Finite Group Databasebeta

← #27170

Subgroups — #27170

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2420C211yesmaximal, characteristic
2#2347C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic
7#1949C211yesmaximal, characteristic, complemented
8#1941C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic, complemented
11#1921C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#397—21no—
32#394—21no—
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#386—21no—
40#383—21no—
41#382C2211yescharacteristic
42#381—21no—
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
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—41no—
69C4×D4—22no—
70C4×D4—41no—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4D411yescharacteristic
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98#107—41no—
99#106—22no—
100#106—41no—
101C4⋊C8—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105#102—22no—
106C22⋊C8—22no—
107C22⋊C8—41no—
108C4×C8—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C24C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—81no—
130C2×D4#4112yes—
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—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
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#4112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
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—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4#4112yes—
181C4⋊C4#4112yes—
182C22⋊C4—24no—
183C22⋊C4—22no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C42—24no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—81no—
197C42—81no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23C22×D411yescharacteristic
213D4—22no—
214D4—22no—
215D4—41no—
216D4—41no—
217D4—84no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
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—84no—
258C2×C4#13512yes—
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C2×C42−1+411yescharacteristic
265C8—22no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#52511yescharacteristic
284C22#54611yescharacteristic
285C22#54711yescharacteristic
286C22C2×#13511yescharacteristic
287C22#57511yescharacteristic
288C22#57611yescharacteristic
289C22#57711yescharacteristic
290C4—22no—
291C4—41no—
292C4—41no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#52911yescharacteristic
303C4#54211yescharacteristic
304C4#55211yescharacteristic
305C4#55311yescharacteristic
306C2—81no—
307C2—82no—
308C2#247011yescharacteristic
309C2#310211yescharacteristic
310C2#311911yescharacteristic
311C2#329711yescharacteristic
312C2#331111yescharacteristic
313C2#334011yescharacteristic
314C2#335511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2347C2semidirect
3. #2101C2semidirect
4. #2075C2semidirect
5. #2065C2semidirect
7. #1949C2semidirect
8. #1941C2semidirect
9. #1925C2semidirect
10. #1923C2semidirect
11. #1921C2semidirect
13. #1862C2semidirect
14. #1854C2semidirect
15. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
#24202
#23472
#21012
#20752
#20652
#20342
#19492
#19412
#19252
#19232
#19212
#18752
#18622
#18542
C4×#1022