Finite Group Databasebeta

← #24195

Subgroups — #24195

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3C2×C4×Q16C211yesmaximal, characteristic, complemented
4#2581C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic
7#2016C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic
9#1984C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19#556—24no—
20#556—24no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#126—21no—
23C2×#126—21no—
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33#464—24no—
34C4×Q16—24no—
35C2×#107C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C4×C8C2211yescharacteristic
48#400—21no—
49#395—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#387—21no—
55#387—21no—
56#382C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8D411yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—41no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84#126—28no—
85#126—28no—
86#124—22no—
87#124—22no—
88#124—22no—
89#124D414yes—
90C22⋊Q8—28no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8D414yes—
99C4×D4—22no—
100C4×D4—22no—
101#117—28no—
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⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—41no—
134#103—41no—
135#103—41no—
136#102—22no—
137#102—22no—
138#102—41no—
139C4×C8—22no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
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—41no—
167C2×Q8C2×D412yes—
168C2×Q8C2×D412yes—
169C2×Q8#4112yes—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4C2×D412yes—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—28no—
194Q16—44no—
195SD16—44no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8#4112yes—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4C2×D412yes—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C42—24no—
232C42—24no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—42no—
238C42C2×D412yes—
239C42C2×D412yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—22no—
253Q8—22no—
254Q8—44no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261D4—22no—
262D4—22no—
263D4—41no—
264D4—41no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—48no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4#12012yes—
303C2×C4#12012yes—
304C2×C4#13512yes—
305C2×C4C22×D411yescharacteristic
306C2×C4C22×D411yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C42−1+411yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22C2×#12011yescharacteristic
323C22#54811yescharacteristic
324C22#55011yescharacteristic
325C22C2×#13511yescharacteristic
326C22#57511yescharacteristic
327C22#57811yescharacteristic
328C22#57811yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#52511yescharacteristic
343C4#53711yescharacteristic
344C4#54611yescharacteristic
345C4#54711yescharacteristic
346C2—82no—
347C2#240711yescharacteristic
348C2#300211yescharacteristic
349C2#301111yescharacteristic
350C2#336411yescharacteristic
351C2#337711yescharacteristic
352C2#339311yescharacteristic
353C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. C2×C4×Q16C2semidirect
4. #2581C2semidirect
5. #2043C2semidirect
7. #2016C2semidirect
9. #1984C2semidirect
11. #1878C2semidirect
13. #1863C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4642
C2×C4×Q162
#25812
#20432
#20422
#20162
#20152
#19842
#19582
#18782
#18772
#18632
#18622
#18052