Finite Group Databasebeta

← #28873

Subgroups — #28873

Order 256.

431 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2512C211yesmaximal, characteristic, complemented
2#2445C211yesmaximal, characteristic, complemented
3#2083C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2011C211yesmaximal, characteristic
7#1970C211yesmaximal, characteristic, complemented
8#1970C211yesmaximal, characteristic, complemented
9#1957C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14C8×C4⋊C4C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#126—21no—
20C2×#123—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#397—21no—
37#396—21no—
38#394—21no—
39#394—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#384C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×D8—22no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#127—22no—
76#127—41no—
77#126—44no—
78#123—44no—
79C4×D4—22no—
80C4×D4—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
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×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—41no—
110#107—41no—
111C4⋊C8—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—41no—
118#102—41no—
119#102—41no—
120C4×C8—22no—
121C4×C8—41no—
122C4×C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24—21no—
129C24—21no—
130C24—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—81no—
153C2×D4—81no—
154C2×D4#4112yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—81no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2411yescharacteristic
170D8—44no—
171D8—44no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8#4112yes—
182C2×C8#4112yes—
183C2×C8#4112yes—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4D814yescomplemented
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C23—22no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
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—
241C23C22×D411yescharacteristic
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—84no—
255D4—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
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—84no—
286C2×C4C2×D812yes—
287C2×C4C2×D812yes—
288C2×C4C2×D812yes—
289C2×C4#13512yes—
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C8—22no—
297C8—22no—
298C8—22no—
299C8—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#53211yescharacteristic
322C22#54611yescharacteristic
323C22#55011yescharacteristic
324C22C22×D811yescharacteristic
325C22C2×#13511yescharacteristic
326C22#57611yescharacteristic
327C22#57711yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4#52911yescharacteristic
340C4#53811yescharacteristic
341C4#55111yescharacteristic
342C4#55311yescharacteristic
343C2—82no—
344C2—161no—
345C2—161no—
346C2#255411yescharacteristic
347C2#316911yescharacteristic
348C2#317911yescharacteristic
349C2#329411yescharacteristic
350C2#329711yescharacteristic
351C2#339111yescharacteristic
352C2#339411yescharacteristic
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. #2512C2semidirect
2. #2445C2semidirect
3. #2083C2semidirect
4. #2067C2semidirect
5. #2054C2semidirect
7. #1970C2semidirect
8. #1970C2semidirect
9. #1957C2semidirect
10. #1934C2semidirect
11. #1934C2semidirect
12. #1925C2semidirect
13. #1879C2semidirect
14. C8×C4⋊C4C2semidirect
15. C4×#102C2semidirect
199. C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#25122
#24452
#20832
#20672
#20542
#20112
#19702
#19572
#19342
#19252
#18792
C8×C4⋊C42
C4×#1022