Finite Group Databasebeta

← #17599

Subgroups — #17599

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2319C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1516C212yesmaximal, complemented
6#1513C212yesmaximal, complemented
7#1510C212yesmaximal, complemented
8#1509C211yesmaximal, characteristic, complemented
9#1505C211yesmaximal, characteristic, complemented
10#1502C211yesmaximal, characteristic, complemented
11#1500C211yesmaximal, characteristic, complemented
12#555C411yescharacteristic, complemented
13#554C411yescharacteristic, complemented
14#553—21no—
15#553C412yescomplemented
16#550C411yescharacteristic, complemented
17#546—22no—
18#546C412yescomplemented
19D42—21no—
20D42C411yescharacteristic, complemented
21#540—21no—
22#538—22no—
23#532C2211yescharacteristic
24#529C2211yescharacteristic
25#525C2212yes—
26C2×#121C2212yes—
27#518—21no—
28C2×C4×D4C2211yescharacteristic
29#436—21no—
30#435—22no—
31C8×D4—21no—
32#432C2211yescharacteristic, complemented
33#431C2211yescharacteristic, complemented
34#407C2212yescomplemented
35C2×C22⋊C8C2212yescomplemented
36#390—21no—
37#389—21no—
38#388—22no—
39#381C2212yescomplemented
40#380C2212yescomplemented
41C4×C4⋊C4C2211yescharacteristic, complemented
42#376C2211yescharacteristic, complemented
43#328C2211yescharacteristic, complemented
44#328C2211yescharacteristic, complemented
45#328C2212yescomplemented
46#327C2211yescharacteristic
47#327C2211yescharacteristic
48#327C2212yes—
49#327C2212yescomplemented
50#327C2212yescomplemented
51#327C2212yescomplemented
52#327C2212yescomplemented
53C2×#41—21no—
54C2×#41C2×C412yes—
55C22×D4—22no—
56C22×D4—21no—
57C22×D4C2×C412yes—
58C22×D4D411yescharacteristic
59C23×C4C2312yes—
60C2×M16—22no—
61C22×C8—22no—
62C4⋊Q8C2×C411yescharacteristic
63#127—21no—
64#127C2×C411yescharacteristic
65#126—22no—
66#126—21no—
67#125—21no—
68#125C2×C411yescharacteristic
69#124—22no—
70#124—21no—
71#124C2×C411yescharacteristic
72#123—22no—
73#123—21no—
74#123—22no—
75#123—22no—
76C22⋊Q8—22no—
77C22⋊Q8C2×C412yes—
78C22⋊Q8C2311yescharacteristic
79#121—22no—
80#121—21no—
81#121—22no—
82#121—22no—
83#121—21no—
84#121—21no—
85#121—22no—
86#121—22no—
87#121C2×C412yes—
88#121C2×C412yes—
89#121C2×C412yes—
90#121C2311yescharacteristic
91#121C2312yes—
92#120—22no—
93#120—22no—
94#120—22no—
95C4×Q8C2×C411yescharacteristic
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4C2×C411yescharacteristic
104C4×D4C2×C412yes—
105C4×D4C2×C411yescharacteristic
106C4×D4C2×C412yes—
107C4×D4C2×C411yescharacteristic
108C4×D4D411yescharacteristic
109C4×D4D412yes—
110C4×D4D411yescharacteristic
111#117—21no—
112#117—21no—
113#117C2×C412yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C2×C412yes—
117C2×C4⋊C4D411yescharacteristic
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4D412yes—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126C4⋊C8—21no—
127C4⋊C8—21no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2311yescharacteristic
131C22⋊C8C2312yes—
132C22⋊C8C2311yescharacteristic
133#97—21no—
134C4×C8—21no—
135#95—21no—
136#95—22no—
137#95C2311yescharacteristic
138#95C2311yescharacteristic
139#95C2312yes—
140C24—22no—
141C24C2×D412yes—
142#41—81no—
143#41—81no—
144#41—82no—
145C2×Q8—21no—
146C2×Q8C22×C411yescharacteristic
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—21no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—22no—
156C2×D4—42no—
157C2×D4—44no—
158C2×D4—82no—
159C2×D4—82no—
160C2×D4—82no—
161C2×D4C22⋊C411yescharacteristic
162C2×D4C22⋊C411yescharacteristic
163C2×D4C22⋊C412yes—
164C2×D4C22×C411yescharacteristic
165C2×D4C22×C412yes—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—82no—
174C22×C4C22×C412yes—
175C22×C4C22×C412yes—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×D412yes—
181C22×C4C2411yescharacteristic
182M16—82no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—22no—
186C2×C8—82no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C4⋊C4C22⋊C411yescharacteristic
197C4⋊C4C22⋊C411yescharacteristic
198C4⋊C4C22⋊C412yes—
199C4⋊C4C22×C411yescharacteristic
200C4⋊C4C22×C412yes—
201C4⋊C4C22×C411yescharacteristic
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—21no—
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—22no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4C22⋊C412yes—
217C22⋊C4C22⋊C412yes—
218C22⋊C4C22⋊C412yes—
219C22⋊C4C22⋊C412yes—
220C42—22no—
221C42—21no—
222C42—42no—
223C42—81no—
224C42C22×C411yescharacteristic
225C42C22×C411yescharacteristic
226C42C2×D411yescharacteristic
227C42C2×D411yescharacteristic
228C23—22no—
229C23—22no—
230C23—21no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—44no—
235C23—82no—
236C23—82no—
237C23C2×C22⋊C412yes—
238C23C2×C22⋊C412yes—
239C23C22×D411yescharacteristic
240Q8—41no—
241Q8—81no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—41no—
246D4—42no—
247D4—44no—
248D4—41no—
249D4—42no—
250D4—41no—
251D4—42no—
252D4—42no—
253D4—82no—
254D4—81no—
255D4—82no—
256D4—82no—
257D4—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×C22⋊C412yes—
285C2×C4C2×C22⋊C411yescharacteristic
286C2×C4C2×C22⋊C411yescharacteristic
287C2×C4C2×C22⋊C411yescharacteristic
288C2×C4C2×C22⋊C412yes—
289C2×C4C2×C22⋊C411yescharacteristic
290C2×C4C23×C411yescharacteristic
291C2×C4C22×D411yescharacteristic
292C8—41no—
293C8—42no—
294C8—41no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22—164no—
312C22C22×C22⋊C411yescharacteristic
313C22#57611yescharacteristic
314C22#57711yescharacteristic
315C4—21no—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—81no—
327C4—82no—
328C4—81no—
329C4#42112yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2—82no—
335C2—81no—
336C2#288311yescharacteristic
337C2#289711yescharacteristic
338C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3519C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2974C2semidirect
3. #2319C2semidirect
4. #1519C2semidirect
5. #1516C2semidirect
6. #1513C2semidirect
7. #1510C2semidirect
8. #1509C2semidirect
9. #1505C2semidirect
10. #1502C2semidirect
11. #1500C2semidirect
12. #555C4semidirect
13. #554C4semidirect
15. #553C4semidirect
16. #550C4semidirect
18. #546C4semidirect
20. D42C4semidirect
32. #432C22semidirect
33. #431C22semidirect
34. #407C22semidirect
35. C2×C22⋊C8C22semidirect
39. #381C22semidirect
40. #380C22semidirect
41. C4×C4⋊C4C22semidirect
42. #376C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
49. #327C22semidirect
50. #327C22semidirect
51. #327C22semidirect
52. #327C22semidirect

Maximal subgroups

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

subgroupindex
#35192
#29742
#23192
#15192
#15162
#15132
#15102
#15092
#15052
#15022
#15002