Finite Group Databasebeta

← #30675696

Subgroups — #30675696

Order 16768.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22554111C211yesmaximal, characteristic
2C2×#16862278C211yesmaximal, characteristic, complemented
3#137×D131C211yesmaximal, characteristic, complemented
4#22554077C211yesmaximal, characteristic, complemented
5#22553586C211yesmaximal, characteristic, complemented
6#22553585C211yesmaximal, characteristic, complemented
7#22553584C211yesmaximal, characteristic, complemented
8#22553578C211yesmaximal, characteristic, complemented
9#22553577C211yesmaximal, characteristic, complemented
10#22553576C211yesmaximal, characteristic, complemented
11#22553575C211yesmaximal, characteristic, complemented
12#22553359C211yesmaximal, characteristic, complemented
13#22553295C211yesmaximal, characteristic, complemented
14#101×D131C211yesmaximal, characteristic, complemented
15#473×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17#41×D131C2211yescharacteristic
18C2×Q8×D131C2211yescharacteristic
19#16862279C2211yescharacteristic
20#16862278—21no—
21#16862278—21no—
22#16862278—21no—
23#16862278—21no—
24#16862278—21no—
25#16862278—21no—
26#16862277C2211yescharacteristic
27#16862277C2211yescharacteristic
28#16862276C2211yescharacteristic
29#16862275—21no—
30#16862274C2211yescharacteristic
31#16862273—21no—
32#16862271C2211yescharacteristic
33#16862269—21no—
34#16862267—21no—
35C2×#14523503C2211yescharacteristic, complemented
36Q16×D131—21no—
37C2×#14523502C2211yescharacteristic, complemented
38C2×#14523500C2211yescharacteristic, complemented
39SD16×D131—21no—
40#16862254—21no—
41#16862254C2211yescharacteristic
42#16862253C2211yescharacteristic
43#16862252—21no—
44#16862252C2211yescharacteristic
45C2×#14523496—21no—
46C2×#14523496C2211yescharacteristic
47M16×D131—21no—
48M16×D131C2211yescharacteristic
49M16×D131C2211yescharacteristic
50M16×D131C2211yescharacteristic
51#16862182C2211yescharacteristic, complemented
52#16862181C2211yescharacteristic, complemented
53#16862181C2211yescharacteristic, complemented
54#16862165C2211yescharacteristic, complemented
55#16862165C2211yescharacteristic, complemented
56#16862164C2211yescharacteristic, complemented
57#16862164C2211yescharacteristic, complemented
58#16862162C2211yescharacteristic, complemented
59#16862162C2211yescharacteristic, complemented
60#137×C131C2211yescharacteristic
61#137×C131C2211yescharacteristic
62C2×Q16×C131C2211yescharacteristic, complemented
63#131×C131C2211yescharacteristic
64#108×C131C2211yescharacteristic, complemented
65#101×C131C2211yescharacteristic, complemented
66#101×C131C2211yescharacteristic, complemented
67#14523514—21no—
68#14523514—21no—
69#14523514—21no—
70#14523514—21no—
71#14523514D411yescharacteristic
72#14523513—21no—
73#14523513C2311yescharacteristic
74#14523512—21no—
75#14523512D411yescharacteristic
76C2×Dic262—21no—
77C2×Dic262C2311yescharacteristic
78Q8×D131—21no—
79Q8×D131—21no—
80Q8×D131—21no—
81Q8×D131—21no—
82Q8×D131D411yescharacteristic
83C2×D524—21no—
84C2×D524C2311yescharacteristic
85D4×D131—21no—
86D4×D131D411yescharacteristic
87C2×C4×D131—21no—
88C2×C4×D131—21no—
89C2×C4×D131—21no—
90C2×C4×D131C2311yescharacteristic
91Dic524—21no—
92#14523503—21no—
93#14523503—21no—
94#14523503—21no—
95#14523503—21no—
96#14523503—21no—
97#14523502—21no—
98#14523502—21no—
99#14523502—21no—
100#14523502—21no—
101#14523501—21no—
102#14523500—21no—
103#14523500—21no—
104#14523500—21no—
105#14523500—21no—
106#14523500—21no—
107D1048—21no—
108#14523498—21no—
109#14523497—21no—
110#14523497C2311yescharacteristic
111#14523497C2311yescharacteristic
112#14523497C2311yescharacteristic
113#14523496—21no—
114#14523496—21no—
115#14523496—21no—
116#14523496—21no—
117#14523496—21no—
118#14523496D411yescharacteristic
119#14523496D411yescharacteristic
120#14523496D411yescharacteristic
121#14523496D411yescharacteristic
122C2×C131⋊C8—21no—
123C2×C131⋊C8C2311yescharacteristic
124C8×D131—21no—
125C8×D131—21no—
126C8×D131—21no—
127C8×D131—21no—
128C8×D131—21no—
129#41×C131C2311yescharacteristic
130C2×Q8×C131C2311yescharacteristic
131C2×Q8×C131C2311yescharacteristic
132Q16×C131—21no—
133Q16×C131—21no—
134Q16×C131—21no—
135Q16×C131—21no—
136SD16×C131—21no—
137SD16×C131—21no—
138M16×C131—21no—
139M16×C131C2311yescharacteristic
140M16×C131C2311yescharacteristic
141M16×C131C2311yescharacteristic
142C2×C8×C131—21no—
143C2×C8×C131C2311yescharacteristic
144C22×D131—21no—
145C22×D131—21no—
146C22×D131#4111yescharacteristic
147Dic262—21no—
148Dic262—21no—
149Dic262—21no—
150Dic262—41no—
151Dic262—41no—
152Dic262C2×D411yescharacteristic
153D524—21no—
154D524—21no—
155D524—21no—
156D524—41no—
157D524—41no—
158D524C2×D411yescharacteristic
159C22⋊D131—21no—
160C22⋊D131—21no—
161C22⋊D131—21no—
162C2×Dic131—21no—
163C2×Dic131—21no—
164C2×Dic131#4111yescharacteristic
165C4×D131—21no—
166C4×D131—21no—
167C4×D131—21no—
168C4×D131—41no—
169C4×D131—41no—
170C4×D131—41no—
171C4×D131—41no—
172C4×D131C2×D411yescharacteristic
173C4×D131C2×D411yescharacteristic
174C4×D131C2×D411yescharacteristic
175C4×D131C2×D411yescharacteristic
176C131⋊C8—21no—
177C131⋊C8—21no—
178C131⋊C8—21no—
179C131⋊C8C2×D411yescharacteristic
180C131⋊C8C2×D411yescharacteristic
181Q8×C131—21no—
182Q8×C131—21no—
183Q8×C131—21no—
184Q8×C131—21no—
185Q8×C131C2×D411yescharacteristic
186D4×C131—21no—
187D4×C131C2×D411yescharacteristic
188C2×C4×C131—21no—
189C2×C4×C131—21no—
190C2×C4×C131—21no—
191C2×C4×C131C2411yescharacteristic
192C8×C131—21no—
193C8×C131—21no—
194C8×C131—21no—
195C8×C131C2×D411yescharacteristic
196C8×C131C2×D411yescharacteristic
197C22×C131—21no—
198C22×C131C2×#4111yescharacteristic
199C2×D131—21no—
200C2×D131—41no—
201C2×D131—41no—
202C2×D131—81no—
203C2×D131—81no—
204C2×D131#12111yescharacteristic
205C2×D131#12111yescharacteristic
206C4×C131—21no—
207C4×C131—41no—
208C4×C131—41no—
209C4×C131C22×D411yescharacteristic
210C4×C131C22×D411yescharacteristic
211Dic131—21no—
212Dic131—41no—
213Dic131#12111yescharacteristic
214Dic131#12111yescharacteristic
215C2×C131—21no—
216C2×C131—41no—
217C2×C131C2×#12111yescharacteristic
218D131—21no—
219D131—21no—
220D131—41no—
221D131—81no—
222C131#306911yescharacteristic, complemented, Sylow
223#3069—1311nomaximal, Sylow
224#575—1311no—
225C2×#137—1311no—
226C2×#137—1311no—
227#568—1311no—
228#473—1311no—
229#473—1311no—
230#473—1311no—
231#473D13111yescharacteristic, complemented
232#471—1311no—
233#471—1311no—
234#471—1311no—
235#471—1311no—
236#430—1311no—
237#413—1311no—
238C2×#101—1311no—
239C2×#41—1311no—
240C2×#41—1311no—
241C22×Q8—1311no—
242#137—2621no—
243#137—2621no—
244#137—2621no—
245#137—2621no—
246#137—2621no—
247#137—2621no—
248#137—2621no—
249#137—1311no—
250#137—1311no—
251#137—1311no—
252#137—1311no—
253#137C2×D13111yescharacteristic
254#137C2×D13111yescharacteristic
255AGL(1,ℤ/8)—2621no—
256AGL(1,ℤ/8)—1311no—
257AGL(1,ℤ/8)—1311no—
258#135—2621no—
259#135—2621no—
260C2×Q16—2621no—
261C2×Q16—1311no—
262C2×Q16C2×D13111yescharacteristic, complemented
263C2×SD16—2621no—
264C2×SD16—1311no—
265C2×SD16—1311no—
266#131—2621no—
267#131—2621no—
268#131—1311no—
269#131—1311no—
270#131—1311no—
271#131C2×D13111yescharacteristic
272C2×M16—2621no—
273C2×M16—2621no—
274C2×M16—1311no—
275C2×M16—1311no—
276C2×M16—1311no—
277C2×M16—1311no—
278#108—1311no—
279#108—1311no—
280#108—1311no—
281#108C2×D13111yescharacteristic, complemented
282#101—1311no—
283#101—1311no—
284#101—1311no—
285#101—1311no—
286#101C2×D13111yescharacteristic, complemented
287#101C2×D13111yescharacteristic, complemented
288#100—1311no—
289#100—1311no—
290#41—2621no—
291#41—2621no—
292#41—2621no—
293#41—2621no—
294#41—2621no—
295#41—2621no—
296#41—1311no—
297#41—1311no—
298#41—1311no—
299#41C22×D13111yescharacteristic
300C2×Q8—2621no—
301C2×Q8—2621no—
302C2×Q8—2621no—
303C2×Q8—2621no—
304C2×Q8—2621no—
305C2×Q8—1311no—
306C2×Q8—1311no—
307C2×Q8C22×D13111yescharacteristic
308C2×Q8C22×D13111yescharacteristic
309C2×D4—2621no—
310C2×D4—2621no—
311C2×D4—1311no—
312C2×D4—1311no—
313C22×C4—2621no—
314C22×C4—2621no—
315C22×C4—2621no—
316C22×C4—1311no—
317Q16—21no—
318Q16—21no—
319Q16—21no—
320Q16—21no—
321Q16—2621no—
322Q16—2621no—
323Q16—2621no—
324Q16—2621no—
325Q16—2621no—
326Q16—2621no—
327SD16—21no—
328SD16—21no—
329SD16—2621no—
330SD16—2621no—
331SD16—2621no—
332SD16—2621no—
333SD16—2621no—
334SD16—2621no—
335SD16—2621no—
336SD16—2621no—
337SD16—2621no—
338SD16—2621no—
339D8—2621no—
340D8—2621no—
341M16—21no—
342M16—2621no—
343M16—2621no—
344M16—2621no—
345M16—2621no—
346M16—2621no—
347M16—2621no—
348M16—1311no—
349M16—1311no—
350M16—1311no—
351M16—1311no—
352M16—1311no—
353M16—1311no—
354M16—1311no—
355M16C22×D13111yescharacteristic
356M16C22×D13111yescharacteristic
357M16C22×D13111yescharacteristic
358C2×C8—21no—
359C2×C8—2621no—
360C2×C8—2621no—
361C2×C8—2621no—
362C2×C8—2621no—
363C2×C8—2621no—
364C2×C8—2621no—
365C2×C8—1311no—
366C2×C8C22×D13111yescharacteristic
367C23—2621no—
368C23—2621no—
369C23—1311no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—2621no—
375Q8—2621no—
376Q8—2621no—
377Q8—5241no—
378Q8—5241no—
379Q8—1311no—
380Q8D4×D13111yescharacteristic
381D4—21no—
382D4—2621no—
383D4—2621no—
384D4—2621no—
385D4—2621no—
386D4—2621no—
387D4—2621no—
388D4—5241no—
389D4—5241no—
390D4—1311no—
391D4D4×D13111yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—2621no—
396C2×C4—2621no—
397C2×C4—2621no—
398C2×C4—2621no—
399C2×C4—2621no—
400C2×C4—5241no—
401C2×C4—5241no—
402C2×C4—5241no—
403C2×C4—5241no—
404C2×C4—1311no—
405C2×C4—1311no—
406C2×C4—1311no—
407C2×C4—1311no—
408C2×C4—1311no—
409C2×C4C23×D13111yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—2621no—
414C8—2621no—
415C8—2621no—
416C8—1311no—
417C8—1311no—
418C8D4×D13111yescharacteristic
419C8D4×D13111yescharacteristic
420C22—21no—
421C22—2621no—
422C22—5241no—
423C22—5241no—
424C22—10481no—
425C22—10481no—
426C22—1311no—
427C22—1311no—
428C22#41×D13111yescharacteristic
429C4—21no—
430C4—41no—
431C4—41no—
432C4—2621no—
433C4—5241no—
434C4—1311no—
435C4—1311no—
436C4C2×D4×D13111yescharacteristic
437C4C2×D4×D13111yescharacteristic
438C2—21no—
439C2—41no—
440C2—2621no—
441C2—2621no—
442C2—5241no—
443C2—10481no—
444C2#121×D13111yescharacteristic
subgroupG/H
centerC2#121×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#473×C131C2
socleC2×C131C2×#121
prime radical#22554111C2
hypercenter#473D131
hypocenterC131#3069
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×#16862278C2semidirect
3. #137×D131C2semidirect
4. #22554077C2semidirect
5. #22553586C2semidirect
6. #22553585C2semidirect
7. #22553584C2semidirect
8. #22553578C2semidirect
9. #22553577C2semidirect
10. #22553576C2semidirect
11. #22553575C2semidirect
12. #22553359C2semidirect
13. #22553295C2semidirect
14. #101×D131C2semidirect
15. #473×C131C2semidirect
35. C2×#14523503C22semidirect
37. C2×#14523502C22semidirect
38. C2×#14523500C22semidirect
51. #16862182C22semidirect
52. #16862181C22semidirect
53. #16862181C22semidirect
54. #16862165C22semidirect
55. #16862165C22semidirect
56. #16862164C22semidirect
57. #16862164C22semidirect
58. #16862162C22semidirect
59. #16862162C22semidirect
62. C2×Q16×C131C22semidirect
64. #108×C131C22semidirect
65. #101×C131C22semidirect
66. #101×C131C22semidirect
222. C131#3069semidirect
231. #473D131semidirect
262. C2×Q16C2×D131semidirect
281. #108C2×D131semidirect
286. #101C2×D131semidirect
287. #101C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541112
C2×#168622782
#137×D1312
#225540772
#225535862
#225535852
#225535842
#225535782
#225535772
#225535762
#225535752
#225533592
#225532952
#101×D1312
#473×C1312
#3069131