Finite Group Databasebeta

← #30675320

Subgroups — #30675320

Order 16768.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862292C211yesmaximal, characteristic, complemented
2#22554112C211yesmaximal, characteristic, complemented
3#22554110C211yesmaximal, characteristic, complemented
4#22553881C211yesmaximal, characteristic, complemented
5#22553491C214yesmaximal, complemented
6#22553481C214yesmaximal, complemented
7#22553330C211yesmaximal, characteristic, complemented
8C2×#16862162C211yesmaximal, characteristic, complemented
9#421×C131C211yesmaximal, characteristic, complemented
10#16862292—22no—
11#16862292—22no—
12#16862292—22no—
13#16862292C2214yes—
14C2×#14523514—21no—
15C2×#14523513—21no—
16C2×#14523513C2211yescharacteristic
17C2×#14523513C2211yescharacteristic, complemented
18C22×D524—21no—
19C22×D524C2211yescharacteristic
20C2×D4×D131—22no—
21C2×D4×D131—21no—
22#16862279—21no—
23#16862279C2212yescomplemented
24#16862277—21no—
25#16862276C2212yescomplemented
26#16862274—21no—
27#16862274C2212yescomplemented
28#16862272—21no—
29#16862271C2212yescomplemented
30#16862268—22no—
31#16862266—22no—
32C2×#14523497C2211yescharacteristic, complemented
33#16862243C2212yescomplemented
34#16862234C2212yescomplemented
35#16862221—22no—
36#16862216—22no—
37#16862174C2214yescomplemented
38#16862162C2214yescomplemented
39C2×#41×C131C2211yescharacteristic
40C2×M16×C131C2211yescharacteristic, complemented
41#117×C131C2211yescharacteristic, complemented
42#104×C131C2214yescomplemented
43C23×D131—22no—
44C23×D131D412yes—
45#14523514—24no—
46#14523514—24no—
47#14523513—24no—
48#14523513—24no—
49#14523513—21no—
50#14523513—21no—
51#14523513—21no—
52#14523513—21no—
53#14523513—21no—
54#14523513—21no—
55#14523513D412yescomplemented
56#14523513C2312yes—
57C2×Dic262D411yescharacteristic, complemented
58C2×Dic262C2311yescharacteristic, complemented
59C2×D524—22no—
60C2×D524—24no—
61C2×D524—24no—
62C2×D524—24no—
63C2×D524—21no—
64C2×D524—21no—
65C2×D524—21no—
66C2×D524—21no—
67C2×D524D412yes—
68C2×D524D411yescharacteristic, complemented
69C2×D524C2312yes—
70C2×D524C2311yescharacteristic, complemented
71C2×C22⋊D131—22no—
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74D4×D131—28no—
75D4×D131—24no—
76D4×D131—28no—
77D4×D131—24no—
78C2×C4×D131—22no—
79C2×C4×D131—21no—
80C2×C4×D131—21no—
81Dic524—22no—
82#14523503—22no—
83#14523502—22no—
84#14523502—22no—
85#14523501—22no—
86#14523500—22no—
87D1048—22no—
88#14523498—22no—
89#14523497—21no—
90#14523497C2312yescomplemented
91C2×C131⋊C8—21no—
92(C2×D131)⋊C4—24no—
93#41×C131—21no—
94#41×C131—21no—
95#41×C131D412yescomplemented
96#41×C131C2312yes—
97C2×Q8×C131D411yescharacteristic, complemented
98C2×D4×C131—21no—
99C2×D4×C131D411yescharacteristic, complemented
100C22×C4×C131—21no—
101C22×C4×C131C2311yescharacteristic
102M16×C131—21no—
103M16×C131C2312yescomplemented
104C2×C8×C131—21no—
105C4⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C42×C131C2312yescomplemented
108C22×D131—22no—
109C22×D131—21no—
110C22×D131—28no—
111C22×D131—22no—
112C22×D131—21no—
113C22×D131—82no—
114C22×D131—84no—
115C22×D131C2×D414yes—
116Dic262—21no—
117Dic262—21no—
118Dic262—21no—
119Dic262C2×D412yes—
120D524—28no—
121D524—22no—
122D524—22no—
123D524—21no—
124D524—22no—
125D524—22no—
126D524—21no—
127D524—21no—
128D524—84no—
129D524C2×D412yes—
130C22⋊D131—28no—
131C22⋊D131—22no—
132C22⋊D131—22no—
133C22⋊D131—41no—
134C22⋊D131—41no—
135C2×Dic131—21no—
136C2×Dic131—21no—
137C4×D131—28no—
138C4×D131—22no—
139C4×D131—22no—
140C4×D131—41no—
141C4×D131—41no—
142C131⋊C8—22no—
143C23×C131—21no—
144C23×C131C2×D411yescharacteristic
145Q8×C131—21no—
146Q8×C131C2×D412yes—
147D4×C131—22no—
148D4×C131—22no—
149D4×C131—21no—
150D4×C131C2×D412yes—
151C2×C4×C131—21no—
152C2×C4×C131—22no—
153C2×C4×C131—22no—
154C2×C4×C131—22no—
155C2×C4×C131C2×D411yescharacteristic
156C2×C4×C131C2×D411yescharacteristic
157C2×C4×C131C2×D411yescharacteristic
158C2×C4×C131C2×D411yescharacteristic
159C2×C4×C131C2×D411yescharacteristic
160C2×C4×C131C2411yescharacteristic
161C8×C131—22no—
162C22×C131—22no—
163C22×C131—41no—
164C22×C131—42no—
165C22×C131#12011yescharacteristic
166C22×C131#12011yescharacteristic
167C22×C131C22×D411yescharacteristic
168C2×D131—24no—
169C2×D131—22no—
170C2×D131—44no—
171C2×D131—42no—
172C2×D131—44no—
173C2×D131—42no—
174C2×D131—41no—
175C2×D131—82no—
176C2×D131—88no—
177C4×C131—22no—
178C4×C131—42no—
179C4×C131#12011yescharacteristic
180C4×C131#12011yescharacteristic
181C4×C131C22×D411yescharacteristic
182C4×C131C22×D411yescharacteristic
183Dic131—22no—
184Dic131—41no—
185C2×C131—21no—
186C2×C131—21no—
187C2×C131—21no—
188C2×C131—42no—
189C2×C131C2×#12011yescharacteristic
190D131—44no—
191D131—42no—
192D131—81no—
193C131#301911yescharacteristic, complemented, Sylow
194#3019—1311nomaximal, Sylow
195C2×H(5,𝔽2)—1311no—
196#575—1311no—
197#575—1311no—
198#535—1311no—
199#455—1314no—
200#453—1314no—
201#421—1311no—
202#421D13111yescharacteristic, complemented
203C2×#100—1311no—
204H(5,𝔽2)—2622no—
205H(5,𝔽2)—2622no—
206H(5,𝔽2)—2622no—
207H(5,𝔽2)—1314no—
208C2×#41—2621no—
209C2×#41—2621no—
210C2×#41—1311no—
211C2×#41—1311no—
212C2×#41C2×D13111yescharacteristic
213C22×D4—2621no—
214C22×D4—2622no—
215C22×D4—2621no—
216C22×D4—1311no—
217#137—2621no—
218#137—2621no—
219#137—1312no—
220#137—1312no—
221AGL(1,ℤ/8)—2621no—
222AGL(1,ℤ/8)—2621no—
223AGL(1,ℤ/8)—1312no—
224AGL(1,ℤ/8)—1312no—
225#135—2622no—
226#135—2622no—
227C2×M16—1311no—
228C2×M16C2×D13111yescharacteristic, complemented
229#127—1312no—
230#124—1312no—
231#121—2622no—
232#120—2622no—
233#117C2×D13111yescharacteristic, complemented
234#104—1314no—
235#104C2×D13114yescomplemented
236#100—1314no—
237C24—2622no—
238C24—1312no—
239#41—21no—
240#41—21no—
241#41—2624no—
242#41—2624no—
243#41—2624no—
244#41—2624no—
245#41—2621no—
246#41—2621no—
247#41—2621no—
248#41—2621no—
249#41—2621no—
250#41—2621no—
251#41—1312no—
252#41—1312no—
253#41D52412yescomplemented
254#41C22×D13112yes—
255C2×Q8—1311no—
256C2×Q8—1311no—
257C2×Q8D52411yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—2622no—
260C2×D4—2624no—
261C2×D4—2624no—
262C2×D4—2624no—
263C2×D4—2628no—
264C2×D4—2624no—
265C2×D4—2628no—
266C2×D4—2622no—
267C2×D4—2621no—
268C2×D4—2624no—
269C2×D4—2621no—
270C2×D4—2621no—
271C2×D4—2621no—
272C2×D4—2621no—
273C2×D4—2621no—
274C2×D4—1312no—
275C2×D4—1311no—
276C2×D4—1311no—
277C2×D4—1312no—
278C2×D4D52411yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—2622no—
281C22×C4—2621no—
282C22×C4—2621no—
283C22×C4C22×D13111yescharacteristic
284Q16—2622no—
285Q16—2622no—
286SD16—2622no—
287SD16—2622no—
288SD16—2622no—
289SD16—2622no—
290D8—2622no—
291D8—2622no—
292M16—21no—
293M16—2621no—
294M16—1312no—
295M16C22×D13112yescomplemented
296C2×C8—21no—
297C2×C8—2621no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2624no—
301C42C22×D13112yescomplemented
302C23—21no—
303C23—2628no—
304C23—2622no—
305C23—2621no—
306C23—2622no—
307C23—2621no—
308C23—10482no—
309C23—10484no—
310C23—1314no—
311C23D4×D13111yescharacteristic
312Q8—21no—
313Q8—2621no—
314Q8—2621no—
315Q8—2621no—
316Q8—1312no—
317Q8C2×D52412yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—2628no—
322D4—2628no—
323D4—2622no—
324D4—2622no—
325D4—2622no—
326D4—2621no—
327D4—2622no—
328D4—2622no—
329D4—2621no—
330D4—2622no—
331D4—2621no—
332D4—5241no—
333D4—5241no—
334D4—10484no—
335D4—1312no—
336D4C2×D52412yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—2628no—
342C2×C4—2621no—
343C2×C4—2622no—
344C2×C4—2622no—
345C2×C4—2621no—
346C2×C4—5241no—
347C2×C4—5241no—
348C2×C4D4×D13111yescharacteristic
349C2×C4D4×D13111yescharacteristic
350C2×C4D4×D13111yescharacteristic
351C2×C4C2×D52411yescharacteristic
352C2×C4C2×D52411yescharacteristic
353C2×C4C23×D13111yescharacteristic
354C8—22no—
355C8—2622no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—2624no—
360C22—2622no—
361C22—5244no—
362C22—5244no—
363C22—5242no—
364C22—5242no—
365C22—5241no—
366C22—10482no—
367C22—10488no—
368C22#1686221611yescharacteristic
369C22#1686221611yescharacteristic
370C22C2×D4×D13111yescharacteristic
371C4—22no—
372C4—42no—
373C4—2622no—
374C4—5241no—
375C4#1686221611yescharacteristic
376C4#1686221611yescharacteristic
377C4C2×D4×D13111yescharacteristic
378C4C22×D52411yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—5244no—
384C2—5242no—
385C2—10481no—
386C2C2×#1686221611yescharacteristic
subgroupG/H
centerC2C2×#16862216
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#421×C131C2
socleC2×C131C2×#120
hypercenter#421D131
hypocenterC131#3019
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#16862292C2semidirect
2. #22554112C2semidirect
3. #22554110C2semidirect
4. #22553881C2semidirect
5. #22553491C2semidirect
6. #22553481C2semidirect
7. #22553330C2semidirect
8. C2×#16862162C2semidirect
9. #421×C131C2semidirect
17. C2×#14523513C22semidirect
23. #16862279C22semidirect
25. #16862276C22semidirect
27. #16862274C22semidirect
29. #16862271C22semidirect
32. C2×#14523497C22semidirect
33. #16862243C22semidirect
34. #16862234C22semidirect
37. #16862174C22semidirect
38. #16862162C22semidirect
40. C2×M16×C131C22semidirect
41. #117×C131C22semidirect
42. #104×C131C22semidirect
55. #14523513D4semidirect
57. C2×Dic262D4semidirect
58. C2×Dic262C23semidirect
68. C2×D524D4semidirect
70. C2×D524C23semidirect
90. #14523497C23semidirect
95. #41×C131D4semidirect
97. C2×Q8×C131D4semidirect
99. C2×D4×C131D4semidirect
103. M16×C131C23semidirect
107. C42×C131C23semidirect
193. C131#3019semidirect
202. #421D131semidirect
228. C2×M16C2×D131semidirect
233. #117C2×D131semidirect
235. #104C2×D131semidirect
253. #41D524semidirect
257. C2×Q8D524semidirect
278. C2×D4D524semidirect
295. M16C22×D131semidirect
301. C42C22×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622922
#225541122
#225541102
#225538812
#225534912
#225534812
#225533302
C2×#168621622
#421×C1312
#3019131