Finite Group Databasebeta

← #24109

Subgroups — #24109

Order 256.

542 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
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1823C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17#554—24no—
18#554—24no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#125—21no—
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#485—24no—
32#428—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#397—21no—
47#394—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×D4D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79#125—28no—
80C22⋊Q8—28no—
81C22⋊Q8—28no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8D414yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×D4—28no—
89C4×D4—22no—
90C4×D4—22no—
91#117—22no—
92#117—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4D414yes—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#107—22no—
127#106—22no—
128#106—22no—
129#106—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×Q8—24no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8C2×D412yes—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
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—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4C2×D412yes—
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187M16—28no—
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×C8#4112yes—
199C2×C8#4112yes—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
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—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4C2×D412yes—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—24no—
232C42—21no—
233C42—21no—
234C42—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23#12012yes—
248C23#13712yes—
249C23C22×D411yescharacteristic
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—84no—
255D4—28no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
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—41no—
297C2×C4—41no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4#12012yes—
301C2×C4#13712yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C42−1+411yescharacteristic
307C2×C42−1+411yescharacteristic
308C8—22no—
309C8—22no—
310C8—41no—
311C8—41no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22C2×#12011yescharacteristic
327C22#54711yescharacteristic
328C22#54711yescharacteristic
329C22C2×#13711yescharacteristic
330C22C2×#13711yescharacteristic
331C22#57611yescharacteristic
332C22#57711yescharacteristic
333C4—22no—
334C4—22no—
335C4—41no—
336C4—42no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#52511yescharacteristic
346C4#53611yescharacteristic
347C4#54711yescharacteristic
348C4#54711yescharacteristic
349C2—42no—
350C2—42no—
351C2#240611yescharacteristic
352C2#300411yescharacteristic
353C2#301211yescharacteristic
354C2#334411yescharacteristic
355C2#334611yescharacteristic
356C2#335611yescharacteristic
357C2#335711yescharacteristic
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×#428C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1894C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect
13. #1853C2semidirect
14. #1852C2semidirect
15. #1823C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4852
C2×#4282
#24852
#20582
#20572
#20362
#20352
#18942
#18702
#18652
#18632
#18532
#18522
#18232