Finite Group Databasebeta

← #22579

Subgroups — #22579

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#417C211yesmaximal, characteristic
2#2345C211yesmaximal, characteristic, complemented
3#2341C211yesmaximal, characteristic
4#1936C211yesmaximal, characteristic, complemented
5#1934C211yesmaximal, characteristic, complemented
6#1926C211yesmaximal, characteristic, complemented
7#1925C211yesmaximal, characteristic, complemented
8#1865C211yesmaximal, characteristic, complemented
9#1863C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23#417C418yescomplemented
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#390—21no—
36#390—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#387—21no—
42#387—21no—
43#380—21no—
44#380—21no—
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41C2×C414yes—
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×M16—41no—
73C2×M16—41no—
74C2×M16C2×C414yes—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—41no—
82C4×Q8—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2×C414yes—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—41no—
111#106—41no—
112#103—22no—
113#103—22no—
114#103C2×C414yes—
115#103C2×C414yes—
116#102—41no—
117#102—41no—
118#102C2×C414yes—
119#102C2×C414yes—
120C22⋊C8—22no—
121#97—41no—
122C4×C8—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C24C2×D411yescharacteristic
133#41—28no—
134#41—44no—
135#41—44no—
136#41—44no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8C22×C412yes—
141C2×Q8#4112yes—
142C2×Q8#4112yes—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4C22×C412yes—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C22×C412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4#4112yes—
176C22×C4C2411yescharacteristic
177M16—44no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C22×C412yes—
189C2×C8C22×C412yes—
190C4⋊C4—24no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4C22×C412yes—
206C4⋊C4C22×C412yes—
207C22⋊C4—22no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42—42no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23C4×D412yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8#11714yes—
242D4—22no—
243D4—22no—
244D4—48no—
245D4—41no—
246D4—41no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
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—48no—
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—41no—
286C2×C4—84no—
287C2×C4C4×D412yes—
288C2×C4#13712yes—
289C2×C4C23×C411yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C8—22no—
296C8—22no—
297C8—41no—
298C8—41no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×C4×D411yescharacteristic
314C22#54611yescharacteristic
315C22#54611yescharacteristic
316C22C2×#13711yescharacteristic
317C22#57511yescharacteristic
318C22#57611yescharacteristic
319C22#57711yescharacteristic
320C4—22no—
321C4—24no—
322C4—22no—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4C2×#11711yescharacteristic
333C4#51811yescharacteristic
334C4#55111yescharacteristic
335C4#55111yescharacteristic
336C2—42no—
337C2—82no—
338C2#234211yescharacteristic
339C2C4×#13711yescharacteristic
340C2#294611yescharacteristic
341C2#330911yescharacteristic
342C2#331611yescharacteristic
343C2#332611yescharacteristic
344C2#333311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2345C2semidirect
4. #1936C2semidirect
5. #1934C2semidirect
6. #1926C2semidirect
7. #1925C2semidirect
8. #1865C2semidirect
9. #1863C2semidirect
10. #1823C2semidirect
11. #1809C2semidirect
13. #1763C2semidirect
14. C4×#103C2semidirect
15. #1753C2semidirect
23. #417C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4172
#23452
#23412
#19362
#19342
#19262
#19252
#18652
#18632
#18232
#18092
#17882
#17632
C4×#1032
#17532