Finite Group Databasebeta

← #30903

Subgroups — #30903

Order 256.

468 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3#2548C211yesmaximal, characteristic, complemented
4#2452C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic
11#1972C211yesmaximal, characteristic, complemented
12#1917C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1865C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#489—24no—
27#463—24no—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#396—21no—
42#394—21no—
43#394—21no—
44#393—21no—
45#390—21no—
46#390—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#385—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—41no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C2×D8—41no—
77C2×D8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—41no—
84#127—41no—
85#127—41no—
86#124—22no—
87#124—22no—
88#124—41no—
89#124—41no—
90C22⋊Q8—44no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×D4—22no—
94C4×D4—41no—
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×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#106—41no—
123C4⋊C8—22no—
124#103—22no—
125#103—41no—
126#103D414yescomplemented
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132#102—41no—
133#102—41no—
134#97—22no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—81no—
150C2×Q8C2×D412yes—
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—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4#4112yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2411yescharacteristic
190SD16—44no—
191D8—44no—
192D8—44no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8C2×D412yes—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4C2×D412yes—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
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—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—84no—
266D4—22no—
267D4—22no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—84no—
279D4—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
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—84no—
313C2×C4AGL(1,ℤ/8)12yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C4H(5,𝔽2)11yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C42−1+411yescharacteristic
320C8—22no—
321C8—41no—
322C8—41no—
323C8—41no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
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#53511yescharacteristic
346C22D4211yescharacteristic
347C22#54811yescharacteristic
348C22C2×AGL(1,ℤ/8)11yescharacteristic
349C22#57511yescharacteristic
350C22#57611yescharacteristic
351C22#57711yescharacteristic
352C4—22no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4#53711yescharacteristic
364C4#53811yescharacteristic
365C4#54611yescharacteristic
366C4#56211yescharacteristic
367C2—82no—
368C2—161no—
369C2—161no—
370C2#267611yescharacteristic
371C2#321311yescharacteristic
372C2#322411yescharacteristic
373C2#327111yescharacteristic
374C2#327411yescharacteristic
375C2#337011yescharacteristic
376C2#337311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
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×#463C2semidirect
3. #2548C2semidirect
4. #2452C2semidirect
5. #2065C2semidirect
6. #2042C2semidirect
7. #2041C2semidirect
8. #2023C2semidirect
9. #2012C2semidirect
11. #1972C2semidirect
12. #1917C2semidirect
13. #1880C2semidirect
14. #1879C2semidirect
15. #1865C2semidirect
126. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4632
#25482
#24522
#20652
#20422
#20412
#20232
#20122
#20112
#19722
#19172
#18802
#18792
#18652