Finite Group Databasebeta

← #24347

Subgroups — #24347

Order 256.

570 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1992C211yesmaximal, characteristic, complemented
10#1985C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic
12#1879C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17#554—24no—
18#554—24no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8—21no—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C22×C4⋊C4—21no—
32#478—24no—
33#442—24no—
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#396—21no—
46#396—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#384C2211yescharacteristic
52#380—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8D411yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8Q814yescomplemented
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C4⋊Q8—22no—
79C4⋊Q8—22no—
80#125—22no—
81#125—22no—
82#125—22no—
83#125D414yes—
84C22⋊Q8—28no—
85C22⋊Q8—28no—
86C22⋊Q8—28no—
87C22⋊Q8—44no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4D414yes—
98C2×C4⋊C4—28no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128#107—22no—
129#106—22no—
130#106—22no—
131C4⋊C8—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—22no—
138#97—22no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24—21no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—41no—
147C2×Q8—81no—
148C2×Q8—81no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4C2×D412yes—
158C2×D4C2×Q812yes—
159C2×D4C2×Q812yes—
160C2×D4#4112yes—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2411yescharacteristic
186D8—28no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8C2×Q812yes—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4C2×D412yes—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42C2×D412yes—
236C42C2×D412yes—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C22×D411yescharacteristic
252Q8—44no—
253Q8—84no—
254D4—22no—
255D4—22no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—41no—
260D4—41no—
261D4C22⋊Q814yescomplemented
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—48no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4#12012yes—
300C2×C4#12012yes—
301C2×C4AGL(1,ℤ/8)12yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C22×Q811yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C42−1+411yescharacteristic
307C2×C42−1+411yescharacteristic
308C8—22no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—24no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#12011yescharacteristic
327C22#54811yescharacteristic
328C22D4×Q811yescharacteristic
329C22C2×AGL(1,ℤ/8)11yescharacteristic
330C22#57511yescharacteristic
331C22#57611yescharacteristic
332C22#57711yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4C2×C22⋊Q811yescharacteristic
346C4#53611yescharacteristic
347C4#54811yescharacteristic
348C4D4×Q811yescharacteristic
349C2—44no—
350C2—82no—
351C2#240911yescharacteristic
352C2#300611yescharacteristic
353C2#301211yescharacteristic
354C2#336611yescharacteristic
355C2#337611yescharacteristic
356C2#338411yescharacteristic
357C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#478C2semidirect
4. #2538C2semidirect
5. #2059C2semidirect
6. #2058C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
9. #1992C2semidirect
10. #1985C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1802C2semidirect
73. C2×D8Q8semidirect
261. D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4782
C2×#4422
#25382
#20592
#20582
#20252
#20232
#19922
#19852
#18792
#18642
#18622
#18022