Finite Group Databasebeta

← #30902

Subgroups — #30902

Order 256.

443 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3#2545C211yesmaximal, characteristic, complemented
4#2452C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic
10#2012C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic, complemented
12#1917C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic
15#1863C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#489—24no—
27#461—24no—
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#396—21no—
42#396—21no—
43#396—21no—
44#393—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#388—21no—
53#385—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—41no—
72C2×SD16—41no—
73C2×D8—22no—
74C2×D8—41no—
75C2×D8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#124—22no—
85#124—41no—
86#124—41no—
87#124—41no—
88C22⋊Q8—44no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×D4—22no—
92C4×D4—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⋊C4D411yescharacteristic
100C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—41no—
121C4⋊C8—22no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103—41no—
126#103—41no—
127#103D414yes—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132#97—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
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×Q8C2×D412yes—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4#4112yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2411yescharacteristic
184SD16—44no—
185SD16—44no—
186D8—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—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4C2×D412yes—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
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—21no—
231C42—21no—
232C42—42no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—84no—
257D4—22no—
258D4—22no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
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×C4—41no—
300C2×C4—84no—
301C2×C4AGL(1,ℤ/8)12yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C42−1+411yescharacteristic
308C8—22no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22#53511yescharacteristic
327C22D4211yescharacteristic
328C22#54811yescharacteristic
329C22C2×AGL(1,ℤ/8)11yescharacteristic
330C22#57511yescharacteristic
331C22#57711yescharacteristic
332C22#57811yescharacteristic
333C4—22no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#53711yescharacteristic
346C4#53811yescharacteristic
347C4#54611yescharacteristic
348C4#56211yescharacteristic
349C2—82no—
350C2—161no—
351C2#267611yescharacteristic
352C2#321411yescharacteristic
353C2#322511yescharacteristic
354C2#326511yescharacteristic
355C2#327711yescharacteristic
356C2#336711yescharacteristic
357C2#337611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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
1. C2×#489C2semidirect
2. C2×#461C2semidirect
3. #2545C2semidirect
4. #2452C2semidirect
5. #2067C2semidirect
6. #2043C2semidirect
7. #2042C2semidirect
8. #2024C2semidirect
11. #1973C2semidirect
12. #1917C2semidirect
13. #1880C2semidirect
15. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4612
#25452
#24522
#20672
#20432
#20422
#20242
#20122
#19732
#19172
#18802
#18792
#18632