9. #2012C2semidirect
10. #2000C2semidirect
11. #1983C2semidirect
12. #1970C2semidirect
13. #1925C2semidirect
14. #1923C2semidirect
15. #1786C2semidirect
17. C22×SD16C22semidirect
20. C2×#125C22semidirect
27. C2×#107C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
38. C2×#102C22semidirect
41. C2×C4×C8C22semidirect
49. #390C22semidirect
50. #390C22semidirect
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×#4882
C2×#4472
#24902
#24352
#20542
#20362
#20252
#20122
#20002
#19832
#19702
#19252
#19232
#17862