Finite Group Databasebeta

← #30937

Subgroups — #30937

Order 256.

439 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic
4#2429C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic
11#1981C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1894C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124—21no—
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26#477—24no—
27#468—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#398—21no—
42#396—21no—
43#394—21no—
44#393—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#389—21no—
49#389C2211yescharacteristic
50#388—21no—
51#388—21no—
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×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×SD16—41no—
73C2×M16—22no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#124—44no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89#121—22no—
90#121—41no—
91C4×Q8—22no—
92C4×Q8—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#107D414yes—
121C4⋊C8—22no—
122C4⋊C8—41no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—41no—
127#103—41no—
128#103—41no—
129#102—22no—
130#102—41no—
131C22⋊C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×Q8#4112yes—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—81no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2411yescharacteristic
184SD16—44no—
185SD16—44no—
186M16—44no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8C2×D412yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4C2×D412yes—
215C4⋊C4C2×D412yes—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—42no—
231C42—42no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—84no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—84no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
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—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4AGL(1,ℤ/8)12yes—
300C2×C4#13712yes—
301C2×C4C22×D411yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C42−1+411yescharacteristic
307C8—22no—
308C8—41no—
309C8—41no—
310C8—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#53711yescharacteristic
326C22D4211yescharacteristic
327C22#54611yescharacteristic
328C22C2×AGL(1,ℤ/8)11yescharacteristic
329C22C2×#13711yescharacteristic
330C22#57711yescharacteristic
331C22#57811yescharacteristic
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—42no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4#53511yescharacteristic
345C4#53811yescharacteristic
346C4#54811yescharacteristic
347C4#56211yescharacteristic
348C2—81no—
349C2—161no—
350C2#267611yescharacteristic
351C2#325411yescharacteristic
352C2#325811yescharacteristic
353C2#327011yescharacteristic
354C2#327211yescharacteristic
355C2#330811yescharacteristic
356C2#332911yescharacteristic
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×#477C2semidirect
4. #2429C2semidirect
5. #2067C2semidirect
6. #2055C2semidirect
7. #2036C2semidirect
8. #2024C2semidirect
9. #2013C2semidirect
11. #1981C2semidirect
12. #1918C2semidirect
13. #1894C2semidirect
14. #1880C2semidirect
15. #1851C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4682
#24912
#24292
#20672
#20552
#20362
#20242
#20132
#20002
#19812
#19182
#18942
#18802
#18512