Finite Group Databasebeta

← #22889

Subgroups — #22889

Order 256.

510 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3002C211yesmaximal, characteristic
2C2×#443C211yesmaximal, characteristic, complemented
3#2933C211yesmaximal, characteristic, complemented
4C2×#421C211yesmaximal, characteristic, complemented
5#2880C211yesmaximal, characteristic, complemented
6#1975C211yesmaximal, characteristic, complemented
7#1937C211yesmaximal, characteristic, complemented
8#1929C211yesmaximal, characteristic, complemented
9#1897C211yesmaximal, characteristic, complemented
10#1876C211yesmaximal, characteristic, complemented
11#1876C211yesmaximal, characteristic, complemented
12#1866C211yesmaximal, characteristic, complemented
13#1855C211yesmaximal, characteristic, complemented
14#1765C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#135C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic
21C2×#131C2211yescharacteristic, complemented
22C22×M16C2211yescharacteristic
23#525C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#452C411yescharacteristic, complemented
26#452C411yescharacteristic, complemented
27#451C411yescharacteristic, complemented
28#450C411yescharacteristic, complemented
29#449C411yescharacteristic, complemented
30#449C411yescharacteristic, complemented
31#448C411yescharacteristic, complemented
32#447C411yescharacteristic, complemented
33#443—24no—
34#436—21no—
35#435—21no—
36#435—21no—
37C8×D4—21no—
38C2×#108C2211yescharacteristic, complemented
39C2×#108C2211yescharacteristic, complemented
40#424C2211yescharacteristic, complemented
41#421—24no—
42C2×#104C2211yescharacteristic
43C2×#104C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic, complemented
47C2×#104C2211yescharacteristic, complemented
48#416C2211yescharacteristic
49#416C2211yescharacteristic
50#408—21no—
51#408—21no—
52#408C2211yescharacteristic
53#408C2211yescharacteristic
54#408C2211yescharacteristic, complemented
55#407—21no—
56#407C2211yescharacteristic
57C2×C22⋊C8—21no—
58C2×C22⋊C8C2211yescharacteristic
59C2×C22⋊C8C2211yescharacteristic
60#405C2211yescharacteristic, complemented
61C2×#97C2211yescharacteristic, complemented
62C2×C4×C8C2211yescharacteristic, complemented
63#341C2211yescharacteristic, complemented
64#341C2211yescharacteristic, complemented
65#339C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67#336C2211yescharacteristic, complemented
68#336C2211yescharacteristic, complemented
69C2×#41—21no—
70C2×#41—21no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41C2×C411yescharacteristic
76C2×#41C2×C411yescharacteristic
77C2×#41C2311yescharacteristic
78C2×#41C2311yescharacteristic
79C2×#41C2311yescharacteristic
80C22×Q8C2×C411yescharacteristic
81C22×D4—21no—
82C22×D4C2×C411yescharacteristic
83C23×C4—21no—
84C23×C4C2311yescharacteristic
85#135—22no—
86#135—22no—
87#135—41no—
88#135—41no—
89C2×Q16C2×C411yescharacteristic
90C2×Q16C2×C411yescharacteristic
91C2×SD16C2×C411yescharacteristic
92C2×SD16C2×C411yescharacteristic
93C2×SD16C2×C411yescharacteristic
94C2×SD16C2×C411yescharacteristic
95C2×D8C2×C411yescharacteristic
96C2×D8C2×C411yescharacteristic
97#131—22no—
98#131—22no—
99#131—22no—
100#131—41no—
101#131—41no—
102#131—41no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C2×M16C2311yescharacteristic
112C2×M16C2311yescharacteristic
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8C2311yescharacteristic
118C22×C8C2311yescharacteristic
119C22×C8C2311yescharacteristic
120C22×C8C2311yescharacteristic
121#123—21no—
122C22⋊Q8C2×C411yescharacteristic
123C22⋊Q8C2×C411yescharacteristic
124#121C2×C411yescharacteristic
125#121C2×C411yescharacteristic
126#120—21no—
127C4×D4—21no—
128C4×D4—21no—
129#117—22no—
130#117—41no—
131#117C2311yescharacteristic
132C2×C4⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136#108—22no—
137#108—41no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140#104—22no—
141#104—22no—
142#104—22no—
143#104—22no—
144#104—41no—
145#104D414yescomplemented
146#103C2×C411yescharacteristic
147#103C2×C411yescharacteristic
148#103C2×C411yescharacteristic
149#103C2×C411yescharacteristic
150#102C2×C411yescharacteristic
151#102C2×C411yescharacteristic
152#102C2×C411yescharacteristic
153#102C2×C411yescharacteristic
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8C2×C411yescharacteristic
163C22⋊C8C2×C411yescharacteristic
164C22⋊C8C2×C411yescharacteristic
165C22⋊C8C2×C411yescharacteristic
166C22⋊C8D411yescharacteristic
167C22⋊C8D411yescharacteristic
168C22⋊C8D411yescharacteristic
169C22⋊C8D411yescharacteristic
170#97—21no—
171#97—41no—
172C4×C8—21no—
173C4×C8—22no—
174C24—21no—
175C24#4111yescharacteristic
176#41—21no—
177#41—21no—
178#41—22no—
179#41—22no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—21no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—42no—
189#41—42no—
190#41—41no—
191#41—41no—
192#41—42no—
193#41—42no—
194#41C2×D412yes—
195#41#4112yes—
196C2×Q8—21no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8C22×C411yescharacteristic
200C2×Q8C22×C411yescharacteristic
201C2×Q8C22×C411yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4C22×C411yescharacteristic
215C2×D4C22×C411yescharacteristic
216C2×D4C22×C411yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22×C411yescharacteristic
232C22×C4C22×C411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4#4111yescharacteristic
236C22×C4C2411yescharacteristic
237Q16—22no—
238Q16—41no—
239SD16—22no—
240SD16—22no—
241SD16—41no—
242SD16—41no—
243D8—22no—
244D8—41no—
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—21no—
250M16—21no—
251M16—21no—
252M16—42no—
253M16—42no—
254M16—42no—
255M16—42no—
256M16C2×D412yes—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8C22×C411yescharacteristic
274C2×C8C22×C411yescharacteristic
275C2×C8C22×C411yescharacteristic
276C2×C8C22×C411yescharacteristic
277C2×C8C2×D411yescharacteristic
278C2×C8C2×D411yescharacteristic
279C2×C8C2×D411yescharacteristic
280C2×C8C2×D411yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—42no—
284C4⋊C4C22×C411yescharacteristic
285C4⋊C4C22×C411yescharacteristic
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—42no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C2×D412yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23C4×D411yescharacteristic
308C23C4×D411yescharacteristic
309C23C2×#4111yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—42no—
316Q8C4×D412yes—
317D4—22no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4C4×D412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
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×C4C4×D411yescharacteristic
365C2×C4C4×D411yescharacteristic
366C2×C4C23×C411yescharacteristic
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C2×#4111yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—21no—
378C8—22no—
379C8—21no—
380C8—21no—
381C8—41no—
382C8—41no—
383C8—41no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—84no—
404C22C2×C4×D411yescharacteristic
405C22#54611yescharacteristic
406C22#55111yescharacteristic
407C4—21no—
408C4—21no—
409C4—22no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—81no—
416C4C2×C4×D411yescharacteristic
417C4#51811yescharacteristic
418C4D4211yescharacteristic
419C4#54611yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—81no—
427C2#235811yescharacteristic
428C2#295511yescharacteristic
429C2#295711yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#3002C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#443C2semidirect
3. #2933C2semidirect
4. C2×#421C2semidirect
5. #2880C2semidirect
6. #1975C2semidirect
7. #1937C2semidirect
8. #1929C2semidirect
9. #1897C2semidirect
10. #1876C2semidirect
11. #1876C2semidirect
12. #1866C2semidirect
13. #1855C2semidirect
14. #1765C2semidirect
15. #1750C2semidirect
21. C2×#131C22semidirect
25. #452C4semidirect
26. #452C4semidirect
27. #451C4semidirect
28. #450C4semidirect
29. #449C4semidirect
30. #449C4semidirect
31. #448C4semidirect
32. #447C4semidirect
38. C2×#108C22semidirect
39. C2×#108C22semidirect
40. #424C22semidirect
46. C2×#104C22semidirect
47. C2×#104C22semidirect
54. #408C22semidirect
60. #405C22semidirect
61. C2×#97C22semidirect
62. C2×C4×C8C22semidirect
63. #341C22semidirect
64. #341C22semidirect
65. #339C22semidirect
66. #339C22semidirect
67. #336C22semidirect
68. #336C22semidirect
145. #104D4semidirect

Maximal subgroups

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

subgroupindex
#30022
C2×#4432
#29332
C2×#4212
#28802
#19752
#19372
#19292
#18972
#18762
#18662
#18552
#17652
#17502