Finite Group Databasebeta

← #30675497

Subgroups — #30675497

Order 16768.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862273C211yesmaximal, characteristic, complemented
2C2×#16862267C211yesmaximal, characteristic, complemented
3#22553875C211yesmaximal, characteristic, complemented
4#22553794C211yesmaximal, characteristic, complemented
5#22553534C211yesmaximal, characteristic, complemented
6#22553532C211yesmaximal, characteristic, complemented
7#22553528C211yesmaximal, characteristic, complemented
8#22553526C211yesmaximal, characteristic, complemented
9#22553510C211yesmaximal, characteristic, complemented
10#22553506C211yesmaximal, characteristic, complemented
11#22553504C211yesmaximal, characteristic, complemented
12#22553337C211yesmaximal, characteristic, complemented
13#22553310C211yesmaximal, characteristic, complemented
14#102×D131C211yesmaximal, characteristic, complemented
15#460×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523512C2211yescharacteristic
18C2×D4×D131C2211yescharacteristic
19#16862273—24no—
20#16862267—24no—
21C2×#14523503C2211yescharacteristic, complemented
22C2×#14523502C2211yescharacteristic, complemented
23C2×#14523500C2211yescharacteristic
24C2×D1048C2211yescharacteristic
25C2×#14523498C2211yescharacteristic
26C2×#14523498C2211yescharacteristic, complemented
27C2×#14523496C2211yescharacteristic
28C2×C8×D131C2211yescharacteristic
29#16862242C2211yescharacteristic
30#16862234C2211yescharacteristic
31#16862233C2211yescharacteristic
32#16862218—21no—
33#16862218—21no—
34#16862214—21no—
35#16862214—21no—
36Q8×Dic131C2211yescharacteristic, complemented
37C4×D524—21no—
38C4×D524C2211yescharacteristic, complemented
39#16862208—21no—
40#16862208C2211yescharacteristic, complemented
41#16862203—21no—
42#16862203C2211yescharacteristic
43C4⋊C4×D131—21no—
44C4⋊C4×D131C2211yescharacteristic
45C4⋊(C131⋊C8)C2211yescharacteristic, complemented
46Dic131⋊C8C2211yescharacteristic, complemented
47Dic131⋊C8C2211yescharacteristic, complemented
48#16862171C2211yescharacteristic, complemented
49#16862169C2211yescharacteristic, complemented
50#16862168C2211yescharacteristic
51#16862167C2211yescharacteristic
52#16862167C2211yescharacteristic, complemented
53#16862166C2211yescharacteristic
54C2×SD16×C131C2211yescharacteristic
55C2×SD16×C131C2211yescharacteristic, complemented
56#127×C131C2211yescharacteristic
57C4×Q8×C131C2211yescharacteristic, complemented
58C4⋊C8×C131C2211yescharacteristic, complemented
59#102×C131C2211yescharacteristic
60#102×C131C2211yescharacteristic, complemented
61C23×D131—21no—
62#14523514—22no—
63#14523514D414yescomplemented
64#14523512—22no—
65#14523512—22no—
66C2×Dic262C2311yescharacteristic
67C2×D524—21no—
68C2×D524C2311yescharacteristic
69C2×D524C2311yescharacteristic
70C2×C22⋊D131—21no—
71C2×C22⋊D131—21no—
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74D4×D131—22no—
75D4×D131—22no—
76C22×Dic131—21no—
77C2×C4×D131—21no—
78C2×C4×D131—21no—
79C2×C4×D131—21no—
80C2×C4×D131—21no—
81C2×C4×D131C2311yescharacteristic
82#14523503—22no—
83#14523502—22no—
84#14523500—22no—
85D1048—22no—
86#14523498—22no—
87#14523498—22no—
88#14523496—22no—
89C2×C131⋊C8C2311yescharacteristic
90C2×C131⋊C8C2311yescharacteristic
91C8×D131—22no—
92C4⋊Dic131—21no—
93C4⋊Dic131C2311yescharacteristic
94Dic131⋊C4—21no—
95Dic131⋊C4D411yescharacteristic
96Dic131⋊C4D411yescharacteristic
97C22⋊Dic131—21no—
98C22⋊Dic131—21no—
99(C2×D131)⋊C4—21no—
100(C2×D131)⋊C4—21no—
101(C2×D131)⋊C4—21no—
102(C2×D131)⋊C4—21no—
103(C2×D131)⋊C4D411yescharacteristic
104(C2×D131)⋊C4D411yescharacteristic
105C4×Dic131—21no—
106C4×Dic131C2311yescharacteristic
107C2×Q8×C131C2311yescharacteristic
108C2×D4×C131—21no—
109C2×D4×C131—21no—
110C2×D4×C131C2311yescharacteristic
111C2×D4×C131C2311yescharacteristic
112SD16×C131—22no—
113SD16×C131—22no—
114C2×C8×C131C2311yescharacteristic
115C2×C8×C131C2311yescharacteristic
116C4⋊C4×C131—21no—
117C4⋊C4×C131C2311yescharacteristic
118C42×C131—21no—
119C42×C131C2311yescharacteristic
120C22×D131—21no—
121C22×D131—21no—
122C22×D131—42no—
123C22×D131—81no—
124C22×D131—81no—
125C22×D131C2×D411yescharacteristic
126Dic262—21no—
127Dic262—21no—
128D524—24no—
129D524—21no—
130D524—21no—
131D524—21no—
132D524C2×D412yes—
133C22⋊D131—42no—
134C22⋊D131—42no—
135C22⋊D131—81no—
136C22⋊D131—81no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C2×Dic131—42no—
140C2×Dic131C2×D411yescharacteristic
141C2×Dic131C2×D411yescharacteristic
142C2×Dic131C2×D411yescharacteristic
143C4×D131—24no—
144C4×D131—81no—
145C4×D131—81no—
146C4×D131—81no—
147C4×D131C2×D412yes—
148C4×D131#4112yes—
149C131⋊C8—21no—
150C131⋊C8—21no—
151C23×C131—21no—
152C23×C131—21no—
153Q8×C131—21no—
154Q8×C131C2×D412yes—
155D4×C131—21no—
156D4×C131—21no—
157D4×C131—21no—
158D4×C131—21no—
159D4×C131—81no—
160D4×C131—81no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131C2×D411yescharacteristic
164C2×C4×C131C2×D411yescharacteristic
165C2×C4×C131C2411yescharacteristic
166C8×C131—21no—
167C8×C131—21no—
168C22×C131—41no—
169C22×C131—41no—
170C22×C131—41no—
171C22×C131—41no—
172C22×C131—41no—
173C22×C131—41no—
174C22×C131C22×D411yescharacteristic
175C2×D131—21no—
176C2×D131—21no—
177C2×D131—22no—
178C2×D131—41no—
179C2×D131—41no—
180C2×D131—41no—
181C2×D131—41no—
182C2×D131—41no—
183C2×D131—84no—
184C2×D131#12112yes—
185C4×C131—22no—
186C4×C131—21no—
187C4×C131—21no—
188C4×C131—41no—
189C4×C131C22×D411yescharacteristic
190C4×C131C2×#4111yescharacteristic
191Dic131—21no—
192Dic131—21no—
193Dic131—41no—
194Dic131—41no—
195Dic131#12112yes—
196C2×C131—81no—
197C2×C131—81no—
198C2×C131C2×#12111yescharacteristic
199C2×C131#57611yescharacteristic
200C2×C131#57711yescharacteristic
201D131—22no—
202D131—42no—
203D131—81no—
204C131#304511yescharacteristic, complemented, Sylow
205#3045—1311nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1311no—
207C2×#135—1311no—
208#535—1311no—
209#520—1311no—
210#464—1311no—
211#464—1311no—
212#463—1311no—
213#463—1311no—
214#460—1311no—
215#460D13111yescharacteristic, complemented
216#459—1311no—
217#459—1311no—
218#424—1311no—
219#417—1311no—
220C2×#102—1311no—
221C2×#41—1311no—
222C2×#41—1311no—
223C22×D4—1311no—
224AGL(1,ℤ/8)—2624no—
225#135—2624no—
226C2×Q16—1311no—
227C2×SD16—1311no—
228C2×SD16—1311no—
229C2×SD16C2×D13111yescharacteristic
230C2×SD16C2×D13111yescharacteristic, complemented
231C2×D8—1311no—
232C2×D8—1311no—
233C2×D8—1311no—
234C2×M16—1311no—
235C22×C8—1311no—
236#127—1311no—
237#127C2×D13111yescharacteristic
238#124—1311no—
239#124—1311no—
240#121—2621no—
241#121—2621no—
242#120—2621no—
243#120—2621no—
244C4×Q8—1311no—
245C4×Q8C2×D13111yescharacteristic, complemented
246C4×D4—2621no—
247C4×D4—2621no—
248C4×D4—1311no—
249C4×D4—1311no—
250#117—2621no—
251#117—1311no—
252C2×C4⋊C4—2621no—
253C2×C4⋊C4—1311no—
254C4⋊C8—1311no—
255C4⋊C8—1311no—
256C4⋊C8—1311no—
257C4⋊C8C2×D13111yescharacteristic, complemented
258#103—1311no—
259#103—1311no—
260#102—1311no—
261#102—1311no—
262#102—1311no—
263#102—1311no—
264#102C2×D13111yescharacteristic
265#102C2×D13111yescharacteristic, complemented
266C24—2621no—
267#41—2622no—
268#41—2622no—
269#41—2622no—
270#41—1314no—
271C2×Q8—1311no—
272C2×Q8C22×D13111yescharacteristic
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—2621no—
276C2×D4—2621no—
277C2×D4—2622no—
278C2×D4—2622no—
279C2×D4—2621no—
280C2×D4—2621no—
281C2×D4—2621no—
282C2×D4—1311no—
283C2×D4—1311no—
284C2×D4C22×D13111yescharacteristic
285C2×D4C22×D13111yescharacteristic
286C22×C4—2621no—
287C22×C4—2621no—
288C22×C4—2621no—
289C22×C4—2621no—
290C22×C4—2621no—
291C22×C4—1311no—
292Q16—2622no—
293SD16—22no—
294SD16—22no—
295SD16—2622no—
296SD16—2622no—
297D8—2622no—
298D8—2622no—
299D8—2622no—
300M16—2622no—
301C2×C8—2622no—
302C2×C8—1311no—
303C2×C8—1311no—
304C2×C8C22×D13111yescharacteristic
305C2×C8C22×D13111yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—2621no—
308C4⋊C4—2621no—
309C4⋊C4—1311no—
310C4⋊C4—1311no—
311C4⋊C4—1311no—
312C4⋊C4C22×D13111yescharacteristic
313C22⋊C4—2621no—
314C22⋊C4—2621no—
315C22⋊C4—2621no—
316C22⋊C4—2621no—
317C22⋊C4—2621no—
318C22⋊C4—2621no—
319C22⋊C4—1311no—
320C22⋊C4—1311no—
321C42—21no—
322C42—2621no—
323C42—1311no—
324C42C22×D13111yescharacteristic
325C23—21no—
326C23—21no—
327C23—2621no—
328C23—2621no—
329C23—5242no—
330C23—10481no—
331C23—10481no—
332C23—1311no—
333Q8—21no—
334Q8—2621no—
335Q8—2621no—
336Q8D4×D13112yes—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—2624no—
342D4—2621no—
343D4—2621no—
344D4—2621no—
345D4—81no—
346D4—81no—
347D4—5242no—
348D4—5242no—
349D4—10481no—
350D4—10481no—
351D4—1312no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—2624no—
355C2×C4—2621no—
356C2×C4—2621no—
357C2×C4—5242no—
358C2×C4—10481no—
359C2×C4—10481no—
360C2×C4—10481no—
361C2×C4—1311no—
362C2×C4—1311no—
363C2×C4—1311no—
364C2×C4—1312no—
365C2×C4—1312no—
366C2×C4D4×D13111yescharacteristic
367C2×C4D4×D13111yescharacteristic
368C2×C4C23×D13111yescharacteristic
369C8—21no—
370C8—21no—
371C8—2621no—
372C8—2621no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2621no—
380C22—2621no—
381C22—2622no—
382C22—5241no—
383C22—5241no—
384C22—5241no—
385C22—5241no—
386C22—5241no—
387C22—10484no—
388C22—1312no—
389C22C2×D4×D13111yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—2621no—
395C4—2621no—
396C4—5241no—
397C4—5241no—
398C4—1312no—
399C4C2×D4×D13111yescharacteristic
400C4#41×D13111yescharacteristic
401C2—2622no—
402C2—81no—
403C2—81no—
404C2—5242no—
405C2—10481no—
406C2#121×D13111yescharacteristic
407C2#2255411611yescharacteristic
408C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#460×C131C2
socleC22×C131C22×D4
hypercenter#460D131
hypocenterC131#3045
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16862273C2semidirect
2. C2×#16862267C2semidirect
3. #22553875C2semidirect
4. #22553794C2semidirect
5. #22553534C2semidirect
6. #22553532C2semidirect
7. #22553528C2semidirect
8. #22553526C2semidirect
9. #22553510C2semidirect
10. #22553506C2semidirect
11. #22553504C2semidirect
12. #22553337C2semidirect
13. #22553310C2semidirect
14. #102×D131C2semidirect
15. #460×C131C2semidirect
21. C2×#14523503C22semidirect
22. C2×#14523502C22semidirect
26. C2×#14523498C22semidirect
36. Q8×Dic131C22semidirect
38. C4×D524C22semidirect
40. #16862208C22semidirect
45. C4⋊(C131⋊C8)C22semidirect
46. Dic131⋊C8C22semidirect
47. Dic131⋊C8C22semidirect
48. #16862171C22semidirect
49. #16862169C22semidirect
52. #16862167C22semidirect
55. C2×SD16×C131C22semidirect
57. C4×Q8×C131C22semidirect
58. C4⋊C8×C131C22semidirect
60. #102×C131C22semidirect
63. #14523514D4semidirect
204. C131#3045semidirect
215. #460D131semidirect
230. C2×SD16C2×D131semidirect
245. C4×Q8C2×D131semidirect
257. C4⋊C8C2×D131semidirect
265. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622732
C2×#168622672
#225538752
#225537942
#225535342
#225535322
#225535282
#225535262
#225535102
#225535062
#225535042
#225533372
#225533102
#102×D1312
#460×C1312
#3045131