Finite Group Databasebeta

← #22422

Subgroups — #22422

Order 256.

765 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#550C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4C2×#416C211yesmaximal, characteristic, complemented
5#2352C211yesmaximal, characteristic, complemented
6#1982C211yesmaximal, characteristic, complemented
7#1961C211yesmaximal, characteristic, complemented
8#1895C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#550—24no—
21#550C418yescomplemented
22C2×#127—21no—
23C2×#127C2211yescharacteristic
24C2×#121—21no—
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#435—24no—
34C2×C4⋊C8C2211yescharacteristic
35#417—24no—
36#416—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×#97C2211yescharacteristic
52#388—21no—
53#387—21no—
54#386—21no—
55#382C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59C4×C22⋊C4—21no—
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41C2×C414yes—
75C2×#41C2×C414yescomplemented
76C22×Q8C2311yescharacteristic
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4D411yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—41no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101#127—28no—
102#127—22no—
103#127C2×C414yescomplemented
104#121—22no—
105#121—22no—
106#121—28no—
107#121—28no—
108#121C2×C414yes—
109#121C2×C414yescomplemented
110C4×Q8—22no—
111C4×Q8C2×C414yescomplemented
112C4×D4—28no—
113C4×D4C2×C414yes—
114C4×D4D414yes—
115#117—22no—
116#117—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4D411yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141#103—22no—
142#103D414yescomplemented
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—41no—
148#102—41no—
149#102—41no—
150#102—41no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153#97—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C24—21no—
160C24—21no—
161C24C2×D411yescharacteristic
162C24C2×D411yescharacteristic
163#41—28no—
164#41—28no—
165#41—44no—
166#41—44no—
167#41—44no—
168#41—44no—
169#41—44no—
170#41—44no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8C22×C412yes—
175C2×Q8C2×D412yes—
176C2×Q8#4112yes—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4C22⋊C414yes—
208C2×D4C22×C412yes—
209C2×D4C22×C412yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4C22×C412yes—
234C22×C4C22×C412yes—
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D412yes—
240C22×C4C2×D412yes—
241C22×C4C2411yescharacteristic
242M16—44no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—44no—
255C2×C8C2×D412yes—
256C4⋊C4—24no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—42no—
266C4⋊C4C22⋊C414yes—
267C4⋊C4C22×C412yes—
268C4⋊C4C2×D412yes—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4C22⋊C414yes—
275C22⋊C4C22⋊C414yes—
276C42—24no—
277C42—42no—
278C42—81no—
279C42C22×C412yes—
280C42C2×D412yes—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C2×C22⋊C412yes—
308C23C2×C22⋊C412yes—
309C23C22×D411yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—22no—
313Q8C4×D414yes—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—22no—
318D4—44no—
319D4—48no—
320D4—48no—
321D4—48no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—84no—
327D4—84no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—84no—
366C2×C4C2×C22⋊C412yes—
367C2×C4C2×C22⋊C412yes—
368C2×C4C2×C22⋊C412yes—
369C2×C4C2×C22⋊C412yes—
370C2×C4C23×C411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C2×#4111yescharacteristic
374C2×C4H(5,𝔽2)11yescharacteristic
375C2×C4H(5,𝔽2)11yescharacteristic
376C8—22no—
377C8—41no—
378C8—41no—
379C8—41no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C22×C22⋊C411yescharacteristic
409C22D4211yescharacteristic
410C22#54611yescharacteristic
411C22#57511yescharacteristic
412C22#57511yescharacteristic
413C22#57611yescharacteristic
414C22#57711yescharacteristic
415C4—22no—
416C4—22no—
417C4—24no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4C2×C4×D411yescharacteristic
426C4#51811yescharacteristic
427C4D4211yescharacteristic
428C4#54611yescharacteristic
429C2—42no—
430C2—42no—
431C2—82no—
432C2—82no—
433C2D4×C22⋊C411yescharacteristic
434C2#289611yescharacteristic
435C2#289711yescharacteristic
436C2#326811yescharacteristic
437C2#327111yescharacteristic
438C2#330911yescharacteristic
439C2#331711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#550C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#435C2semidirect
3. C2×#417C2semidirect
4. C2×#416C2semidirect
5. #2352C2semidirect
6. #1982C2semidirect
7. #1961C2semidirect
8. #1895C2semidirect
9. #1893C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1805C2semidirect
13. #1788C2semidirect
14. #1788C2semidirect
15. #1763C2semidirect
21. #550C4semidirect
75. C2×#41C2×C4semidirect
103. #127C2×C4semidirect
109. #121C2×C4semidirect
111. C4×Q8C2×C4semidirect
142. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4352
C2×#4172
C2×#4162
#23522
#19822
#19612
#18952
#18932
#18752
#18742
#18052
#17882
#17632