Finite Group Databasebeta

← #22172

Subgroups — #22172

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic, complemented
3C4×#121C211yesmaximal, characteristic
4#1924C211yesmaximal, characteristic, complemented
5#1923C211yesmaximal, characteristic, complemented
6#1917C211yesmaximal, characteristic, complemented
7#1916C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1869C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1797C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C411yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25#435—24no—
26C8×D4—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#390C411yescharacteristic
42#389C411yescharacteristic
43#388C411yescharacteristic
44#388C411yescharacteristic
45#386C411yescharacteristic
46#386C411yescharacteristic
47#380C411yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#336C2211yescharacteristic
52#336C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4C2×C411yescharacteristic
62C22×D4C2×C411yescharacteristic
63C22×D4C2×C411yescharacteristic
64C23×C4C2×C411yescharacteristic
65C23×C4C2×C411yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×M16—22no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8D414yescomplemented
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—24no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—41no—
95C4×D4—41no—
96C2×C4⋊C4C2×C411yescharacteristic
97C2×C4⋊C4C2×C411yescharacteristic
98C2×C4⋊C4C2×C411yescharacteristic
99C2×C4⋊C4C2×C411yescharacteristic
100C2×C4⋊C4C2×C411yescharacteristic
101C2×C22⋊C4C2×C411yescharacteristic
102C2×C22⋊C4C2×C411yescharacteristic
103C2×C22⋊C4C2×C411yescharacteristic
104C2×C22⋊C4C2×C411yescharacteristic
105C2×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C22⋊C4C2×C411yescharacteristic
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C2×C411yescharacteristic
114C2×C42C2×C411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131#97—22no—
132C4×C8—22no—
133#95C2×C411yescharacteristic
134#95C2×C411yescharacteristic
135#95C2×C411yescharacteristic
136#95C2×C411yescharacteristic
137#95C2×C411yescharacteristic
138#95C2×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141C24C22×C411yescharacteristic
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—41no—
150C2×D4M1614yescomplemented
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C22×C411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4#4112yes—
186C22×C4C2411yescharacteristic
187M16—44no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—41no—
239C42—41no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C4×D412yes—
258C23C2×M1612yes—
259C23C2×M1612yes—
260C23#13112yes—
261C23C23×C411yescharacteristic
262D4—28no—
263D4—44no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4C4×D412yes—
309C2×C4C2×M1612yes—
310C2×C4#13112yes—
311C2×C4#13112yes—
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C42−1+411yescharacteristic
318C8—22no—
319C8—22no—
320C8—41no—
321C8—41no—
322C8—41no—
323C8—41no—
324C8—41no—
325C8—41no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22#43514yes—
347C22C2×C4×D411yescharacteristic
348C22#51811yescharacteristic
349C22#52011yescharacteristic
350C22C22×M1611yescharacteristic
351C22C2×#13111yescharacteristic
352C22C2×#13111yescharacteristic
353C22C2×#13111yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4#53411yescharacteristic
366C4#53711yescharacteristic
367C4#53811yescharacteristic
368C4#53911yescharacteristic
369C2—24no—
370C2—42no—
371C2—81no—
372C2#232511yescharacteristic
373C2C2×#43511yescharacteristic
374C2#293111yescharacteristic
375C2#297411yescharacteristic
376C2#297711yescharacteristic
377C2#299011yescharacteristic
378C2#299411yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#121C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#435C2semidirect
2. C2×C8×D4C2semidirect
4. #1924C2semidirect
5. #1923C2semidirect
6. #1917C2semidirect
7. #1916C2semidirect
8. #1870C2semidirect
9. #1869C2semidirect
10. #1869C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1797C2semidirect
15. C2×#336C2semidirect
80. C22×C8D4semidirect
150. C2×D4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×C8×D42
C4×#1212
#19242
#19232
#19172
#19162
#18702
#18692
#18512
#18502
#17972
C2×#3362