Finite Group Databasebeta

← #32172

Subgroups — #32172

Order 256.

413 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2544C211yesmaximal, characteristic, complemented
2#2513C211yesmaximal, characteristic, complemented
3#2073C211yesmaximal, characteristic, complemented
4#2055C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2012C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic
9#1982C211yesmaximal, characteristic, complemented
10#1970C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1916C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#397—21no—
37#397—21no—
38#394—21no—
39#394—21no—
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#386—21no—
49#386—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×SD16—41no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×D8—22no—
70C2×D8—22no—
71C2×D8—41no—
72C2×D8—41no—
73C2×D8—41no—
74C2×D8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#127—22no—
80#127—41no—
81#124—22no—
82#124—41no—
83#124—41no—
84#124—41no—
85C4×D4—41no—
86C4×D4—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
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×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112C4⋊C8—41no—
113#103—22no—
114#103—41no—
115#103—41no—
116#102—22no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121#102—41no—
122#102—41no—
123C4×C8—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C24—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
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—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4#4112yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
180SD16—44no—
181D8—44no—
182D8—44no—
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—
194C2×C8#4112yes—
195C4⋊C4—22no—
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—41no—
205C4⋊C4—81no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—81no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
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—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252D4—22no—
253D4—22no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—84no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
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—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—84no—
299C2×C4#13512yes—
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4H(5,𝔽2)11yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C8—22no—
307C8—41no—
308C8—41no—
309C8—41no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#53511yescharacteristic
332C22#54611yescharacteristic
333C22#55011yescharacteristic
334C22C2×#13511yescharacteristic
335C22#57511yescharacteristic
336C22#57611yescharacteristic
337C22#57611yescharacteristic
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4#53811yescharacteristic
350C4#53811yescharacteristic
351C4#55311yescharacteristic
352C4#56011yescharacteristic
353C2—82no—
354C2—161no—
355C2—161no—
356C2#271411yescharacteristic
357C2#321211yescharacteristic
358C2#322011yescharacteristic
359C2#329711yescharacteristic
360C2#331911yescharacteristic
361C2#339611yescharacteristic
362C2#340311yescharacteristic
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. #2544C2semidirect
2. #2513C2semidirect
3. #2073C2semidirect
4. #2055C2semidirect
5. #2042C2semidirect
6. #2012C2semidirect
7. #2012C2semidirect
9. #1982C2semidirect
10. #1970C2semidirect
11. #1958C2semidirect
12. #1928C2semidirect
13. #1916C2semidirect
14. #1879C2semidirect
15. #1862C2semidirect

Maximal subgroups

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

subgroupindex
#25442
#25132
#20732
#20552
#20422
#20122
#20112
#19822
#19702
#19582
#19282
#19162
#18792
#18622