Finite Group Databasebeta

← #61735

Subgroups — #61735

Order 256.

898 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#3520C211yesmaximal, characteristic
2C2×#554C211yesmaximal, characteristic, complemented
3#3356C212yesmaximal, complemented
4#3344C214yesmaximal, complemented
5#3335C212yesmaximal, complemented
6#3331C212yesmaximal, complemented
7#3316C214yesmaximal, complemented
8#3299C212yesmaximal, complemented
9#3238C211yesmaximal, characteristic, complemented
10#3229C211yesmaximal, characteristic, complemented
11#3195C212yesmaximal, complemented
12#3190C212yesmaximal, complemented
13#3113C211yesmaximal, characteristic, complemented
14#3108C211yesmaximal, characteristic, complemented
15#3092C212yesmaximal, complemented
16C2×#483C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18#556C2211yescharacteristic
19#554—21no—
20#554—22no—
21#554—21no—
22#554—21no—
23#554C2212yes—
24#554C2212yes—
25#554C2214yes—
26#552C2212yes—
27#551—22no—
28D4×Q8C2211yescharacteristic
29#548C2212yes—
30#547C2211yescharacteristic
31#547C2211yescharacteristic
32#546—21no—
33#542—21no—
34#539—22no—
35#533C2211yescharacteristic
36C2×C4⋊Q8C2211yescharacteristic
37#529C2211yescharacteristic
38C2×#125—21no—
39#525C2212yes—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8C2212yes—
42C2×C22⋊Q8C2212yes—
43#520—21no—
44C2×C4×Q8C2211yescharacteristic
45C2×C4×D4—21no—
46C2×C4×D4C2211yescharacteristic
47C22×C4⋊C4C2212yes—
48#501C2212yescomplemented
49#498C2212yescomplemented
50#489C2212yescomplemented
51#487C2212yescomplemented
52#485C2214yescomplemented
53#484C2214yescomplemented
54#483C2211yescharacteristic
55#483C2212yes—
56#483C2212yes—
57#483C2212yes—
58#483C2212yes—
59#483C2214yescomplemented
60#483C2214yescomplemented
61#483C2214yescomplemented
62#481C2214yescomplemented
63#480C2211yescharacteristic
64#480C2212yes—
65#479C2212yescomplemented
66#477C2212yescomplemented
67#476C2212yescomplemented
68#475C2212yescomplemented
69#475C2214yescomplemented
70#470C2214yescomplemented
71#470C2214yescomplemented
72#467C2214yescomplemented
73#465C2214yescomplemented
74#464C2212yescomplemented
75#462C2212yescomplemented
76#452C2214yescomplemented
77#451C2211yescharacteristic
78#451C2212yes—
79#451C2214yescomplemented
80#448C2211yescharacteristic
81#441C2214yescomplemented
82C4×Q16C2212yescomplemented
83C4×SD16C2212yescomplemented
84#436C2212yes—
85#435C2214yes—
86C8×D4C2212yes—
87#432C2211yescharacteristic
88#431C2211yescharacteristic
89#428C2214yescomplemented
90C2×#106C2214yescomplemented
91#417C2214yescomplemented
92C2×#103C2214yescomplemented
93#407C2212yes—
94C2×C22⋊C8C2212yes—
95C2×#41—21no—
96C2×#41C2312yes—
97C22×Q8—21no—
98C22×Q8C2311yescharacteristic
99C22×Q8C2312yes—
100C22×D4D411yescharacteristic
101C23×C4—22no—
102C23×C4C2312yes—
103C2×Q16C2312yes—
104C2×Q16C2314yes—
105C2×SD16C2312yes—
106C2×M16C2314yes—
107C22×C8C2314yes—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8C2311yescharacteristic
112C4⋊Q8C2312yes—
113C4⋊Q8C2312yes—
114C4⋊Q8C2311yescharacteristic
115#126—22no—
116#126—22no—
117#126—21no—
118#125—22no—
119#125—24no—
120#125—22no—
121#125—22no—
122#125—21no—
123#125C2311yescharacteristic
124#124—21no—
125#124—22no—
126#124C2311yescharacteristic
127#123—22no—
128#123—22no—
129#123—21no—
130#123—22no—
131#123—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—21no—
137C22⋊Q8—22no—
138C22⋊Q8—21no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—22no—
144C22⋊Q8C2311yescharacteristic
145C22⋊Q8C2312yes—
146C22⋊Q8C2312yes—
147C22⋊Q8C2312yes—
148C22⋊Q8C2314yes—
149C22⋊Q8C2314yes—
150C22⋊Q8C2312yes—
151#121—22no—
152#121—21no—
153#121C2311yescharacteristic
154#121C2312yes—
155#120—22no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8C2312yes—
161C4×Q8C2312yes—
162C4×Q8C2311yescharacteristic
163C4×Q8C2312yes—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4D411yescharacteristic
173C4×D4D412yes—
174C4×D4D411yescharacteristic
175C4×D4C2311yescharacteristic
176C4×D4C2311yescharacteristic
177C4×D4C2311yescharacteristic
178C4×D4C2312yes—
179#117—21no—
180#117—22no—
181#117—21no—
182#117C2312yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4D411yescharacteristic
192C2×C4⋊C4C2311yescharacteristic
193C2×C4⋊C4C2312yes—
194C2×C4⋊C4C2314yes—
195C2×C4⋊C4C2314yes—
196C2×C4⋊C4C2312yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D412yes—
200C2×C42—21no—
201C2×C42C2311yescharacteristic
202#107C2312yes—
203#107C2314yes—
204#106C2312yes—
205#106C2314yes—
206#106C2314yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2312yes—
211#103C2312yes—
212#103C2314yes—
213#103C2314yes—
214#103C2314yes—
215#103C2314yes—
216#103C2312yes—
217#103C2312yes—
218#103C2314yes—
219#102C2312yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2312yes—
223C22⋊C8C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2314yes—
226#97C2312yes—
227C4×C8C2312yes—
228C24C2×D412yes—
229#41—44no—
230#41—44no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—82no—
239C2×Q8C2411yescharacteristic
240C2×Q8C2412yes—
241C2×Q8C2412yes—
242C2×D4—24no—
243C2×D4—21no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4C2×D412yes—
247C2×D4C2×D411yescharacteristic
248C2×D4C2×D411yescharacteristic
249C2×D4C2411yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4—84no—
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D412yes—
264C22×C4C2×D412yes—
265C22×C4C2×D412yes—
266C22×C4C2411yescharacteristic
267C22×C4C2412yes—
268C22×C4C2412yes—
269Q16—44no—
270Q16—42no—
271SD16—42no—
272M16—44no—
273C2×C8—44no—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C2×C8C2414yes—
277C4⋊C4—21no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4C2×D411yescharacteristic
301C4⋊C4C2×D412yes—
302C4⋊C4C2×D411yescharacteristic
303C4⋊C4C2412yes—
304C4⋊C4C2414yes—
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2412yes—
307C4⋊C4C2411yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4C2×D412yes—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C22⋊C4C2×D412yes—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—21no—
330C42C2×D411yescharacteristic
331C42C2×D411yescharacteristic
332C42C2411yescharacteristic
333C42C2411yescharacteristic
334C23—24no—
335C23—24no—
336C23—21no—
337C23—42no—
338C23C22×D412yes—
339C23C22×D412yes—
340C23C22×D411yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—22no—
344Q8—24no—
345Q8—42no—
346Q8—42no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350D4—24no—
351D4—22no—
352D4—22no—
353D4—44no—
354D4—42no—
355D4—82no—
356D4#13714yes—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D411yescharacteristic
384C2×C4C22×D412yes—
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D411yescharacteristic
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C2511yescharacteristic
390C8—22no—
391C8—24no—
392C8—22no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—48no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22C2×#13714yes—
405C22#57712yes—
406C22C23×D411yescharacteristic
407C4—21no—
408C4—22no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—41no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—44no—
420C4—41no—
421C4C2×#13712yes—
422C4#58512yes—
423C2—22no—
424C2—24no—
425C2—42no—
426C2—81no—
427C2#348511yescharacteristic
428C2C22×#13711yescharacteristic
429C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3520C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#554C2semidirect
3. #3356C2semidirect
4. #3344C2semidirect
5. #3335C2semidirect
6. #3331C2semidirect
7. #3316C2semidirect
8. #3299C2semidirect
9. #3238C2semidirect
10. #3229C2semidirect
11. #3195C2semidirect
12. #3190C2semidirect
13. #3113C2semidirect
14. #3108C2semidirect
15. #3092C2semidirect
16. C2×#483C2semidirect
17. #2974C2semidirect
48. #501C22semidirect
49. #498C22semidirect
50. #489C22semidirect
51. #487C22semidirect
52. #485C22semidirect
53. #484C22semidirect
59. #483C22semidirect
60. #483C22semidirect
61. #483C22semidirect
62. #481C22semidirect
65. #479C22semidirect
66. #477C22semidirect
67. #476C22semidirect
68. #475C22semidirect
69. #475C22semidirect
70. #470C22semidirect
71. #470C22semidirect
72. #467C22semidirect
73. #465C22semidirect
74. #464C22semidirect
75. #462C22semidirect
76. #452C22semidirect
79. #451C22semidirect
81. #441C22semidirect
82. C4×Q16C22semidirect
83. C4×SD16C22semidirect
89. #428C22semidirect
90. C2×#106C22semidirect
91. #417C22semidirect
92. C2×#103C22semidirect

Maximal subgroups

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

subgroupindex
#35202
C2×#5542
#33562
#33442
#33352
#33312
#33162
#32992
#32382
#32292
#31952
#31902
#31132
#31082
#30922
C2×#4832
#29742