Finite Group Databasebeta

← #24207

Subgroups — #24207

Order 256.

538 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3C2×#441C211yesmaximal, characteristic, complemented
4#2582C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic
7#2016C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic
9#1985C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic
15#1803C211yesmaximal, 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—
34#441—24no—
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#97C2211yescharacteristic
48#398—21no—
49#398—21no—
50#391—21no—
51#391C2211yescharacteristic
52#391C2211yescharacteristic
53#389—21no—
54#389C2211yescharacteristic
55#389C2211yescharacteristic
56#384C2211yescharacteristic
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⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#106—22no—
130C4⋊C8—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—41no—
136#103—41no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—41no—
141#97—22no—
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—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4C2×D412yes—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—24no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—42no—
240C42C2×D412yes—
241C42C2×D412yes—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23C22×D411yescharacteristic
250Q8—22no—
251Q8—22no—
252Q8—22no—
253Q8—22no—
254Q8—22no—
255Q8—22no—
256Q8—44no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263D4—22no—
264D4—22no—
265D4—41no—
266D4—41no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—48no—
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#12012yes—
305C2×C4#12012yes—
306C2×C4#13712yes—
307C2×C4C22×D411yescharacteristic
308C2×C4C22×D411yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C42−1+411yescharacteristic
313C8—22no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×#12011yescharacteristic
325C22#54811yescharacteristic
326C22#55011yescharacteristic
327C22C2×#13711yescharacteristic
328C22#57511yescharacteristic
329C22#57711yescharacteristic
330C22#57811yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4#52511yescharacteristic
345C4#53711yescharacteristic
346C4#54611yescharacteristic
347C4#54711yescharacteristic
348C2—82no—
349C2#240711yescharacteristic
350C2#300711yescharacteristic
351C2#301311yescharacteristic
352C2#336911yescharacteristic
353C2#337711yescharacteristic
354C2#339811yescharacteristic
355C2#340111yescharacteristic
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×#441C2semidirect
4. #2582C2semidirect
5. #2043C2semidirect
7. #2016C2semidirect
9. #1985C2semidirect
11. #1878C2semidirect
13. #1865C2semidirect
15. #1803C2semidirect

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×#4412
#25822
#20432
#20422
#20162
#20152
#19852
#19592
#18782
#18772
#18652
#18642
#18032