Finite Group Databasebeta

← #22174

Subgroups — #22174

Order 256.

482 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×#436C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, 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#1870C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1797C211yesmaximal, characteristic, complemented
15#1728C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C411yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25#436—24no—
26#435—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—22no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16D414yescomplemented
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
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—41no—
149C2×D4—41no—
150C2×D4—41no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
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—41no—
170C22×C4—41no—
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—
188M16—44no—
189M16—44no—
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—21no—
211C2×C8—21no—
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—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4M1614yescomplemented
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—
257C23—21no—
258C23—21no—
259C23C4×D412yes—
260C23C2×M1612yes—
261C23C23×C411yescharacteristic
262D4—44no—
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×C4C2×M1612yes—
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—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22C2×C4×D411yescharacteristic
348C22#51811yescharacteristic
349C22#52011yescharacteristic
350C22C22×M1611yescharacteristic
351C22C2×#13111yescharacteristic
352C22#56811yescharacteristic
353C22#56811yescharacteristic
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—42no—
370C2—42no—
371C2—81no—
372C2#232511yescharacteristic
373C2D4×M1611yescharacteristic
374C2#293511yescharacteristic
375C2#297411yescharacteristic
376C2#297811yescharacteristic
377C2#298911yescharacteristic
378C2#299311yescharacteristic
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×#436C2semidirect
2. C2×#435C2semidirect
4. #1924C2semidirect
5. #1923C2semidirect
6. #1917C2semidirect
7. #1916C2semidirect
8. #1870C2semidirect
9. #1870C2semidirect
10. #1870C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1797C2semidirect
15. #1728C2semidirect
77. C2×M16D4semidirect
229. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4352
C4×#1212
#19242
#19232
#19172
#19162
#18702
#18692
#18512
#18502
#17972
#17282