Finite Group Databasebeta

← #25542

Subgroups — #25542

Order 256.

546 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#467C211yesmaximal, characteristic, complemented
4#2415C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1878C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#556—24no—
19#556—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#126—21no—
23C2×#124—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C2×#117C2211yescharacteristic
34#485—24no—
35#467—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#397—21no—
49#394—21no—
50#391C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—41no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81#126—28no—
82#126—28no—
83#124—28no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—41no—
91C22⋊Q8D414yes—
92C4×Q8—28no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×D4—22no—
96C4×D4—22no—
97#117—22no—
98#117—22no—
99#117—22no—
100#117—22no—
101#117—41no—
102#117D414yes—
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⋊C4—21no—
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×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#107—22no—
134#106—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×Q8—24no—
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—42no—
163C2×Q8—42no—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—44no—
194C2×C8—21no—
195C2×C8—21no—
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—44no—
206C2×C8#4112yes—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4C2×D412yes—
224C4⋊C4C2×D412yes—
225C4⋊C4C2×D412yes—
226C4⋊C4#4112yes—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C42—24no—
238C42—24no—
239C42—24no—
240C42—21no—
241C42—21no—
242C42—42no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#12012yes—
257C23C22×D411yescharacteristic
258Q8—22no—
259Q8—22no—
260Q8—48no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
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—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4#12012yes—
311C2×C4#13512yes—
312C2×C4C22×D411yescharacteristic
313C2×C4C22×D411yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C42−1+411yescharacteristic
318C8—22no—
319C8—41no—
320C8—41no—
321C8—41no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22C2×#12011yescharacteristic
337C22#54611yescharacteristic
338C22#54711yescharacteristic
339C22C2×#13511yescharacteristic
340C22#57511yescharacteristic
341C22#57811yescharacteristic
342C22#57811yescharacteristic
343C4—22no—
344C4—22no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4#53211yescharacteristic
356C4#53711yescharacteristic
357C4#54611yescharacteristic
358C4#54711yescharacteristic
359C2—42no—
360C2—81no—
361C2#243011yescharacteristic
362C2#300211yescharacteristic
363C2#301111yescharacteristic
364C2#330211yescharacteristic
365C2#331311yescharacteristic
366C2#334211yescharacteristic
367C2#334911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
2. C2×#485C2semidirect
3. C2×#467C2semidirect
5. #2057C2semidirect
6. #2036C2semidirect
7. #2024C2semidirect
8. #2001C2semidirect
9. #1878C2semidirect
10. #1873C2semidirect
11. #1873C2semidirect
13. #1865C2semidirect
14. #1854C2semidirect
15. #1735C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4852
C2×#4672
#24152
#20572
#20362
#20242
#20012
#18782
#18732
#18692
#18652
#18542
#17352