Finite Group Databasebeta

← #30675634

Subgroups — #30675634

Order 16768.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862273C211yesmaximal, characteristic, complemented
2C2×#16862252C211yesmaximal, characteristic, complemented
3#22553892C211yesmaximal, characteristic, complemented
4#22553878C211yesmaximal, characteristic, complemented
5#22553566C211yesmaximal, characteristic, complemented
6#22553563C211yesmaximal, characteristic, complemented
7#22553560C211yesmaximal, characteristic, complemented
8#22553555C211yesmaximal, characteristic, complemented
9#22553548C211yesmaximal, characteristic, complemented
10#22553547C211yesmaximal, characteristic, complemented
11#22553544C211yesmaximal, characteristic, complemented
12#22553351C211yesmaximal, characteristic, complemented
13#22553307C211yesmaximal, characteristic, complemented
14#102×D131C211yesmaximal, characteristic, complemented
15#465×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523513C2211yescharacteristic
18C2×D4×D131C2211yescharacteristic
19#16862273—24no—
20C2×#14523500C2211yescharacteristic
21C2×D1048C2211yescharacteristic
22C2×#14523498C2211yescharacteristic
23#16862252—24no—
24C2×#14523497C2211yescharacteristic, complemented
25C2×#14523496C2211yescharacteristic
26C2×#14523496C2211yescharacteristic, complemented
27C2×C8×D131C2211yescharacteristic
28C2×C8×D131C2211yescharacteristic, complemented
29#16862242—21no—
30#16862232—21no—
31#16862224—21no—
32#16862222C2211yescharacteristic, complemented
33#16862221—21no—
34#16862221C2211yescharacteristic
35#16862221C2211yescharacteristic, complemented
36#16862220C2211yescharacteristic
37#16862220C2211yescharacteristic, complemented
38#16862219—21no—
39#16862219—21no—
40#16862217C2211yescharacteristic
41#16862216—21no—
42#16862214—21no—
43#16862204C2211yescharacteristic
44C4⋊C4×D131C2211yescharacteristic
45#16862180C2211yescharacteristic, complemented
46#16862179C2211yescharacteristic, complemented
47#16862177C2211yescharacteristic, complemented
48#16862170C2211yescharacteristic, complemented
49#16862168C2211yescharacteristic
50#16862168C2211yescharacteristic, complemented
51#16862167C2211yescharacteristic
52#16862166C2211yescharacteristic
53#16862166C2211yescharacteristic, complemented
54C2×SD16×C131C2211yescharacteristic
55C22×C8×C131C2211yescharacteristic, complemented
56C22⋊Q8×C131C2211yescharacteristic, complemented
57#121×C131C2211yescharacteristic
58#106×C131C2211yescharacteristic, complemented
59#103×C131C2211yescharacteristic, complemented
60#102×C131C2211yescharacteristic
61C23×D131—21no—
62#14523514—22no—
63#14523514—22no—
64#14523513—22no—
65#14523513—22no—
66C2×Dic262D411yescharacteristic
67C2×D524—21no—
68C2×D524—21no—
69C2×D524—21no—
70C2×D524D411yescharacteristic
71C2×D524C2311yescharacteristic
72C2×D524C2311yescharacteristic
73C2×C22⋊D131—21no—
74C2×C22⋊D131—21no—
75C2×C22⋊D131—21no—
76C2×C22⋊D131—21no—
77C2×C22⋊D131D411yescharacteristic
78C2×C22⋊D131D411yescharacteristic
79D4×D131—22no—
80D4×D131—22no—
81C2×C4×D131—21no—
82C2×C4×D131—21no—
83C2×C4×D131C2311yescharacteristic
84C2×C4×D131C2311yescharacteristic
85#14523500—22no—
86D1048—22no—
87#14523498—22no—
88#14523497—22no—
89#14523496—22no—
90#14523496D414yescomplemented
91C2×C131⋊C8C2311yescharacteristic
92C2×C131⋊C8C2311yescharacteristic
93C8×D131—22no—
94C8×D131—22no—
95C4⋊Dic131C2311yescharacteristic
96C4⋊Dic131C2311yescharacteristic
97Dic131⋊C4—21no—
98Dic131⋊C4—21no—
99C22⋊Dic131—21no—
100(C2×D131)⋊C4—21no—
101(C2×D131)⋊C4—21no—
102(C2×D131)⋊C4—21no—
103(C2×D131)⋊C4—21no—
104(C2×D131)⋊C4—21no—
105(C2×D131)⋊C4—21no—
106C4×Dic131—21no—
107C2×Q8×C131C2311yescharacteristic
108C2×D4×C131—21no—
109C2×D4×C131C2311yescharacteristic
110C22×C4×C131C2311yescharacteristic
111SD16×C131—22no—
112C2×C8×C131—22no—
113C2×C8×C131C2311yescharacteristic
114C2×C8×C131C2311yescharacteristic
115C4⋊C4×C131—21no—
116C4⋊C4×C131C2311yescharacteristic
117C4⋊C4×C131C2311yescharacteristic
118C22⋊C4×C131—21no—
119C22⋊C4×C131—21no—
120C22×D131—21no—
121C22×D131—21no—
122C22×D131—42no—
123C22×D131—81no—
124C22×D131—81no—
125C22×D131C2×D411yescharacteristic
126C22×D131C2×D411yescharacteristic
127Dic262—22no—
128D524—22no—
129D524—21no—
130D524—21no—
131D524—21no—
132D524—21no—
133D524—42no—
134D524—81no—
135D524—81no—
136C22⋊D131—22no—
137C22⋊D131—22no—
138C22⋊D131—42no—
139C22⋊D131—81no—
140C22⋊D131—81no—
141C22⋊D131—81no—
142C2×Dic131—21no—
143C2×Dic131—21no—
144C2×Dic131C2×D411yescharacteristic
145C2×Dic131C2×D411yescharacteristic
146C4×D131—21no—
147C4×D131—21no—
148C4×D131—42no—
149C4×D131—81no—
150C4×D131C2×D412yes—
151C4×D131#4112yes—
152C131⋊C8—21no—
153C131⋊C8C2×D412yes—
154C23×C131—21no—
155C23×C131C2×D411yescharacteristic
156Q8×C131—21no—
157Q8×C131—21no—
158D4×C131—21no—
159D4×C131—21no—
160D4×C131—81no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131C2×D411yescharacteristic
167C2×C4×C131C2411yescharacteristic
168C8×C131—21no—
169C8×C131C2×D412yes—
170C22×C131—21no—
171C22×C131—21no—
172C22×C131—21no—
173C22×C131—41no—
174C22×C131—41no—
175C22×C131—41no—
176C22×C131C22×D411yescharacteristic
177C2×D131—21no—
178C2×D131—21no—
179C2×D131—21no—
180C2×D131—21no—
181C2×D131—21no—
182C2×D131—41no—
183C2×D131—41no—
184C2×D131—41no—
185C2×D131—41no—
186C2×D131—41no—
187C2×D131—41no—
188C2×D131—84no—
189C2×D131#12112yes—
190C4×C131—21no—
191C4×C131—41no—
192C4×C131—41no—
193C4×C131—41no—
194C4×C131C22×D411yescharacteristic
195C4×C131C2×#4111yescharacteristic
196Dic131—21no—
197Dic131—41no—
198Dic131—41no—
199Dic131#12112yes—
200C2×C131—41no—
201C2×C131—81no—
202C2×C131C2×#12111yescharacteristic
203C2×C131#57611yescharacteristic
204C2×C131#57711yescharacteristic
205D131—22no—
206D131—41no—
207D131—81no—
208D131—81no—
209C131#306211yescharacteristic, complemented, Sylow
210#3062—1311nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1311no—
212C2×#131—1311no—
213#537—1311no—
214#535—1311no—
215#469—1311no—
216#469—1311no—
217#468—1311no—
218#468—1311no—
219#466—1311no—
220#466—1311no—
221#465—1311no—
222#465D13111yescharacteristic, complemented
223#428—1311no—
224#416—1311no—
225C2×#102—1311no—
226C2×#41—1311no—
227C2×#41—1311no—
228C22×D4—1311no—
229AGL(1,ℤ/8)—2624no—
230C2×SD16—1311no—
231C2×SD16C2×D13111yescharacteristic
232C2×D8—1311no—
233C2×D8—1311no—
234#131—2624no—
235C2×M16—1311no—
236C2×M16—1311no—
237C2×M16—1311no—
238C22×C8—1311no—
239C22×C8—1311no—
240C22×C8C2×D13111yescharacteristic, complemented
241#127—2621no—
242#124—2621no—
243C22⋊Q8—2621no—
244C22⋊Q8—1311no—
245C22⋊Q8C2×D13111yescharacteristic, complemented
246#121—2621no—
247#121—2621no—
248#121—2621no—
249#121—1311no—
250#121—1311no—
251#121—1311no—
252#121—1311no—
253#121—1311no—
254#121C2×D13111yescharacteristic
255#120—2621no—
256#120—2621no—
257#117—1311no—
258C2×C4⋊C4—1311no—
259#107—1311no—
260#107—1311no—
261#106—1311no—
262#106C2×D13111yescharacteristic, complemented
263#103—1311no—
264#103C2×D13111yescharacteristic, complemented
265#102—1311no—
266#102—1311no—
267#102—1311no—
268#102—1311no—
269#102—1311no—
270#102C2×D13111yescharacteristic
271C24—2621no—
272#41—2622no—
273#41—2622no—
274#41—2622no—
275#41—2622no—
276C2×Q8—1311no—
277C2×Q8C22×D13111yescharacteristic
278C2×D4—21no—
279C2×D4—2621no—
280C2×D4—2621no—
281C2×D4—2621no—
282C2×D4—2621no—
283C2×D4—2621no—
284C2×D4—2621no—
285C2×D4—2622no—
286C2×D4—2622no—
287C2×D4—2621no—
288C2×D4—1311no—
289C2×D4—1311no—
290C2×D4—1311no—
291C2×D4—1311no—
292C2×D4—1311no—
293C2×D4C22×D13111yescharacteristic
294C22×C4—2621no—
295C22×C4—2621no—
296C22×C4—1311no—
297C22×C4—1311no—
298C22×C4C22×D13111yescharacteristic
299SD16—22no—
300SD16—2622no—
301D8—2622no—
302D8—2622no—
303M16—2622no—
304M16—2622no—
305M16—1314no—
306C2×C8—22no—
307C2×C8—2622no—
308C2×C8—2622no—
309C2×C8—1311no—
310C2×C8—1311no—
311C2×C8C22×D13111yescharacteristic
312C2×C8C22×D13111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2621no—
315C4⋊C4—2621no—
316C4⋊C4—1311no—
317C4⋊C4—1311no—
318C4⋊C4C22×D13111yescharacteristic
319C4⋊C4C22×D13111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2621no—
323C22⋊C4—2621no—
324C22⋊C4—2621no—
325C22⋊C4—2621no—
326C22⋊C4—2621no—
327C22⋊C4—2621no—
328C22⋊C4—2621no—
329C42—2621no—
330C23—21no—
331C23—2621no—
332C23—2621no—
333C23—5242no—
334C23—10481no—
335C23—10481no—
336C23—1311no—
337C23—1311no—
338C23D4×D13111yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—2622no—
342D4—21no—
343D4—21no—
344D4—2622no—
345D4—2622no—
346D4—2622no—
347D4—2621no—
348D4—2621no—
349D4—2621no—
350D4—2621no—
351D4—81no—
352D4—5242no—
353D4—5242no—
354D4—10481no—
355D4—10481no—
356D4—10481no—
357D4—10481no—
358D4—10481no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2621no—
365C2×C4—2621no—
366C2×C4—2621no—
367C2×C4—2621no—
368C2×C4—5242no—
369C2×C4—10481no—
370C2×C4—1311no—
371C2×C4—1311no—
372C2×C4—1312no—
373C2×C4—1312no—
374C2×C4D4×D13111yescharacteristic
375C2×C4C23×D13111yescharacteristic
376C8—21no—
377C8—2621no—
378C8—1312no—
379C8D4×D13112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2621no—
387C22—2621no—
388C22—2621no—
389C22—2621no—
390C22—2621no—
391C22—5241no—
392C22—5241no—
393C22—5241no—
394C22—5241no—
395C22—5241no—
396C22—5241no—
397C22—10484no—
398C22—1312no—
399C22C2×D4×D13111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—2621no—
405C4—5241no—
406C4—5241no—
407C4—1312no—
408C4C2×D4×D13111yescharacteristic
409C4#41×D13111yescharacteristic
410C2—41no—
411C2—2622no—
412C2—81no—
413C2—5241no—
414C2—10481no—
415C2—10481no—
416C2#121×D13111yescharacteristic
417C2#2255411411yescharacteristic
418C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#465×C131C2
socleC22×C131C22×D4
hypercenter#465D131
hypocenterC131#3062
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16862273C2semidirect
2. C2×#16862252C2semidirect
3. #22553892C2semidirect
4. #22553878C2semidirect
5. #22553566C2semidirect
6. #22553563C2semidirect
7. #22553560C2semidirect
8. #22553555C2semidirect
9. #22553548C2semidirect
10. #22553547C2semidirect
11. #22553544C2semidirect
12. #22553351C2semidirect
13. #22553307C2semidirect
14. #102×D131C2semidirect
15. #465×C131C2semidirect
24. C2×#14523497C22semidirect
26. C2×#14523496C22semidirect
28. C2×C8×D131C22semidirect
32. #16862222C22semidirect
35. #16862221C22semidirect
37. #16862220C22semidirect
45. #16862180C22semidirect
46. #16862179C22semidirect
47. #16862177C22semidirect
48. #16862170C22semidirect
50. #16862168C22semidirect
53. #16862166C22semidirect
55. C22×C8×C131C22semidirect
56. C22⋊Q8×C131C22semidirect
58. #106×C131C22semidirect
59. #103×C131C22semidirect
90. #14523496D4semidirect
209. C131#3062semidirect
222. #465D131semidirect
240. C22×C8C2×D131semidirect
245. C22⋊Q8C2×D131semidirect
262. #106C2×D131semidirect
264. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622732
C2×#168622522
#225538922
#225538782
#225535662
#225535632
#225535602
#225535552
#225535482
#225535472
#225535442
#225533512
#225533072
#102×D1312
#465×C1312
#3062131