Finite Group Databasebeta

← #21460

Subgroups — #21460

Order 256.

738 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic, complemented
4C2×C4×D8C211yesmaximal, characteristic, complemented
5#2324C211yesmaximal, characteristic, complemented
6#1880C211yesmaximal, characteristic, complemented
7#1879C211yesmaximal, characteristic, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1863C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1852C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20#520—24no—
21#520—24no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30C2×#117C2211yescharacteristic
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2211yescharacteristic
33#449C418yescomplemented
34#440—24no—
35C4×D8—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#390—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#386—21no—
55#385—21no—
56#380C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#336C2211yescharacteristic
63#336C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41C2×C414yescomplemented
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C2311yescharacteristic
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16C2×C414yescomplemented
84C2×D8—22no—
85C2×D8—22no—
86C2×D8C2×C414yescomplemented
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91#121—22no—
92#121C2×C414yescomplemented
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8D414yescomplemented
97C4×D4—28no—
98C4×D4—28no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4D414yescomplemented
105#117—28no—
106#117—22no—
107#117—22no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4D411yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136#107—22no—
137#107—22no—
138#103—22no—
139#103—22no—
140#103C2×C414yescomplemented
141#102—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102C2×C414yescomplemented
146C22⋊C8—22no—
147C22⋊C8C2×C414yescomplemented
148#97—22no—
149C4×C8—22no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—21no—
154C24—21no—
155C24C2×D411yescharacteristic
156#41—28no—
157#41—28no—
158#41—28no—
159#41—28no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C22×C412yes—
164C2×Q8C2×D412yes—
165C2×Q8#4112yes—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4C22×C412yes—
181C2×D4C22×C412yes—
182C2×D4C2×D412yes—
183C2×D4#4112yes—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4C22×C412yes—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4#4112yes—
216C22×C4C2411yescharacteristic
217SD16—28no—
218D8—28no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8C22×C412yes—
230C2×C8C22×C412yes—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—81no—
244C4⋊C4C22×C412yes—
245C4⋊C4C2×D412yes—
246C4⋊C4C2×D412yes—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—24no—
257C42—24no—
258C42—21no—
259C42—21no—
260C42—81no—
261C42C2×D412yes—
262C42C2×D412yes—
263C23—22no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C4×D412yes—
284C23C22×D411yescharacteristic
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8C4×D414yes—
289D4—216no—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—22no—
295D4—41no—
296D4—41no—
297D4—84no—
298D4C4×D414yes—
299C2×C4—216no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4C4×D412yes—
339C2×C4#12012yes—
340C2×C4#12012yes—
341C2×C4C23×C411yescharacteristic
342C2×C4C22×D411yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C2×#4111yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4C2×#4111yescharacteristic
347C8—22no—
348C8—22no—
349C8—41no—
350C8—41no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—41no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C2×C4×D411yescharacteristic
373C22C2×#12011yescharacteristic
374C22#52511yescharacteristic
375C22#57611yescharacteristic
376C22#57611yescharacteristic
377C22#57711yescharacteristic
378C22#57811yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—24no—
383C4—42no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4C2×C4×D411yescharacteristic
391C4C2×C4×D411yescharacteristic
392C4#52511yescharacteristic
393C4#52511yescharacteristic
394C2—42no—
395C2—44no—
396C2—82no—
397C2C4×#12011yescharacteristic
398C2#294611yescharacteristic
399C2#295111yescharacteristic
400C2#300811yescharacteristic
401C2#301311yescharacteristic
402C2#311911yescharacteristic
403C2#312411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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
1. C2×#520C2semidirect
3. C2×#440C2semidirect
4. C2×C4×D8C2semidirect
5. #2324C2semidirect
6. #1880C2semidirect
7. #1879C2semidirect
8. #1874C2semidirect
9. #1863C2semidirect
10. #1862C2semidirect
11. #1852C2semidirect
12. #1788C2semidirect
13. #1762C2semidirect
14. C4×#103C2semidirect
15. #1750C2semidirect
33. #449C4semidirect
69. C2×#41C2×C4semidirect
83. C2×SD16C2×C4semidirect
86. C2×D8C2×C4semidirect
92. #121C2×C4semidirect
96. C4×Q8D4semidirect
104. C4×D4D4semidirect
140. #103C2×C4semidirect
145. #102C2×C4semidirect
147. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4492
C2×#4402
C2×C4×D82
#23242
#18802
#18792
#18742
#18632
#18622
#18522
#17882
#17622
C4×#1032
#17502