Finite Group Databasebeta

← #21774

Subgroups — #21774

Order 256.

743 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3#2280C211yesmaximal, characteristic, complemented
4#1897C214yesmaximal, complemented
5#1871C214yesmaximal, complemented
6#1826C212yesmaximal, complemented
7#1789C212yesmaximal, complemented
8C22×#41C2211yescharacteristic
9C24×C4C2211yescharacteristic
10C22×M16C2212yescomplemented
11#525—21no—
12#525—21no—
13#525—21no—
14#525—21no—
15#525C412yescomplemented
16#525C412yescomplemented
17#525C2212yes—
18#525C2212yes—
19C2×#123—21no—
20C2×C22⋊Q8C411yescharacteristic, complemented
21C2×C22⋊Q8C411yescharacteristic, complemented
22C2×#121C411yescharacteristic, complemented
23C2×#121C411yescharacteristic, complemented
24C2×#120—21no—
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×#117C2211yescharacteristic
28C2×#117C2212yescomplemented
29C2×#104C2218yescomplemented
30#407—21no—
31#407—21no—
32#407C2212yes—
33#407C2212yescomplemented
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36#381—21no—
37#380—21no—
38#380—21no—
39#379—21no—
40C4×C22⋊C4C2212yes—
41C4×C22⋊C4C2212yescomplemented
42#339C2218yescomplemented
43C25D411yescharacteristic
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—44no—
47C2×#41C2×C412yes—
48C2×#41C2312yes—
49C22×Q8C2×C411yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2×C411yescharacteristic
54C23×C4—21no—
55C23×C4—44no—
56C23×C4C2×C411yescharacteristic
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—44no—
72C2×M16C2314yes—
73C22×C8—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8C2×C412yes—
83C22⋊Q8C2×C412yes—
84C22⋊Q8C2×C412yes—
85C22⋊Q8C2×C412yes—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121C2×C412yes—
91#121C2×C412yes—
92#121C2×C412yes—
93#121C2×C412yes—
94#120—22no—
95#120—22no—
96#120—22no—
97#120—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106#117—21no—
107#117—21no—
108#117—44no—
109#117C2×C412yes—
110#117C2312yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C2×C411yescharacteristic
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C42—21no—
125C2×C42C2314yes—
126#104—48no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131#95—22no—
132#95—22no—
133C24—21no—
134C24—42no—
135C24—42no—
136C24—42no—
137C24C22⋊C411yescharacteristic
138C24C22⋊C411yescharacteristic
139C24C22⋊C411yescharacteristic
140C24C22⋊C411yescharacteristic
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144#41—28no—
145#41—44no—
146#41—44no—
147#41—44no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×Q8C22×C412yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4C22×C412yes—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4C22⋊C411yescharacteristic
185C22×C4C22⋊C411yescharacteristic
186C22×C4C22⋊C411yescharacteristic
187C22×C4C22⋊C411yescharacteristic
188C22×C4C22⋊C411yescharacteristic
189C22×C4C22⋊C411yescharacteristic
190C22×C4C22⋊C411yescharacteristic
191C22×C4C22⋊C411yescharacteristic
192C22×C4C22⋊C411yescharacteristic
193C22×C4C22⋊C411yescharacteristic
194C22×C4C22⋊C411yescharacteristic
195C22×C4C22⋊C411yescharacteristic
196C22×C4C22×C411yescharacteristic
197C22×C4C22×C411yescharacteristic
198C22×C4C22×C411yescharacteristic
199C22×C4C22×C411yescharacteristic
200C22×C4C22×C411yescharacteristic
201C22×C4C22×C411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2411yescharacteristic
212M16—28no—
213M16—44no—
214C2×C8—24no—
215C2×C8—44no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—44no—
223C4⋊C4C22×C412yes—
224C4⋊C4C22×C412yes—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—22no—
239C42—22no—
240C42—28no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23—416no—
266C23—42no—
267C23C2×C22⋊C411yescharacteristic
268C23C2×C22⋊C411yescharacteristic
269C23C2×C22⋊C411yescharacteristic
270C23C2×C22⋊C411yescharacteristic
271C23C2×C22⋊C411yescharacteristic
272C23C2×C22⋊C411yescharacteristic
273C23C22×D411yescharacteristic
274Q8—24no—
275Q8—42no—
276D4—24no—
277D4—42no—
278D4—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283D4—44no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4C2×C22⋊C411yescharacteristic
323C2×C4C2×C22⋊C411yescharacteristic
324C2×C4C2×C22⋊C411yescharacteristic
325C2×C4C2×C22⋊C411yescharacteristic
326C2×C4C2×C22⋊C411yescharacteristic
327C2×C4C2×C22⋊C411yescharacteristic
328C2×C4C23×C411yescharacteristic
329C2×C4C22×D411yescharacteristic
330C2×C4H(5,𝔽2)14yes—
331C8—44no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—84no—
361C22#42112yes—
362C22#42112yes—
363C22C22×C22⋊C411yescharacteristic
364C22#53412yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—84no—
375C4#51812yes—
376C4#53412yes—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—41no—
381C2—41no—
382C2—41no—
383C2—41no—
384C2—82no—
385C2#230811yescharacteristic
386C2C2×#42111yescharacteristic
387C2C2×#42111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radicalC2×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. #2280C2semidirect
4. #1897C2semidirect
5. #1871C2semidirect
6. #1826C2semidirect
7. #1789C2semidirect
10. C22×M16C22semidirect
15. #525C4semidirect
16. #525C4semidirect
20. C2×C22⋊Q8C4semidirect
21. C2×C22⋊Q8C4semidirect
22. C2×#121C4semidirect
23. C2×#121C4semidirect
28. C2×#117C22semidirect
29. C2×#104C22semidirect
33. #407C22semidirect
41. C4×C22⋊C4C22semidirect
42. #339C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4072
#22802
#18972
#18712
#18262
#17892