Finite Group Databasebeta

← #33103

Subgroups — #33103

Order 256.

387 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2624C211yesmaximal, characteristic
2#2274C211yesmaximal, characteristic, complemented
3#1926C211yesmaximal, characteristic, complemented
4#1925C211yesmaximal, characteristic, complemented
5#1925C211yesmaximal, characteristic, complemented
6#1925C211yesmaximal, characteristic, complemented
7#1921C211yesmaximal, characteristic, complemented
8#1921C211yesmaximal, characteristic, complemented
9#1919C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic
12#1874C211yesmaximal, characteristic
13#1854C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1818C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#399—21no—
31#397—21no—
32#394—21no—
33#394—21no—
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#380C2211yescharacteristic
39C4×C4⋊C4—21no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4—21no—
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#121—22no—
68#121—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—22no—
104#107—22no—
105#107—22no—
106#106—22no—
107#102—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120C24—21no—
121C24C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—81no—
130C2×D4—81no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4#4112yes—
151C22×C4#4112yes—
152C22×C4C2411yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4#4112yes—
181C4⋊C4#4112yes—
182C4⋊C4#4112yes—
183C4⋊C4#4112yes—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—42no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C42—24no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—81no—
199C42—81no—
200C42—81no—
201C42—81no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23#12312yes—
216C23C22×D411yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—84no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4#12312yes—
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22C2×#12311yescharacteristic
288C22#52511yescharacteristic
289C22#52511yescharacteristic
290C22#57611yescharacteristic
291C22#57611yescharacteristic
292C22#57711yescharacteristic
293C22#57811yescharacteristic
294C4—22no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4#52911yescharacteristic
307C4#52911yescharacteristic
308C4#52911yescharacteristic
309C4#52911yescharacteristic
310C2—42no—
311C2—161no—
312C2#285311yescharacteristic
313C2#309611yescharacteristic
314C2#309811yescharacteristic
315C2#311911yescharacteristic
316C2#312111yescharacteristic
317C2#312211yescharacteristic
318C2#312411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2274C2semidirect
3. #1926C2semidirect
4. #1925C2semidirect
5. #1925C2semidirect
6. #1925C2semidirect
7. #1921C2semidirect
8. #1921C2semidirect
9. #1919C2semidirect
10. #1918C2semidirect
13. #1854C2semidirect
14. #1852C2semidirect
15. #1818C2semidirect

Maximal subgroups

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

subgroupindex
#26242
#22742
#19262
#19252
#19212
#19192
#19182
#18752
#18742
#18542
#18522
#18182