Finite Group Databasebeta

← #20286404

Subgroups — #20286404

Order 6784.

546 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
1C22×#11719924C211yesmaximal, characteristic, complemented
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3Q8×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811257C211yesmaximal, characteristic
5#16810994C211yesmaximal, characteristic, complemented
6#16810855C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810827C211yesmaximal, characteristic, complemented
9#16810821C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810763C211yesmaximal, characteristic, complemented
13#16810708C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic, complemented
15C53×#448C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic, complemented
17C22×D212C2211yescharacteristic
18C2×D4×D53—21no—
19C2×D4×D53C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719913C2211yescharacteristic, complemented
22C2×#11719923C2211yescharacteristic, complemented
23C2×#11719924—24no—
24C2×#11719924C2211yescharacteristic
25C2×#11719924C2211yescharacteristic
26C2×#11719924C2211yescharacteristic, complemented
27C2×#11719924C2211yescharacteristic, complemented
28SD16×D53—24no—
29C22×#88322C2211yescharacteristic, complemented
30C2×C8×D53C2211yescharacteristic, complemented
31#13133215—21no—
32#13133215C2211yescharacteristic, complemented
33#13133213C2211yescharacteristic
34#13133196—21no—
35#13133196C2211yescharacteristic, complemented
36#13133193—21no—
37#13133193C2211yescharacteristic, complemented
38#13133192C2211yescharacteristic
39#13133190—21no—
40#13133187—21no—
41#13133185—21no—
42Q8×Dic53C2211yescharacteristic, complemented
43#13133179C2211yescharacteristic
44C4×C22⋊D53—21no—
45C4×C22⋊D53C2211yescharacteristic
46#13133148C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53C8×Dic53C2211yescharacteristic, complemented
54C22×Q8×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic, complemented
56C2×SD16×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60C22⋊C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53C2311yescharacteristic
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53D411yescharacteristic
76Q8×D53—22no—
77Q8×D53—41no—
78D4×D53—22no—
79D4×D53—41no—
80D4×D53—44no—
81C2×D212—22no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—41no—
85C2×D212C2311yescharacteristic
86C2×D212C2311yescharacteristic
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53C2311yescharacteristic
90C2×Dic106—21no—
91C2×Dic106C2311yescharacteristic
92SD16×C53—22no—
93SD16×C53—41no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53D411yescharacteristic
100#11719924—22no—
101#11719924—22no—
102#11719924—22no—
103#11719924D414yescomplemented
104#11719923—22no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921D411yescharacteristic
109C4⋊Dic53—21no—
110C4⋊Dic53C2311yescharacteristic
111#11719919—21no—
112#11719919—21no—
113#11719919—21no—
114#11719919—21no—
115#11719919D411yescharacteristic
116C4×Dic53—21no—
117C4×Dic53C2311yescharacteristic
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121#11719913—22no—
122#11719913—41no—
123C8×D53—22no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—81no—
133C22×D53—81no—
134C22×D53C2×D411yescharacteristic
135Q8×C53—21no—
136Q8×C53—21no—
137Q8×C53—21no—
138Q8×C53—42no—
139Q8×C53C2×D412yescomplemented
140D4×C53—21no—
141D4×C53—21no—
142D4×C53—42no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—42no—
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2411yescharacteristic
151C22⋊D53—42no—
152C22⋊D53—42no—
153C22⋊D53—81no—
154C22⋊D53SD1614yescomplemented
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159C2×Dic53C2×D411yescharacteristic
160D212—21no—
161D212—21no—
162D212—21no—
163D212—42no—
164D212—42no—
165D212C2×D412yes—
166C4×D53—21no—
167C4×D53—21no—
168C4×D53—42no—
169C4×D53—42no—
170Dic106—21no—
171Dic106—21no—
172Dic106—42no—
173C8×C53—21no—
174C8×C53—21no—
175#88322—21no—
176#88322C2×D412yescomplemented
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C2×SD1612yescomplemented
183C22×C53C22×D411yescharacteristic
184C2×D53—21no—
185C2×D53—21no—
186C2×D53—22no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—84no—
193C2×D53—84no—
194C2×D53C2×SD1612yescomplemented
195C4×C53—22no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53C22×D411yescharacteristic
200C4×C53H(5,𝔽2)11yescharacteristic
201Dic53—21no—
202Dic53—21no—
203Dic53—41no—
204Dic53—41no—
205Dic53C2×SD1612yescomplemented
206D4×SD16—531nomaximal, Sylow
207C2×C53—22no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53C22×SD1611yescharacteristic
211C2×C53#57711yescharacteristic
212D53—22no—
213D53—42no—
214D53—81no—
215C22×SD16—531no—
216C22×SD16—531no—
217D4×Q8—531no—
218D42—531no—
219#492—531no—
220#465—531no—
221#465—531no—
222#461—531no—
223#460—531no—
224#450—531no—
225#450—531no—
226#448—531no—
227#448D5311yescharacteristic, complemented
228C4×SD16—531no—
229C8×D4—531no—
230C53D4×SD1611yescharacteristic, complemented, Sylow
231C22×Q8—531no—
232C22×Q8C2×D5311yescharacteristic, complemented
233C22×D4—531no—
234C22×D4—531no—
235C22×D4—1061no—
236C2×SD16—531no—
237C2×SD16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16—531no—
242C2×SD16—531no—
243C2×SD16—1064no—
244C2×SD16—1064no—
245C2×SD16C2×D5311yescharacteristic, complemented
246C2×SD16C2×D5311yescharacteristic, complemented
247C22×C8—531no—
248C22×C8—531no—
249C4⋊Q8—531no—
250C4⋊Q8—1061no—
251#127—531no—
252C22⋊Q8—531no—
253C22⋊Q8—531no—
254C22⋊Q8—1061no—
255C22⋊Q8—1061no—
256#121—531no—
257#121—1061no—
258#121C2×D5311yescharacteristic
259#120—1061no—
260#120—1061no—
261C4×Q8—531no—
262C4×D4—531no—
263C4×D4—531no—
264C4×D4—1061no—
265#106—531no—
266C4⋊C8—531no—
267#103—531no—
268#103C2×D5311yescharacteristic, complemented
269#103C2×D5311yescharacteristic, complemented
270#102—531no—
271#102—531no—
272#102—531no—
273C22⋊C8—531no—
274C22⋊C8C2×D5311yescharacteristic, complemented
275C4×C8—531no—
276C24—1061no—
277C24—1061no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—531no—
282C2×Q8—1061no—
283C2×Q8—1062no—
284C2×Q8—2121no—
285C2×Q8C22×D5311yescharacteristic
286C2×Q8C22×D5311yescharacteristic
287C2×D4—21no—
288C2×D4—531no—
289C2×D4—531no—
290C2×D4—531no—
291C2×D4—1062no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1062no—
297C2×D4—1061no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2124no—
301C2×D4C22×D5311yescharacteristic
302C22×C4—21no—
303C22×C4—531no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4C22×D5311yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—534no—
310SD16—1062no—
311SD16—1062no—
312SD16—1062no—
313SD16—1062no—
314SD16—1062no—
315SD16—2121no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C4⋊C4—1061no—
328C4⋊C4—1061no—
329C4⋊C4C22×D5311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—531no—
332C22⋊C4—531no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C42—531no—
338C42—1061no—
339C23—21no—
340C23—531no—
341C23—1061no—
342C23—1061no—
343C23—4241no—
344C23—4241no—
345C23—2122no—
346C23—2122no—
347C23—2122no—
348C23—2122no—
349C23D4×D5311yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—21no—
353Q8—42no—
354Q8—1061no—
355Q8—1061no—
356Q8—2122no—
357Q8D4×D5312yescomplemented
358D4—21no—
359D4—21no—
360D4—42no—
361D4—534no—
362D4—532no—
363D4—1061no—
364D4—1061no—
365D4—1061no—
366D4—4241no—
367D4—2122no—
368D4—2122no—
369D4—2122no—
370D4—2122no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—531no—
378C2×C4—531no—
379C2×C4—531no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4D4×D5311yescharacteristic
387C2×C4C23×D5311yescharacteristic
388C8—21no—
389C8—21no—
390C8—532no—
391C8—1061no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—532no—
398C22—1061no—
399C22—1061no—
400C22—1062no—
401C22—4244no—
402C22—4244no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22SD16×D5312yescomplemented
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—532no—
415C4—1061no—
416C4—1061no—
417C4—2121no—
418C4—2121no—
419C4C2×D4×D5311yescharacteristic
420C4#1313326211yescharacteristic
421C2—22no—
422C2—81no—
423C2—1062no—
424C2—4241no—
425C2—2122no—
426C2#1681125511yescharacteristic
427C2C2×SD16×D5311yescharacteristic
428C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53D4×SD16
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#11719924C2semidirect
2. C2×SD16×D53C2semidirect
3. Q8×C22⋊D53C2semidirect
5. #16810994C2semidirect
6. #16810855C2semidirect
7. #16810850C2semidirect
8. #16810827C2semidirect
9. #16810821C2semidirect
10. #16810776C2semidirect
11. #16810772C2semidirect
12. #16810763C2semidirect
13. #16810708C2semidirect
14. #16810686C2semidirect
15. C53×#448C2semidirect
16. C2×Q8×D53C22semidirect
21. C2×#11719913C22semidirect
22. C2×#11719923C22semidirect
26. C2×#11719924C22semidirect
27. C2×#11719924C22semidirect
29. C22×#88322C22semidirect
30. C2×C8×D53C22semidirect
32. #13133215C22semidirect
35. #13133196C22semidirect
37. #13133193C22semidirect
42. Q8×Dic53C22semidirect
46. #13133148C22semidirect
47. #13133146C22semidirect
48. #13133140C22semidirect
50. #13133139C22semidirect
51. #13133138C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C22×Q8×C53C22semidirect
55. C2×SD16×C53C22semidirect
56. C2×SD16×C53C22semidirect
58. #103×C53C22semidirect
59. #103×C53C22semidirect
60. C22⋊C8×C53C22semidirect
103. #11719924D4semidirect
139. Q8×C53C2×D4semidirect
154. C22⋊D53SD16semidirect
176. #88322C2×D4semidirect
182. C22×C53C2×SD16semidirect
194. C2×D53C2×SD16semidirect
205. Dic53C2×SD16semidirect
227. #448D53semidirect
230. C53D4×SD16semidirect
232. C22×Q8C2×D53semidirect
245. C2×SD16C2×D53semidirect
246. C2×SD16C2×D53semidirect
268. #103C2×D53semidirect
269. #103C2×D53semidirect
274. C22⋊C8C2×D53semidirect
357. Q8D4×D53semidirect
408. C22SD16×D53semidirect