Finite Group Databasebeta

← #16857251

Subgroups — #16857251

Order 3968.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758491C211yesmaximal, characteristic, complemented
2#14521718C211yesmaximal, characteristic, complemented
3#14521687C211yesmaximal, characteristic, complemented
4#14521435C211yesmaximal, characteristic, complemented
5#14521292C211yesmaximal, characteristic, complemented
6#14521285C211yesmaximal, characteristic, complemented
7#14521273C211yesmaximal, characteristic, complemented
8#14521242C211yesmaximal, characteristic, complemented
9#14521210C211yesmaximal, characteristic, complemented
10#14521194C211yesmaximal, characteristic, complemented
11#14521194C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521150C211yesmaximal, characteristic, complemented
14#14521120C211yesmaximal, characteristic, complemented
15C2×C31×#135C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×#41×C31C2211yescharacteristic, complemented
18C2×#92499C2211yescharacteristic
19C2×D4×D31C2211yescharacteristic
20C31×#135—24no—
21C2×Q16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic
23C2×SD16×C31C2211yescharacteristic, complemented
24C2×D8×C31C2211yescharacteristic
25C22×C8×C31C2211yescharacteristic, complemented
26#11758524C2211yescharacteristic
27#11758523C2211yescharacteristic
28Q8×Dic31C2211yescharacteristic, complemented
29C2×#92481C2211yescharacteristic
30C2×#92481C2211yescharacteristic, complemented
31#11758517—21no—
32#11758517C2211yescharacteristic, complemented
33#11758516—21no—
34#11758515—21no—
35#11758515C2211yescharacteristic
36#11758514—21no—
37D4×Dic31C2211yescharacteristic
38C2×#92479C2211yescharacteristic
39C2×#92479C2211yescharacteristic
40#11758507—21no—
41#11758507C2211yescharacteristic
42#11758507C2211yescharacteristic, complemented
43#11758506—21no—
44C4×C22⋊D31—21no—
45C4×C22⋊D31C2211yescharacteristic
46C2×(C2×D31)⋊C4—21no—
47#11758491—24no—
48C2×D248C2211yescharacteristic
49C2×#92469C2211yescharacteristic
50#11758423C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic
53#11758420C2211yescharacteristic, complemented
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic
56#11758409C2211yescharacteristic, complemented
57(C2×D31)⋊C8C2211yescharacteristic
58#11758406C2211yescharacteristic, complemented
59#11758403C2211yescharacteristic, complemented
60Dic31⋊C8C2211yescharacteristic, complemented
61C23×D31—21no—
62#41×C31—22no—
63#41×C31—22no—
64#41×C31—41no—
65#41×C31—41no—
66C2×Q8×C31C2311yescharacteristic
67C2×Q8×C31C2311yescharacteristic
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31C2311yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31C2311yescharacteristic
75C2×C22⋊D31—21no—
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79C2×C22⋊D31—21no—
80C2×C22⋊D31—21no—
81C2×C22⋊D31D411yescharacteristic
82C22×Dic31—21no—
83#92499—22no—
84#92499—41no—
85D4×D31—22no—
86D4×D31—41no—
87C2×D124—21no—
88C2×D124C2311yescharacteristic
89C2×D124C2311yescharacteristic
90C2×C4×D31—21no—
91C2×C4×D31C2311yescharacteristic
92Q16×C31—22no—
93SD16×C31—22no—
94SD16×C31D414yescomplemented
95D8×C31—22no—
96C2×C8×C31—22no—
97C2×C8×C31C2311yescharacteristic
98C2×C8×C31C2311yescharacteristic
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31D411yescharacteristic
102#92481—22no—
103#92481—41no—
104#92479—22no—
105#92479—41no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31—21no—
113C4⋊Dic31C2311yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4D411yescharacteristic
117C4×Dic31—21no—
118C4×Dic31C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121D248—22no—
122#92469—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—81no—
131C22×D31C2×D411yescharacteristic
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135Q8×C31C2×D412yes—
136D4×C31—21no—
137D4×C31—21no—
138D4×C31—21no—
139D4×C31—42no—
140D4×C31—42no—
141D4×C31C2×D412yes—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2411yescharacteristic
150C22⋊D31—41no—
151C22⋊D31—42no—
152C22⋊D31—42no—
153C22⋊D31—81no—
154C22⋊D31—81no—
155C22⋊D31—81no—
156C22⋊D31—81no—
157C2×Dic31—21no—
158C2×Dic31—21no—
159C2×Dic31—81no—
160C2×Dic31C2×D411yescharacteristic
161C2×Dic31C2×D411yescharacteristic
162C2×Dic31C2×D411yescharacteristic
163D124—21no—
164D124—21no—
165D124—21no—
166D124—21no—
167D124—42no—
168C4×D31—21no—
169C4×D31—21no—
170C4×D31—42no—
171C8×C31—21no—
172C8×C31C2×D412yes—
173C31⋊C8—21no—
174C31⋊C8—21no—
175#3268—311nomaximal, Sylow
176C22×C31—22no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31C22×D411yescharacteristic
186C2×D31—21no—
187C2×D31—21no—
188C2×D31—21no—
189C2×D31—41no—
190C2×D31—41no—
191C2×D31—41no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—84no—
196C4×C31—22no—
197C4×C31—21no—
198C4×C31—41no—
199C4×C31C22×D411yescharacteristic
200C4×C31H(5,𝔽2)11yescharacteristic
201Dic31—21no—
202Dic31—21no—
203Dic31—21no—
204Dic31—41no—
205Dic31—41no—
206C2×AGL(1,ℤ/8)—311no—
207C2×#135D3111yescharacteristic, complemented
208#550—311no—
209#546—311no—
210#496—311no—
211#468—311no—
212#466—311no—
213#464—311no—
214#459—311no—
215#452—311no—
216#449—311no—
217#449—311no—
218#447—311no—
219#440—311no—
220#435—311no—
221C2×C31—42no—
222C2×C31—41no—
223C2×C31—81no—
224C2×C31D4211yescharacteristic
225C2×C31#57511yescharacteristic
226C2×C31#57611yescharacteristic
227D31—41no—
228D31—81no—
229D31—81no—
230C2×#41—311no—
231C2×#41C2×D3111yescharacteristic
232C2×#41C2×D3111yescharacteristic, complemented
233C22×D4—311no—
234AGL(1,ℤ/8)—624no—
235#135—24no—
236C2×Q16C2×D3111yescharacteristic, complemented
237C2×SD16—311no—
238C2×SD16—311no—
239C2×SD16C2×D3111yescharacteristic
240C2×SD16C2×D3111yescharacteristic, complemented
241C2×D8—311no—
242C2×D8—311no—
243C2×D8—311no—
244C2×D8C2×D3111yescharacteristic
245C2×M16—311no—
246C22×C8C2×D3111yescharacteristic, complemented
247#127—311no—
248#127—621no—
249#124—311no—
250#123—621no—
251C22⋊Q8—311no—
252#121—311no—
253#121—311no—
254#121—311no—
255#121—621no—
256#121—621no—
257#121—621no—
258#120—621no—
259C4×Q8—311no—
260C4×D4—311no—
261C4×D4—311no—
262C4×D4—621no—
263C2×C22⋊C4—621no—
264#107—311no—
265C4⋊C8—311no—
266#103—311no—
267#103—311no—
268#102—311no—
269#102—311no—
270#102—311no—
271#102—311no—
272C22⋊C8—311no—
273C22⋊C8—311no—
274#97—311no—
275C31#326811yescharacteristic, complemented, Sylow
276C24—621no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—622no—
282#41—1241no—
283C2×Q8C22×D3111yescharacteristic
284C2×Q8C22×D3111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—311no—
288C2×D4—311no—
289C2×D4—311no—
290C2×D4—621no—
291C2×D4—621no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—622no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—1241no—
299C2×D4C22×D3111yescharacteristic
300C2×D4C22×D3111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—311no—
304C22×C4—621no—
305C22×C4—621no—
306C22×C4C22×D3111yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—622no—
310SD16—1241no—
311SD16C22⋊D3114yescomplemented
312D8—22no—
313D8—622no—
314D8—622no—
315D8—1241no—
316M16—622no—
317C2×C8—22no—
318C2×C8—311no—
319C2×C8—311no—
320C2×C8C22×D3111yescharacteristic
321C2×C8C22×D3111yescharacteristic
322C4⋊C4—311no—
323C4⋊C4—311no—
324C4⋊C4—311no—
325C4⋊C4—621no—
326C4⋊C4—621no—
327C22⋊C4—311no—
328C22⋊C4—311no—
329C22⋊C4—621no—
330C22⋊C4—621no—
331C22⋊C4—621no—
332C22⋊C4—621no—
333C22⋊C4—621no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C42—311no—
337C42—621no—
338C23—21no—
339C23—21no—
340C23—311no—
341C23—621no—
342C23—621no—
343C23—1242no—
344C23—1242no—
345C23—2481no—
346C23D4×D3111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D3112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—621no—
357D4—621no—
358D4—621no—
359D4—621no—
360D4—1242no—
361D4—1241no—
362D4—1242no—
363D4—1242no—
364D4—2481no—
365D4—2481no—
366D4—2481no—
367D4—2481no—
368D4C2×C22⋊D3112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—311no—
376C2×C4—311no—
377C2×C4—311no—
378C2×C4—621no—
379C2×C4—621no—
380C2×C4—621no—
381C2×C4—621no—
382C2×C4—1242no—
383C2×C4—2481no—
384C2×C4D4×D3111yescharacteristic
385C2×C4C23×D3111yescharacteristic
386C8—21no—
387C8—621no—
388C8—621no—
389C8C2×C22⋊D3112yes—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—621no—
400C22—621no—
401C22—621no—
402C22—1241no—
403C22—1241no—
404C22—1241no—
405C22—1241no—
406C22—1241no—
407C22—1241no—
408C22—2484no—
409C22C2×D4×D3111yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—621no—
414C4—621no—
415C4—621no—
416C4—1241no—
417C4—1241no—
418C4#1175856111yescharacteristic
419C4C22×C22⋊D3111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—1241no—
424C2—2481no—
425C2—2481no—
426C2D4×C22⋊D3111yescharacteristic
427C2#1452184611yescharacteristic
428C2#1452185111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#135C2
socleC22×C31C22×D4
hypercenterC2×#135D31
hypocenterC31#3268
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. #14521718C2semidirect
3. #14521687C2semidirect
4. #14521435C2semidirect
5. #14521292C2semidirect
6. #14521285C2semidirect
7. #14521273C2semidirect
8. #14521242C2semidirect
9. #14521210C2semidirect
10. #14521194C2semidirect
11. #14521194C2semidirect
12. #14521181C2semidirect
13. #14521150C2semidirect
14. #14521120C2semidirect
15. C2×C31×#135C2semidirect
17. C2×#41×C31C22semidirect
21. C2×Q16×C31C22semidirect
23. C2×SD16×C31C22semidirect
25. C22×C8×C31C22semidirect
28. Q8×Dic31C22semidirect
30. C2×#92481C22semidirect
32. #11758517C22semidirect
42. #11758507C22semidirect
50. #11758423C22semidirect
51. #11758423C22semidirect
53. #11758420C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
56. #11758409C22semidirect
58. #11758406C22semidirect
59. #11758403C22semidirect
60. Dic31⋊C8C22semidirect
94. SD16×C31D4semidirect
207. C2×#135D31semidirect
232. C2×#41C2×D31semidirect
236. C2×Q16C2×D31semidirect
240. C2×SD16C2×D31semidirect
246. C22×C8C2×D31semidirect
275. C31#3268semidirect
311. SD16C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
#145217182
#145216872
#145214352
#145212922
#145212852
#145212732
#145212422
#145212102
#145211942
#145211812
#145211502
#145211202
C2×C31×#1352
#326831