Finite Group Databasebeta

← #32078

Subgroups — #32078

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2541C211yesmaximal, characteristic
2#2506C211yesmaximal, characteristic, complemented
3#2093C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1957C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1916C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#125—21no—
20C2×#121—21no—
21C2×#121—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#398—21no—
41#392—21no—
42#392—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#387—21no—
50#384C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#127—22no—
74#127—22no—
75#127—41no—
76#127—41no—
77#127—41no—
78#127—41no—
79#125—44no—
80#121—44no—
81#121—44no—
82C4×D4—41no—
83C4×D4—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—41no—
107#107—41no—
108C4⋊C8—41no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113#102—41no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121C4×C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95D411yescharacteristic
128#95D411yescharacteristic
129C24—21no—
130C24—21no—
131C24—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4#4112yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2411yescharacteristic
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—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4D814yes—
212C4⋊C4#4112yes—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—42no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C22×D411yescharacteristic
247D4—22no—
248D4—22no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—84no—
260D4—84no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—84no—
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—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4C2×D812yes—
296C2×C4C2×D812yes—
297C2×C4C2×D812yes—
298C2×C4AGL(1,ℤ/8)12yes—
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C2×C4H(5,𝔽2)11yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C42−1+411yescharacteristic
305C8—41no—
306C8—41no—
307C8—41no—
308C8—41no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#53511yescharacteristic
331C22#54611yescharacteristic
332C22#54811yescharacteristic
333C22C22×D811yescharacteristic
334C22C2×AGL(1,ℤ/8)11yescharacteristic
335C22#57711yescharacteristic
336C22#57711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4#53911yescharacteristic
349C4#54011yescharacteristic
350C4#55311yescharacteristic
351C4#56211yescharacteristic
352C2—82no—
353C2—161no—
354C2—161no—
355C2#271111yescharacteristic
356C2#320511yescharacteristic
357C2#322211yescharacteristic
358C2#329411yescharacteristic
359C2#331011yescharacteristic
360C2#335811yescharacteristic
361C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2506C2semidirect
3. #2093C2semidirect
4. #2067C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2041C2semidirect
9. #1982C2semidirect
10. #1957C2semidirect
11. #1934C2semidirect
12. #1925C2semidirect
13. #1916C2semidirect
14. #1862C2semidirect
15. #1862C2semidirect

Maximal subgroups

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

subgroupindex
#25412
#25062
#20932
#20672
#20412
#19822
#19572
#19342
#19252
#19162
#18622