Finite Group Databasebeta

← #29542

Subgroups — #29542

Order 256.

373 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2319C211yesmaximal, characteristic, complemented
2#2319C211yesmaximal, characteristic, complemented
3#2088C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic
7#1926C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1920C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic
12#1879C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1832C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#388—21no—
41#388—21no—
42#381—21no—
43#381—21no—
44#380—21no—
45#380—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48#376C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C2×D8—22no—
64C2×D8—22no—
65C2×D8—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—22no—
100#106—22no—
101#106—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#102—22no—
105#102—22no—
106#102—22no—
107#102—22no—
108#102—41no—
109#102—41no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95D411yescharacteristic
117#95D411yescharacteristic
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C24—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4#4112yes—
133C2×D4#4112yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
153D8—44no—
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—
165C2×C8—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—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4#4112yes—
181C4⋊C4#4112yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—81no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—81no—
199C42—81no—
200C23—22no—
201C23—22no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23C22×D411yescharacteristic
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—41no—
258C2×C4—41no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#52511yescharacteristic
286C22#52511yescharacteristic
287C22#53711yescharacteristic
288C22#57611yescharacteristic
289C22#57611yescharacteristic
290C22#57711yescharacteristic
291C22#57711yescharacteristic
292C4—41no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#55111yescharacteristic
305C4#55211yescharacteristic
306C4#55211yescharacteristic
307C4#55511yescharacteristic
308C2—82no—
309C2—82no—
310C2#260911yescharacteristic
311C2#311811yescharacteristic
312C2#311911yescharacteristic
313C2#312211yescharacteristic
314C2#312511yescharacteristic
315C2#325711yescharacteristic
316C2#325711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2319C2semidirect
2. #2319C2semidirect
3. #2088C2semidirect
4. #2067C2semidirect
5. #2065C2semidirect
7. #1926C2semidirect
8. #1926C2semidirect
9. #1920C2semidirect
10. #1918C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1832C2semidirect

Maximal subgroups

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

subgroupindex
#23192
#20882
#20672
#20652
#20542
#19262
#19202
#19182
#18792
#18642
#18622
#18322