Finite Group Databasebeta

← #30715

Subgroups — #30715

Order 256.

512 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1921C211yesmaximal, characteristic, complemented
13#1892C211yesmaximal, characteristic, complemented
14#1874C211yesmaximal, characteristic, complemented
15#1869C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29#484—24no—
30#466—24no—
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44#399—21no—
45#396—21no—
46#394—21no—
47#392—21no—
48#390—21no—
49#390—21no—
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#386—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×D4D411yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—41no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×D8—22no—
76C2×D8—41no—
77C2×D8—41no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#124—41no—
86#124—41no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—41no—
95#121—41no—
96#121—41no—
97#121—41no—
98#121—41no—
99#121—44no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
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⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129#107—22no—
130#107D414yescomplemented
131#103—22no—
132#103—22no—
133#103—41no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—41no—
138#102—41no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—41no—
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—22no—
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—21no—
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—81no—
180C2×D4—81no—
181C2×D4—81no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
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—42no—
201C22×C4—42no—
202C22×C4—81no—
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2411yescharacteristic
208SD16—44no—
209D8—44no—
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×C8—21no—
221C2×C8—44no—
222C2×C8C2×D412yes—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4C2×D412yes—
236C4⋊C4C2×D412yes—
237C4⋊C4#4112yes—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—81no—
251C42—21no—
252C42—21no—
253C42—81no—
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—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#13512yes—
281C23AGL(1,ℤ/8)12yes—
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—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
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×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—84no—
339C2×C4C22×D411yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C8—22no—
346C8—41no—
347C8—41no—
348C8—41no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
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#53411yescharacteristic
378C22D4211yescharacteristic
379C22#54611yescharacteristic
380C22C2×#13511yescharacteristic
381C22C2×AGL(1,ℤ/8)11yescharacteristic
382C22#57611yescharacteristic
383C22#57811yescharacteristic
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4#53511yescharacteristic
395C4#54011yescharacteristic
396C4#55011yescharacteristic
397C4#56011yescharacteristic
398C2—42no—
399C2—42no—
400C2—161no—
401C2—161no—
402C2#267311yescharacteristic
403C2#318811yescharacteristic
404C2#320311yescharacteristic
405C2#326411yescharacteristic
406C2#328511yescharacteristic
407C2#329711yescharacteristic
408C2#330811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#484C2semidirect
2. C2×#466C2semidirect
3. #2540C2semidirect
4. #2416C2semidirect
5. #2042C2semidirect
6. #2033C2semidirect
7. #2012C2semidirect
8. #2000C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1925C2semidirect
12. #1921C2semidirect
13. #1892C2semidirect
14. #1874C2semidirect
15. #1869C2semidirect
130. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4662
#25402
#24162
#20422
#20332
#20122
#20002
#19992
#19252
#19212
#18922
#18742
#18692