Finite Group Databasebeta

← #30114

Subgroups — #30114

Order 256.

410 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2850C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2097C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2015C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic
11#1927C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15#1817C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C22×C42C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#387—21no—
43#386—21no—
44#386—21no—
45#385—21no—
46#382C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4—21no—
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4C2311yescharacteristic
66C2×Q16—22no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×SD16—22no—
70C2×SD16—41no—
71C2×SD16—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#124—41no—
77#124—41no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82#121—41no—
83#121—41no—
84C4×Q8—41no—
85C4×Q8—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
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×C42—21no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42D411yescharacteristic
108C2×C42D411yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—41no—
113#103—22no—
114#103—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#102—22no—
119#102—41no—
120#102—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8#4112yes—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
180Q16—44no—
181SD16—44no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—24no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4#4112yes—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—41no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23#13712yes—
238C23C22×D411yescharacteristic
239Q8—22no—
240Q8—22no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—84no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—44no—
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×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4#13512yes—
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#52511yescharacteristic
319C22#53411yescharacteristic
320C22#53511yescharacteristic
321C22C2×#13511yescharacteristic
322C22C2×#13711yescharacteristic
323C22#57511yescharacteristic
324C22#57511yescharacteristic
325C4—22no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4#53811yescharacteristic
338C4#53811yescharacteristic
339C4#54011yescharacteristic
340C4#54011yescharacteristic
341C2—42no—
342C2—161no—
343C2#263511yescharacteristic
344C2#310211yescharacteristic
345C2#311711yescharacteristic
346C2#319011yescharacteristic
347C2#319711yescharacteristic
348C2#321211yescharacteristic
349C2#321511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Q8C2semidirect
3. #2101C2semidirect
4. #2097C2semidirect
5. #2042C2semidirect
6. #2036C2semidirect
8. #2015C2semidirect
9. #2001C2semidirect
11. #1927C2semidirect
12. #1925C2semidirect
13. #1880C2semidirect
14. #1878C2semidirect
15. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28502
C4×C22⋊Q82
#21012
#20972
#20422
#20362
#20342
#20152
#20012
#20002
#19272
#19252
#18802
#18782
#18172