Finite Group Databasebeta

← #29120

Subgroups — #29120

Order 256.

425 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2628C211yesmaximal, characteristic, complemented
2#2311C211yesmaximal, characteristic
3#2074C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#1936C211yesmaximal, characteristic, complemented
8#1935C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12#1823C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic
14#1788C211yesmaximal, characteristic
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×#103C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#400—21no—
33#399—21no—
34#395—21no—
35#393—21no—
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#386—21no—
43#386—21no—
44#382C2211yescharacteristic
45#380—21no—
46#380—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8—21no—
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78#117—41no—
79#117—41no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#103—22no—
105#103—22no—
106#103—41no—
107#103—41no—
108#102—22no—
109#102—22no—
110#102—41no—
111#102—41no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114#97—41no—
115C4×C8—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128#41—44no—
129#41—44no—
130#41—44no—
131#41—44no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8#4112yes—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4#4112yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4#4112yes—
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23#12412yes—
231C23C22×D411yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—41no—
235Q8—41no—
236Q8—84no—
237D4—22no—
238D4—22no—
239D4—48no—
240D4—41no—
241D4—41no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—48no—
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—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—84no—
283C2×C4#12412yes—
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C8—41no—
291C8—41no—
292C8—41no—
293C8—41no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#52511yescharacteristic
309C22C2×#12411yescharacteristic
310C22#53411yescharacteristic
311C22#57611yescharacteristic
312C22#57711yescharacteristic
313C22#57711yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#55111yescharacteristic
328C4#55111yescharacteristic
329C4#55111yescharacteristic
330C4#55111yescharacteristic
331C2—42no—
332C2—82no—
333C2#256111yescharacteristic
334C2#312411yescharacteristic
335C2#312511yescharacteristic
336C2#313611yescharacteristic
337C2#313711yescharacteristic
338C2#319811yescharacteristic
339C2#320111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
1. #2628C2semidirect
3. #2074C2semidirect
4. #2073C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #1936C2semidirect
8. #1935C2semidirect
9. #1934C2semidirect
10. #1933C2semidirect
11. #1824C2semidirect
12. #1823C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
#26282
#23112
#20742
#20732
#20242
#20232
#19362
#19352
#19342
#19332
#18242
#18232
#17882
#17502