Finite Group Databasebeta

← #30664

Subgroups — #30664

Order 256.

421 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic
4#2423C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic
11#1948C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1850C211yesmaximal, characteristic, complemented
15#1807C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#475—24no—
24#465—24no—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#399—21no—
40#398—21no—
41#394—21no—
42#393—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#387—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#124—44no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81#121—22no—
82#121—41no—
83C4×Q8—22no—
84C4×Q8—41no—
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⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#106—22no—
112#106—22no—
113#106—22no—
114#106—41no—
115#106—41no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118#103—22no—
119#103—41no—
120#103D414yes—
121#102—22no—
122#102—41no—
123#102—41no—
124C22⋊C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—81no—
143C2×Q8C2×D412yes—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—81no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—44no—
173C2×C8—21no—
174C2×C8—21no—
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—44no—
184C2×C8C2×D412yes—
185C2×C8#4112yes—
186C4⋊C4—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—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4C2×D412yes—
204C22⋊C4—41no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4SD1614yescomplemented
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C42—42no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C2×SD1612yes—
234C23C22×D411yescharacteristic
235Q8—22no—
236Q8—22no—
237Q8—41no—
238Q8—41no—
239Q8—84no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
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—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—41no—
283C2×C4C2×SD1612yes—
284C2×C4C2×SD1612yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C2×C42−1+411yescharacteristic
291C8—22no—
292C8—22no—
293C8—41no—
294C8—41no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#53711yescharacteristic
310C22D4211yescharacteristic
311C22#54711yescharacteristic
312C22C22×SD1611yescharacteristic
313C22#57511yescharacteristic
314C22#57711yescharacteristic
315C22#57711yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—41no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#53611yescharacteristic
329C4#54211yescharacteristic
330C4#54611yescharacteristic
331C4#56211yescharacteristic
332C2—42no—
333C2—161no—
334C2#267211yescharacteristic
335C2#324611yescharacteristic
336C2#326311yescharacteristic
337C2#327111yescharacteristic
338C2D4×SD1611yescharacteristic
339C2#333511yescharacteristic
340C2#334811yescharacteristic
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
1. C2×#475C2semidirect
4. #2423C2semidirect
5. #2094C2semidirect
6. #2065C2semidirect
7. #2058C2semidirect
8. #2055C2semidirect
9. #2035C2semidirect
11. #1948C2semidirect
12. #1927C2semidirect
13. #1880C2semidirect
14. #1850C2semidirect
15. #1807C2semidirect
213. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4652
#24912
#24232
#20942
#20652
#20582
#20552
#20352
#20342
#19482
#19272
#18802
#18502
#18072