Finite Group Databasebeta

← #20746

Subgroups — #20746

Order 256.

649 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2314C211yesmaximal, characteristic, complemented
4#1824C211yesmaximal, characteristic, complemented
5#1823C211yesmaximal, characteristic, complemented
6#1805C211yesmaximal, characteristic, complemented
7#1804C211yesmaximal, characteristic, complemented
8#1803C211yesmaximal, characteristic, complemented
9#1802C211yesmaximal, characteristic, complemented
10#1788C211yesmaximal, characteristic
11#1788C211yesmaximal, characteristic
12#1763C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17#520—24no—
18#520C418yescomplemented
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26C2×#117C2211yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2211yescharacteristic
29#433—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#387—21no—
48#384C2211yescharacteristic
49#382C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41C2×C414yes—
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8C2×C414yes—
83C4×Q8C2×C414yes—
84C4×D4—28no—
85C4×D4—28no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4C2×C414yes—
89C4×D4C2×C414yes—
90#117—28no—
91#117C2×C414yes—
92#117D414yes—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2×C414yes—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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
121C4⋊C8—22no—
122C4⋊C8—22no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133#97—22no—
134C4×C8—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143#41—28no—
144#41—28no—
145#41—28no—
146#41—28no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C22×C412yes—
151C2×Q8#4112yes—
152C2×Q8#4112yes—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4C22×C412yes—
160C2×D4#4112yes—
161C2×D4#4112yes—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4C22×C412yes—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4C2411yescharacteristic
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×C8—21no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4C22⋊C414yes—
221C4⋊C4C22⋊C414yes—
222C4⋊C4C22×C412yes—
223C4⋊C4C22×C412yes—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4C22⋊C414yes—
229C22⋊C4C22⋊C414yes—
230C42—24no—
231C42—24no—
232C42—81no—
233C42—81no—
234C42C22×C412yes—
235C42C22×C412yes—
236C42C2×D412yes—
237C42C2×D412yes—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C2×C22⋊C412yes—
252C23C22×D411yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—22no—
256Q8#11714yes—
257D4—216no—
258D4—22no—
259D4—22no—
260D4—22no—
261D4#11714yes—
262C2×C4—216no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
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—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
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—84no—
300C2×C4—84no—
301C2×C4C2×C22⋊C412yes—
302C2×C4C2×C22⋊C412yes—
303C2×C4C2×C22⋊C412yes—
304C2×C4C2×C22⋊C412yes—
305C2×C4C2×C22⋊C412yes—
306C2×C4C23×C411yescharacteristic
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—24no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—42no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22C22×C22⋊C411yescharacteristic
331C22#52511yescharacteristic
332C22#52511yescharacteristic
333C22#57611yescharacteristic
334C22#57711yescharacteristic
335C22#57711yescharacteristic
336C22#57811yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—24no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4C2×#11711yescharacteristic
350C4C2×#11711yescharacteristic
351C4#52511yescharacteristic
352C4#52511yescharacteristic
353C2—42no—
354C2—44no—
355C2#228011yescharacteristic
356C2#289711yescharacteristic
357C2#289811yescharacteristic
358C2#311911yescharacteristic
359C2#312011yescharacteristic
360C2#312411yescharacteristic
361C2#312511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#433C2semidirect
3. #2314C2semidirect
4. #1824C2semidirect
5. #1823C2semidirect
6. #1805C2semidirect
7. #1804C2semidirect
8. #1803C2semidirect
9. #1802C2semidirect
12. #1763C2semidirect
13. #1762C2semidirect
14. C4×#103C2semidirect
15. C4×#102C2semidirect
18. #520C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4332
#23142
#18242
#18232
#18052
#18042
#18032
#18022
#17882
#17632
#17622
C4×#1032
C4×#1022