Finite Group Databasebeta

← #30892

Subgroups — #30892

Order 256.

494 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic
4#2453C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1917C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#125C2211yescharacteristic
23C2×#121—21no—
24C2×#121—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#490—24no—
28#460—24no—
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#400—21no—
43#398—21no—
44#392—21no—
45#392—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389—21no—
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#387—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#127—22no—
86#127—22no—
87#127—41no—
88#127—41no—
89#127—41no—
90#127—41no—
91#125—22no—
92#125—41no—
93#121—44no—
94#121—44no—
95C4×Q8—22no—
96C4×Q8—41no—
97C4×D4—41no—
98C4×D4—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—41no—
120C4⋊C8—22no—
121#103—41no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129#102—41no—
130#102D414yescomplemented
131#97—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95D411yescharacteristic
139#95D411yescharacteristic
140C24—21no—
141C24—21no—
142C24—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8#4112yes—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
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—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4C2×D412yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2411yescharacteristic
200SD16—44no—
201SD16—44no—
202SD16—44no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8C2×D412yes—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4C2×D412yes—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C22×D411yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268D4—22no—
269D4—22no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—84no—
281D4—84no—
282D4—84no—
283D4—84no—
284D4—84no—
285D4—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—84no—
318C2×C4AGL(1,ℤ/8)12yes—
319C2×C4AGL(1,ℤ/8)12yes—
320C2×C4C22×D411yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C42−1+411yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22#53511yescharacteristic
352C22D4211yescharacteristic
353C22#54811yescharacteristic
354C22C2×AGL(1,ℤ/8)11yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22#57711yescharacteristic
357C22#57711yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4#53711yescharacteristic
370C4#53811yescharacteristic
371C4#54611yescharacteristic
372C4#56211yescharacteristic
373C2—82no—
374C2—161no—
375C2—161no—
376C2#267611yescharacteristic
377C2#321711yescharacteristic
378C2#322211yescharacteristic
379C2#326511yescharacteristic
380C2#327211yescharacteristic
381C2#336811yescharacteristic
382C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
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×#490C2semidirect
2. C2×#460C2semidirect
4. #2453C2semidirect
5. #2067C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2025C2semidirect
9. #2014C2semidirect
10. #2014C2semidirect
11. #1973C2semidirect
12. #1917C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. #1862C2semidirect
130. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4602
#25412
#24532
#20672
#20412
#20252
#20142
#19732
#19172
#18802
#18772
#18622