Finite Group Databasebeta

← #16519

Subgroups — #16519

Order 256.

845 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#396C212yesmaximal, complemented
2C2×#394C211yesmaximal, characteristic
3C2×#390C211yesmaximal, characteristic
4C2×#387C211yesmaximal, characteristic, complemented
5C2×#382C211yesmaximal, characteristic, complemented
6#1444C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1440C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11#1438C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C42C2211yescharacteristic
25#396—28no—
26#394—24no—
27#390—24no—
28#387—24no—
29#382—24no—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2312yes—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—42no—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4D414yescomplemented
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42D414yes—
91#95—24no—
92#95—22no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95D414yes—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24#4114yes—
116C24C2411yescharacteristic
117C2×D4—28no—
118C2×D4—48no—
119C2×D4—48no—
120C2×D4—44no—
121C2×D4—44no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D414yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4#4112yes—
181C22×C4#4112yes—
182C4⋊C4—28no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—44no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C42—28no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23C22×D411yescharacteristic
230C23C22×D411yescharacteristic
231C23C22×D411yescharacteristic
232C23C2×#4111yescharacteristic
233C23C2×#4111yescharacteristic
234C23C2×#4112yes—
235C23H(5,𝔽2)12yes—
236C23H(5,𝔽2)12yes—
237C23H(5,𝔽2)11yescharacteristic
238C232−1+412yes—
239C232−1+411yescharacteristic
240D4—416no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4#12114yes—
300C2×C4#12114yes—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22C2×#12111yescharacteristic
317C22C2×#12111yescharacteristic
318C22#52511yescharacteristic
319C22#53511yescharacteristic
320C22#53511yescharacteristic
321C22#53512yes—
322C22#53611yescharacteristic
323C22#53711yescharacteristic
324C22#53712yes—
325C22#53812yes—
326C22#53912yes—
327C22#54012yes—
328C22#54212yes—
329C22D4211yescharacteristic
330C22#54612yes—
331C22#54711yescharacteristic
332C22#54811yescharacteristic
333C22#54812yes—
334C22#55112yes—
335C22#55212yes—
336C22#55311yescharacteristic
337C22#56212yes—
338C22#56312yes—
339C4—24no—
340C4—24no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—44no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4—82no—
349C2—84no—
350C2#241111yescharacteristic
351C2#241611yescharacteristic
352C2#253611yescharacteristic
353C2#261911yescharacteristic
354C2#262011yescharacteristic
355C2#263611yescharacteristic
356C2#267612yes—
357C2#270712yes—
358C2#272312yes—
359C2#276112yes—
360C2#282011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#396C2semidirect
4. C2×#387C2semidirect
5. C2×#382C2semidirect
6. #1444C2semidirect
7. #1441C2semidirect
8. #1440C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
69. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3962
C2×#3942
C2×#3902
C2×#3872
C2×#3822
#14442
#14412
#14402
#14392
#14382