Finite Group Databasebeta

← #17578

Subgroups — #17578

Order 256.

709 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic
3#2280C211yesmaximal, characteristic
4#1520C214yesmaximal, complemented
5#1511C214yesmaximal, complemented
6#1508C214yesmaximal, complemented
7C23×Q8C411yescharacteristic, complemented
8C24×C4C2211yescharacteristic
9C22×M16—21no—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C412yescomplemented
17C2×C22⋊Q8C412yescomplemented
18C2×C22⋊Q8C412yescomplemented
19C2×C22⋊Q8C2212yes—
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2212yes—
22C2×#117—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C411yescharacteristic, complemented
25C22×C22⋊C4—21no—
26#407—21no—
27#407—21no—
28#407C2212yes—
29#407C2212yes—
30C2×C22⋊C8—21no—
31C2×C22⋊C8C2212yes—
32#381C2212yes—
33#380—21no—
34#380C2212yes—
35#379—21no—
36C4×C22⋊C4—21no—
37C4×C22⋊C4C2212yes—
38#328C22116yescomplemented
39C25D411yescharacteristic
40C22×Q8—21no—
41C22×Q8—21no—
42C22×Q8—21no—
43C22×Q8—44no—
44C22×Q8C2×C412yes—
45C22×Q8C2×C412yes—
46C22×Q8C2×C412yes—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—44no—
50C23×C4C2×C411yescharacteristic
51C23×C4C2×C411yescharacteristic
52C23×C4C2×C411yescharacteristic
53C23×C4C2×C411yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4D411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—44no—
67C22×C8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8C2×C414yes—
81C22⋊Q8C2×C414yes—
82C22⋊Q8C2×C414yes—
83C22⋊Q8C2314yes—
84#117—44no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4C2×C412yes—
96C2×C4⋊C4C2×C412yes—
97C2×C4⋊C4C2×C412yes—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C42—22no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C22⋊C8—22no—
111C22⋊C8C2314yes—
112#95—22no—
113#95C2314yes—
114C24—24no—
115C24—42no—
116C24—42no—
117C24C22⋊C411yescharacteristic
118C24C22⋊C411yescharacteristic
119C24C22⋊C411yescharacteristic
120C24C22⋊C411yescharacteristic
121C24C2×D411yescharacteristic
122C24C2×D411yescharacteristic
123C24C2×D411yescharacteristic
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8C22×C414yes—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4C22⋊C411yescharacteristic
158C22×C4C22⋊C411yescharacteristic
159C22×C4C22⋊C411yescharacteristic
160C22×C4C22⋊C411yescharacteristic
161C22×C4C22⋊C411yescharacteristic
162C22×C4C22⋊C411yescharacteristic
163C22×C4C22⋊C411yescharacteristic
164C22×C4C22⋊C411yescharacteristic
165C22×C4C22⋊C411yescharacteristic
166C22×C4C22⋊C411yescharacteristic
167C22×C4C22⋊C411yescharacteristic
168C22×C4C22⋊C411yescharacteristic
169C22×C4C22×C411yescharacteristic
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C411yescharacteristic
172C22×C4C22×C411yescharacteristic
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2411yescharacteristic
185M16—44no—
186M16—44no—
187C2×C8—24no—
188C2×C8—44no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—44no—
197C4⋊C4C22×C414yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C42—44no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—416no—
227C23C2×C22⋊C411yescharacteristic
228C23C2×C22⋊C411yescharacteristic
229C23C2×C22⋊C411yescharacteristic
230C23C2×C22⋊C411yescharacteristic
231C23C2×C22⋊C411yescharacteristic
232C23C2×C22⋊C411yescharacteristic
233C23#13714yes—
234C23C22×D411yescharacteristic
235Q8—44no—
236Q8—44no—
237Q8—44no—
238Q8—44no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4#9914yes—
278C2×C4C2×C22⋊C411yescharacteristic
279C2×C4C2×C22⋊C411yescharacteristic
280C2×C4C2×C22⋊C411yescharacteristic
281C2×C4C2×C22⋊C411yescharacteristic
282C2×C4C2×C22⋊C411yescharacteristic
283C2×C4C2×C22⋊C411yescharacteristic
284C2×C4C23×C411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C8—44no—
287C22—28no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22C2×#9912yes—
311C22#41912yes—
312C22#41912yes—
313C22#42112yes—
314C22C22×C22⋊C411yescharacteristic
315C22C2×#13712yes—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—44no—
323C4—44no—
324C4—84no—
325C4C2×#9912yes—
326C4C2×#9912yes—
327C2—22no—
328C2—22no—
329C2—22no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—41no—
334C2C22×#9911yescharacteristic
335C2C2×#41911yescharacteristic
336C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1520C2semidirect
5. #1511C2semidirect
6. #1508C2semidirect
7. C23×Q8C4semidirect
16. C2×C22⋊Q8C4semidirect
17. C2×C22⋊Q8C4semidirect
18. C2×C22⋊Q8C4semidirect
24. C22×C4⋊C4C4semidirect
38. #328C22semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4072
#22802
#15202
#15112
#15082