Finite Group Databasebeta

← #31636

Subgroups — #31636

Order 256.

395 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2542C211yesmaximal, characteristic
2#2446C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2072C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic
6#2024C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic
9#1956C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic
11#1916C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic
14#1877C211yesmaximal, characteristic
15#1803C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#400—21no—
35#399—21no—
36#399—21no—
37#398—21no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#387—21no—
45#387—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×SD16—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C22⋊Q8—44no—
79C22⋊Q8—44no—
80C4×Q8—22no—
81C4×Q8—41no—
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⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C4⋊C8—41no—
110#103—41no—
111#103—41no—
112#103—41no—
113#103—41no—
114#103—41no—
115#103—41no—
116#102—22no—
117#102—41no—
118C4×C8—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8#4112yes—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4#4112yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167SD16—44no—
168SD16—44no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4SD1614yescomplemented
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22×D411yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—84no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
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—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—84no—
271C2×C4C2×SD1612yes—
272C2×C4C2×SD1612yes—
273C2×C4C2×SD1612yes—
274C2×C4AGL(1,ℤ/8)12yes—
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—41no—
282C8—41no—
283C8—41no—
284C8—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22#53511yescharacteristic
293C22#54611yescharacteristic
294C22#54811yescharacteristic
295C22C22×SD1611yescharacteristic
296C22C2×AGL(1,ℤ/8)11yescharacteristic
297C22#57811yescharacteristic
298C22#57811yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#53811yescharacteristic
313C4#53911yescharacteristic
314C4#55111yescharacteristic
315C4#56211yescharacteristic
316C2—82no—
317C2#269611yescharacteristic
318C2#320811yescharacteristic
319C2#322311yescharacteristic
320C2#329811yescharacteristic
321C2#330811yescharacteristic
322C2#335911yescharacteristic
323C2#336711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #2446C2semidirect
3. #2099C2semidirect
4. #2072C2semidirect
6. #2024C2semidirect
7. #2024C2semidirect
9. #1956C2semidirect
11. #1916C2semidirect
15. #1803C2semidirect
201. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#25422
#24462
#20992
#20722
#20552
#20242
#20132
#19562
#19342
#19162
#18802
#18772
#18032