Finite Group Databasebeta

← #24194

Subgroups — #24194

Order 256.

569 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2581C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic
7#2015C211yesmaximal, characteristic, complemented
8#2014C211yesmaximal, characteristic
9#1984C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19#559—24no—
20#559—24no—
21C2×#127C2211yescharacteristic
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×#123—21no—
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33#464—24no—
34C4×SD16—24no—
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
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×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—41no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#127—22no—
86#127—22no—
87#124—28no—
88#124—22no—
89#124—22no—
90#124—22no—
91#124D414yes—
92#123—28no—
93#123—28no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×D4—28no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4D414yes—
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×C22⋊C4—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127#106—22no—
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—41no—
137#102—41no—
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—
149C24—21no—
150C2×Q8—22no—
151C2×Q8—22no—
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—41no—
163C2×Q8C2×D412yes—
164C2×Q8#4112yes—
165C2×D4—24no—
166C2×D4—24no—
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—81no—
180C2×D4—81no—
181C2×D4C2×D412yes—
182C2×D4C2×D412yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
202Q16—44no—
203SD16—28no—
204SD16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8#4112yes—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
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—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C42—24no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42C2×D412yes—
245C42C2×D412yes—
246C23—22no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—44no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270D4—22no—
271D4—22no—
272D4—22no—
273D4—22no—
274D4—48no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4#12012yes—
316C2×C4#12012yes—
317C2×C4#13512yes—
318C2×C4C22×D411yescharacteristic
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C42−1+411yescharacteristic
324C8—22no—
325C8—41no—
326C8—41no—
327C8—41no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C2×#12011yescharacteristic
343C22#54811yescharacteristic
344C22#55011yescharacteristic
345C22C2×#13511yescharacteristic
346C22#57511yescharacteristic
347C22#57711yescharacteristic
348C22#57711yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4#52511yescharacteristic
362C4#53711yescharacteristic
363C4#54611yescharacteristic
364C4#54711yescharacteristic
365C2—82no—
366C2—82no—
367C2#240711yescharacteristic
368C2#300211yescharacteristic
369C2#300911yescharacteristic
370C2#336511yescharacteristic
371C2#337111yescharacteristic
372C2#339411yescharacteristic
373C2#340111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
2. C2×#464C2semidirect
3. C2×C4×SD16C2semidirect
4. #2581C2semidirect
5. #2042C2semidirect
7. #2015C2semidirect
9. #1984C2semidirect
10. #1958C2semidirect
11. #1878C2semidirect
12. #1877C2semidirect
13. #1865C2semidirect
14. #1864C2semidirect
15. #1804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4642
C2×C4×SD162
#25812
#20422
#20412
#20152
#20142
#19842
#19582
#18782
#18772
#18652
#18642
#18042