Finite Group Databasebeta

← #30464

Subgroups — #30464

Order 256.

598 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic
4C2×#450C211yesmaximal, characteristic
5#2435C211yesmaximal, characteristic
6#2429C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2001C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1880C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×Q16C2211yescharacteristic, complemented
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic, complemented
30#497—24no—
31#464—24no—
32#450—24no—
33#450—24no—
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic, complemented
46#398—21no—
47#396—21no—
48#394—21no—
49#394—21no—
50#391C2211yescharacteristic, complemented
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389—21no—
54#389C2211yescharacteristic, complemented
55#388—21no—
56#379—21no—
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—41no—
89C2×SD16—41no—
90C2×SD16—41no—
91C2×SD16—41no—
92C2×SD16—41no—
93C2×SD16—41no—
94C2×SD16D414yes—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8—41no—
101#124—22no—
102#124—22no—
103#124—41no—
104#124—41no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C4×Q8—22no—
112C4×Q8—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4D411yescharacteristic
135C2×C22⋊C4D411yescharacteristic
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C2×C42C2311yescharacteristic
139C4⋊C8—22no—
140#103—22no—
141#103—41no—
142#103—41no—
143#103—41no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102D414yes—
148C22⋊C8—22no—
149C22⋊C8—22no—
150#97—22no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95D411yescharacteristic
155#95D411yescharacteristic
156C24—21no—
157C24—21no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24—42no—
163C24—81no—
164C24—81no—
165C24C2×D411yescharacteristic
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8C2×D412yes—
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—21no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4C2×D412yes—
202C2×D4C2×D412yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2411yescharacteristic
221Q16—44no—
222Q16—44no—
223SD16—44no—
224SD16—44no—
225SD16—44no—
226SD16—44no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8C2×D412yes—
238C2×C8C2×D412yes—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4C2×D412yes—
252C22⋊C4—22no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—42no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—84no—
292C23—84no—
293C23—84no—
294C23—84no—
295C23—84no—
296C23—84no—
297C23#13712yes—
298C23C22×D411yescharacteristic
299Q8—22no—
300Q8—22no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—48no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4H(5,𝔽2)11yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—816no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#53511yescharacteristic
390C22D4211yescharacteristic
391C22D4211yescharacteristic
392C22C2×#13711yescharacteristic
393C22#57511yescharacteristic
394C22#57711yescharacteristic
395C22#57711yescharacteristic
396C4—41no—
397C4—42no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—41no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4#53411yescharacteristic
408C4#53511yescharacteristic
409C4D4211yescharacteristic
410C4#56011yescharacteristic
411C2—42no—
412C2—82no—
413C2—82no—
414C2#266911yescharacteristic
415C2#321511yescharacteristic
416C2#322211yescharacteristic
417C2#326611yescharacteristic
418C2#326611yescharacteristic
419C2#326711yescharacteristic
420C2#327511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×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
1. C2×#497C2semidirect
2. C2×#464C2semidirect
6. #2429C2semidirect
7. #2055C2semidirect
8. #2042C2semidirect
9. #2015C2semidirect
10. #2013C2semidirect
11. #2001C2semidirect
12. #2001C2semidirect
13. #1936C2semidirect
14. #1924C2semidirect
15. #1880C2semidirect
17. C22×Q16C22semidirect
18. C22×Q16C22semidirect
23. C2×C4⋊Q8C22semidirect
29. C2×C4×Q8C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
45. C2×#97C22semidirect
50. #391C22semidirect
54. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4642
C2×#4502
#24352
#24292
#20552
#20422
#20152
#20132
#20012
#19362
#19242
#18802