Finite Group Databasebeta

← #17608

Subgroups — #17608

Order 256.

649 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3489C211yesmaximal, characteristic
2#2880C211yesmaximal, characteristic, complemented
3#2339C211yesmaximal, characteristic, complemented
4#1520C211yesmaximal, characteristic, complemented
5#1516C212yesmaximal, complemented
6#1513C211yesmaximal, characteristic, complemented
7#1511C212yesmaximal, complemented
8#1508C211yesmaximal, characteristic, complemented
9#1504C212yesmaximal, complemented
10#1501C211yesmaximal, characteristic, complemented
11C2×#328C212yesmaximal, complemented
12C2×2−1+4C411yescharacteristic, complemented
13C22×#41C2211yescharacteristic
14C2×#131—21no—
15D4×Q8—21no—
16#548—21no—
17#547—21no—
18#547—22no—
19#546—21no—
20#546—22no—
21#537C412yescomplemented
22#537C411yescharacteristic, complemented
23#536C411yescharacteristic, complemented
24#536C412yescomplemented
25#525C2211yescharacteristic
26#525C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2211yescharacteristic
29#520C411yescharacteristic, complemented
30C2×C4×D4—21no—
31#408—22no—
32#408—21no—
33#407C2212yes—
34#407C2211yescharacteristic
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2211yescharacteristic
37#391—21no—
38#386—22no—
39#381C2212yes—
40#380C2211yescharacteristic
41C4×C22⋊C4C2211yescharacteristic
42C2×#95C2212yes—
43#328C2211yescharacteristic
44#328C2212yes—
45#328C2212yes—
46#328C2212yes—
47#328C2211yescharacteristic
48#328C2212yes—
49#328C2212yes—
50#327C2211yescharacteristic
51#327C2211yescharacteristic
52#327C2212yes—
532−1+4—44no—
54C2×#41—22no—
55C2×#41—21no—
56C2×#41—44no—
57C2×#41C2×C411yescharacteristic
58C2×#41C2×C412yes—
59C2×#41C2×C411yescharacteristic
60C2×#41C2×C412yescomplemented
61C2×#41C2×C411yescharacteristic, complemented
62C2×#41D411yescharacteristic
63C2×#41D412yes—
64C2×#41D411yescharacteristic
65C22×Q8—21no—
66C22×Q8C2×C411yescharacteristic, complemented
67C22×Q8C2×C412yescomplemented
68C22×Q8D411yescharacteristic
69C22×D4D411yescharacteristic
70C22×D4D412yes—
71C23×C4—22no—
72C23×C4C2312yes—
73C23×C4C2311yescharacteristic
74#131—44no—
75C2×M16—22no—
76C2×M16—21no—
77C22×C8—22no—
78C22×C8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—22no—
81#124—21no—
82#124—22no—
83#123—22no—
84#123—22no—
85#123—22no—
86#123—22no—
87#123—21no—
88C22⋊Q8—21no—
89C22⋊Q8—22no—
90C22⋊Q8—21no—
91C22⋊Q8—22no—
92C22⋊Q8—21no—
93C22⋊Q8—22no—
94C22⋊Q8C2×C412yescomplemented
95C22⋊Q8C2×C412yescomplemented
96C22⋊Q8C2×C412yescomplemented
97C22⋊Q8C2×C411yescharacteristic, complemented
98C22⋊Q8C2×C412yescomplemented
99C22⋊Q8C2311yescharacteristic
100C22⋊Q8C2312yes—
101#121—22no—
102#121—22no—
103#121—22no—
104#121—21no—
105#121—22no—
106#121C2×C411yescharacteristic, complemented
107#121C2×C412yescomplemented
108#121C2311yescharacteristic
109#120—22no—
110#120—21no—
111C4×Q8—21no—
112C4×D4—22no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—22no—
116C4×D4—21no—
117C4×D4—22no—
118C4×D4—44no—
119#117C2×C411yescharacteristic, complemented
120#117C2×C412yescomplemented
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4C2×C412yescomplemented
125C2×C4⋊C4C2×C411yescharacteristic, complemented
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C22⋊C8—21no—
133C22⋊C8—22no—
134C22⋊C8—21no—
135C22⋊C8—22no—
136C22⋊C8C2311yescharacteristic
137C22⋊C8C2311yescharacteristic
138C22⋊C8C2312yes—
139#95—22no—
140#95—22no—
141#95C2311yescharacteristic
142#95C2312yes—
143#95C2311yescharacteristic
144C24C2×D412yes—
145C24C2×D411yescharacteristic
146#41—24no—
147#41—24no—
148#41—44no—
149#41—44no—
150#41—42no—
151#41—42no—
152#41—44no—
153#41—42no—
154#41—48no—
155#41—44no—
156C2×Q8—22no—
157C2×Q8—21no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8C22⋊C411yescharacteristic
163C2×Q8C22⋊C411yescharacteristic
164C2×Q8C22⋊C412yes—
165C2×Q8C22×C412yes—
166C2×Q8C22×C411yescharacteristic
167C2×D4—24no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4C22⋊C412yes—
175C2×D4C22⋊C412yes—
176C2×D4C22⋊C412yes—
177C2×D4C22⋊C411yescharacteristic
178C2×D4C22⋊C412yes—
179C2×D4C22⋊C412yes—
180C2×D4C22⋊C411yescharacteristic
181C2×D4C22×C411yescharacteristic
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—82no—
196C22×C4C22×C412yes—
197C22×C4C22×C411yescharacteristic
198C22×C4C22×C412yes—
199C22×C4C22×C411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D412yes—
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D412yes—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D412yes—
206C22×C4C2411yescharacteristic
207M16—44no—
208M16—42no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—22no—
212C2×C8—44no—
213C2×C8—42no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4C22×C411yescharacteristic
225C4⋊C4C22×C412yes—
226C4⋊C4C22×C411yescharacteristic
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—21no—
231C22⋊C4—22no—
232C22⋊C4—21no—
233C22⋊C4—22no—
234C22⋊C4—21no—
235C22⋊C4—22no—
236C22⋊C4—21no—
237C22⋊C4—22no—
238C22⋊C4—21no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—81no—
242C42—22no—
243C42—21no—
244C42—42no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊C411yescharacteristic
255C23C2×C22⋊C412yes—
256C23C2×C22⋊C411yescharacteristic
257C23C2×C22⋊C412yes—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—42no—
262Q8—42no—
263Q8—41no—
264Q8—44no—
265Q8—42no—
266D4—24no—
267D4—24no—
268D4—24no—
269D4—22no—
270D4—42no—
271D4—42no—
272D4—41no—
273D4—42no—
274D4—41no—
275D4—41no—
276D4—44no—
277D4—82no—
278D4—81no—
279D4#9914yes—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—88no—
312C2×C4C2×C22⋊C412yes—
313C2×C4C2×C22⋊C411yescharacteristic
314C2×C4C2×C22⋊C412yes—
315C2×C4C2×C22⋊C411yescharacteristic
316C2×C4C23×C411yescharacteristic
317C2×C4C22×D411yescharacteristic
318C8—41no—
319C8—42no—
320C8—41no—
321C22—21no—
322C22—22no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—44no—
335C22—44no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22—81no—
341C22C2×#9914yes—
342C22C22×C22⋊C411yescharacteristic
343C22#57711yescharacteristic
344C22#57811yescharacteristic
345C4—21no—
346C4—21no—
347C4—22no—
348C4—21no—
349C4—22no—
350C4—42no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—81no—
357C4—82no—
358C4C2×#9912yes—
359C2—21no—
360C2—21no—
361C2—24no—
362C2—42no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2C22×#9911yescharacteristic
367C2#289811yescharacteristic
368C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3489C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2880C2semidirect
3. #2339C2semidirect
4. #1520C2semidirect
5. #1516C2semidirect
6. #1513C2semidirect
7. #1511C2semidirect
8. #1508C2semidirect
9. #1504C2semidirect
10. #1501C2semidirect
11. C2×#328C2semidirect
12. C2×2−1+4C4semidirect
21. #537C4semidirect
22. #537C4semidirect
23. #536C4semidirect
24. #536C4semidirect
29. #520C4semidirect
60. C2×#41C2×C4semidirect
61. C2×#41C2×C4semidirect
66. C22×Q8C2×C4semidirect
67. C22×Q8C2×C4semidirect
94. C22⋊Q8C2×C4semidirect
95. C22⋊Q8C2×C4semidirect
96. C22⋊Q8C2×C4semidirect
97. C22⋊Q8C2×C4semidirect
98. C22⋊Q8C2×C4semidirect
106. #121C2×C4semidirect
107. #121C2×C4semidirect
119. #117C2×C4semidirect
120. #117C2×C4semidirect
124. C2×C4⋊C4C2×C4semidirect
125. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34892
#28802
#23392
#15202
#15162
#15132
#15112
#15082
#15042
#15012
C2×#3282