Finite Group Databasebeta

← #22865

Subgroups — #22865

Order 256.

557 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#448C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×#419C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6#1968C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1865C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#448C418yescomplemented
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29#419—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#390—21no—
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#380—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2×C414yescomplemented
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16C2×C414yes—
81C2×SD16C2×C414yes—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—22no—
90#121C2×C414yes—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—22no—
96C4×D4—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—41no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103C2×C414yescomplemented
127#103C2×C414yescomplemented
128#103D414yescomplemented
129#102—22no—
130#102—41no—
131C22⋊C8C2×C414yescomplemented
132#97—22no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95D411yescharacteristic
142#95D411yescharacteristic
143C24—21no—
144C24C2×D411yescharacteristic
145C2×Q8—24no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—28no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8C22×C412yes—
161C2×Q8C22×C412yes—
162C2×Q8C2×D412yes—
163C2×Q8#4112yes—
164C2×D4—24no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4C22×C412yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—81no—
190C22×C4C22×C412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
198SD16—28no—
199SD16—44no—
200M16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8C22×C412yes—
211C2×C8C22×C412yes—
212C2×C8C2×D412yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4C22×C412yes—
225C4⋊C4C2×D412yes—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—81no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C4×D412yes—
256C23C22×D411yescharacteristic
257Q8—22no—
258Q8—22no—
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—48no—
263Q8—41no—
264Q8—41no—
265Q8C4×D414yes—
266D4—22no—
267D4—22no—
268D4—41no—
269D4—41no—
270D4—84no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—48no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
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—84no—
311C2×C4C4×D412yes—
312C2×C4#13712yes—
313C2×C4C23×C411yescharacteristic
314C2×C4C22×D411yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4H(5,𝔽2)11yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C8—22no—
320C8—22no—
321C8—22no—
322C8—41no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C2×C4×D411yescharacteristic
338C22D4211yescharacteristic
339C22#54611yescharacteristic
340C22C2×#13711yescharacteristic
341C22#57511yescharacteristic
342C22#57711yescharacteristic
343C22#57711yescharacteristic
344C4—22no—
345C4—24no—
346C4—22no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4C2×C4×D411yescharacteristic
357C4#51811yescharacteristic
358C4#54611yescharacteristic
359C4#55111yescharacteristic
360C2—42no—
361C2—82no—
362C2#235811yescharacteristic
363C2C4×#13711yescharacteristic
364C2#294911yescharacteristic
365C2#327511yescharacteristic
366C2#327811yescharacteristic
367C2#332011yescharacteristic
368C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. C2×#440C2semidirect
3. C2×#419C2semidirect
5. Q8×C22⋊C4C2semidirect
6. #1968C2semidirect
7. #1935C2semidirect
8. #1928C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1865C2semidirect
13. #1854C2semidirect
14. C4×#103C2semidirect
15. #1753C2semidirect
25. #448C4semidirect
68. C22×Q8C2×C4semidirect
126. #103C2×C4semidirect
127. #103C2×C4semidirect
128. #103D4semidirect
131. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
C2×#4402
C2×#4192
#23632
Q8×C22⋊C42
#19682
#19352
#19282
#18952
#18802
#18772
#18652
#18542
C4×#1032
#17532