Finite Group Databasebeta

← #30730

Subgroups — #30730

Order 256.

507 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic
4#2415C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2014C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1927C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic, complemented
15#1870C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#127—21no—
20C2×#127C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#482—24no—
30#468—24no—
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44#397—21no—
45#394—21no—
46#392—21no—
47#392—21no—
48#391C2211yescharacteristic
49#390—21no—
50#389—21no—
51#389C2211yescharacteristic
52#388—21no—
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×M16—22no—
80C2×M16—41no—
81C2×M16—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#127—41no—
87#127—41no—
88#127—44no—
89C22⋊Q8—22no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101C4×D4—41no—
102C4×D4—41no—
103#117—22no—
104#117—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127#107D414yescomplemented
128#106—22no—
129#103—22no—
130#103—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—41no—
135#102—41no—
136#102—41no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C24C2×D411yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
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×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211SD16—44no—
212SD16—44no—
213M16—44no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8C2×D412yes—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4C2×D412yes—
238C4⋊C4C2×D412yes—
239C4⋊C4#4112yes—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
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—
282C23C22×D411yescharacteristic
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287D4—44no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—84no—
297D4—84no—
298D4—84no—
299D4—84no—
300D4—84no—
301D4—84no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4AGL(1,ℤ/8)12yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C2×#4111yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—22no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
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#53411yescharacteristic
379C22D4211yescharacteristic
380C22#54611yescharacteristic
381C22C2×AGL(1,ℤ/8)11yescharacteristic
382C22#57511yescharacteristic
383C22#57711yescharacteristic
384C22#57711yescharacteristic
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4#53511yescharacteristic
396C4#54011yescharacteristic
397C4#55011yescharacteristic
398C4#56011yescharacteristic
399C2—81no—
400C2—81no—
401C2—161no—
402C2—161no—
403C2#267311yescharacteristic
404C2#319411yescharacteristic
405C2#320011yescharacteristic
406C2#327111yescharacteristic
407C2#327211yescharacteristic
408C2#331711yescharacteristic
409C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
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×#482C2semidirect
2. C2×#468C2semidirect
4. #2415C2semidirect
5. #2041C2semidirect
6. #2034C2semidirect
7. #2014C2semidirect
8. #2000C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1927C2semidirect
12. #1918C2semidirect
13. #1895C2semidirect
14. #1875C2semidirect
15. #1870C2semidirect
127. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4682
#25412
#24152
#20412
#20342
#20142
#20002
#19272
#19182
#18952
#18752
#18702