Finite Group Databasebeta

← #21412

Subgroups — #21412

Order 256.

567 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×#542C211yesmaximal, characteristic
2C2×#405C211yesmaximal, characteristic, complemented
3#2314C211yesmaximal, characteristic, complemented
4#1993C211yesmaximal, characteristic, complemented
5#1990C211yesmaximal, characteristic, complemented
6#1959C211yesmaximal, characteristic, complemented
7#1958C211yesmaximal, characteristic, complemented
8#1895C211yesmaximal, characteristic
9#1894C211yesmaximal, characteristic, complemented
10#1893C211yesmaximal, characteristic
11#1892C211yesmaximal, characteristic, complemented
12#1763C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#542—24no—
19#542C418yescomplemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39#405—24no—
40C2×#97C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#387—21no—
48#384C2211yescharacteristic
49#382C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C4⋊Q8—22no—
79C4⋊Q8C2×C414yes—
80#126—28no—
81#124—28no—
82#124—22no—
83#124C2×C414yes—
84#123—28no—
85C22⋊Q8—22no—
86C22⋊Q8—28no—
87C22⋊Q8C2×C414yes—
88#121—22no—
89#121C2×C414yes—
90C4×Q8—22no—
91C4×Q8C2×C414yes—
92C4×D4—22no—
93C4×D4C2×C414yes—
94#117C2×C414yes—
95#117D414yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#103—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#97—22no—
132#97—22no—
133C4×C8—22no—
134C4×C8—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C2×Q8—24no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—81no—
150C2×Q8—81no—
151C2×Q8C22×C412yes—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—81no—
159C2×D4—81no—
160C2×D4C22×C412yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C22×C412yes—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D412yes—
182C22×C4C2411yescharacteristic
183M16—28no—
184C2×C8—28no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8#4112yes—
194C2×C8#4112yes—
195C2×C8#4112yes—
196C2×C8#4112yes—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4C22⋊C414yes—
213C4⋊C4C22⋊C414yes—
214C4⋊C4C22×C412yes—
215C4⋊C4C22×C412yes—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C22⋊C4C22⋊C414yes—
224C22⋊C4C22⋊C414yes—
225C42—24no—
226C42—81no—
227C42—81no—
228C42C22×C412yes—
229C42C22×C412yes—
230C42C2×D412yes—
231C42C2×D412yes—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C2×C22⋊C412yes—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—41no—
250Q8—41no—
251Q8—84no—
252D4—22no—
253D4—22no—
254D4—41no—
255D4—41no—
256D4—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
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—84no—
293C2×C4C2×C22⋊C412yes—
294C2×C4C2×C22⋊C412yes—
295C2×C4C2×C22⋊C412yes—
296C2×C4C2×C22⋊C412yes—
297C2×C4C2×C22⋊C412yes—
298C2×C4C23×C411yescharacteristic
299C2×C4C22×D411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C8—22no—
305C8—22no—
306C8—22no—
307C8—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22C22×C22⋊C411yescharacteristic
323C22#53211yescharacteristic
324C22#53211yescharacteristic
325C22#57611yescharacteristic
326C22#57711yescharacteristic
327C22#57711yescharacteristic
328C22#57811yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4C4×#4111yescharacteristic
342C4C4×#4111yescharacteristic
343C4#52511yescharacteristic
344C4#52511yescharacteristic
345C2—42no—
346C2—82no—
347C2#229711yescharacteristic
348C2#289711yescharacteristic
349C2#289811yescharacteristic
350C2#317311yescharacteristic
351C2#317511yescharacteristic
352C2#317811yescharacteristic
353C2#317911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#405C2semidirect
3. #2314C2semidirect
4. #1993C2semidirect
5. #1990C2semidirect
6. #1959C2semidirect
7. #1958C2semidirect
9. #1894C2semidirect
11. #1892C2semidirect
12. #1763C2semidirect
13. #1762C2semidirect
14. C4×#103C2semidirect
15. C4×#102C2semidirect
19. #542C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4052
#23142
#19932
#19902
#19592
#19582
#18952
#18942
#18932
#18922
#17632
#17622
C4×#1032
C4×#1022