Finite Group Databasebeta

← #23329

Subgroups — #23329

Order 256.

427 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#121C211yesmaximal, characteristic
2#1955C211yesmaximal, characteristic, complemented
3#1954C211yesmaximal, characteristic, complemented
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#1869C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1817C211yesmaximal, characteristic, complemented
15#1797C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121C411yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#390C411yescharacteristic
40#389C411yescharacteristic
41#388C411yescharacteristic
42#388C411yescharacteristic
43#386C411yescharacteristic
44#386C411yescharacteristic
45#380C411yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C22⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#336C2211yescharacteristic
50#336C2211yescharacteristic
51#336C2211yescharacteristic
52#336C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2×C411yescharacteristic
60C22×D4C2×C411yescharacteristic
61C22×D4C2×C411yescharacteristic
62C23×C4C2×C411yescharacteristic
63C23×C4C2×C411yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#121—24no—
82C4×D4—41no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—41no—
87C4×D4—41no—
88C2×C4⋊C4C2×C411yescharacteristic
89C2×C4⋊C4C2×C411yescharacteristic
90C2×C4⋊C4C2×C411yescharacteristic
91C2×C4⋊C4C2×C411yescharacteristic
92C2×C4⋊C4C2×C411yescharacteristic
93C2×C22⋊C4C2×C411yescharacteristic
94C2×C22⋊C4C2×C411yescharacteristic
95C2×C22⋊C4C2×C411yescharacteristic
96C2×C22⋊C4C2×C411yescharacteristic
97C2×C22⋊C4C2×C411yescharacteristic
98C2×C22⋊C4C2×C411yescharacteristic
99C2×C22⋊C4C2×C411yescharacteristic
100C2×C22⋊C4C2×C411yescharacteristic
101C2×C22⋊C4C2×C411yescharacteristic
102C2×C22⋊C4C2×C411yescharacteristic
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C2×C411yescharacteristic
106C2×C42C2×C411yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125#97—41no—
126C4×C8—41no—
127#95C2×C411yescharacteristic
128#95C2×C411yescharacteristic
129#95C2×C411yescharacteristic
130#95C2×C411yescharacteristic
131#95C2×C411yescharacteristic
132#95C2×C411yescharacteristic
133C24C22×C411yescharacteristic
134C24C22×C411yescharacteristic
135C24C22×C411yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
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—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C22×C411yescharacteristic
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C411yescharacteristic
172C22×C4C22×C411yescharacteristic
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C2411yescharacteristic
181M16—44no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
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—44no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C22⋊C4—22no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4M1614yescomplemented
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—41no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23C2×M1612yes—
252C23#13112yes—
253C23C23×C411yescharacteristic
254D4—44no—
255D4—44no—
256D4—44no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
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—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
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—44no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4C2×M1612yes—
302C2×C4C2×M1612yes—
303C2×C4#13112yes—
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—41no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#51811yescharacteristic
340C22#51811yescharacteristic
341C22#52011yescharacteristic
342C22C22×M1611yescharacteristic
343C22C2×#13111yescharacteristic
344C22C2×#13111yescharacteristic
345C22#56811yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4#56011yescharacteristic
358C4#56011yescharacteristic
359C4#56211yescharacteristic
360C4#56211yescharacteristic
361C2—42no—
362C2—42no—
363C2—81no—
364C2#237711yescharacteristic
365C2#297411yescharacteristic
366C2#297411yescharacteristic
367C2#297611yescharacteristic
368C2#298111yescharacteristic
369C2#298911yescharacteristic
370C2#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
2. #1955C2semidirect
3. #1954C2semidirect
4. #1924C2semidirect
5. #1923C2semidirect
6. #1917C2semidirect
7. #1916C2semidirect
8. #1870C2semidirect
9. #1870C2semidirect
10. #1869C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1817C2semidirect
15. #1797C2semidirect
221. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#19552
#19542
#19242
#19232
#19172
#19162
#18702
#18692
#18512
#18502
#18172
#17972