Finite Group Databasebeta

← #27180750

Subgroups — #27180750

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692279C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265687C211yesmaximal, characteristic, complemented
4#121×D101C211yesmaximal, characteristic, complemented
5#20265428C211yesmaximal, characteristic, complemented
6#20265423C211yesmaximal, characteristic, complemented
7#20265421C211yesmaximal, characteristic, complemented
8#20265412C211yesmaximal, characteristic, complemented
9#20265410C211yesmaximal, characteristic, complemented
10#20265407C211yesmaximal, characteristic, complemented
11#20265405C211yesmaximal, characteristic, complemented
12#20265216C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#20265175C211yesmaximal, characteristic, complemented
15#469×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101—21no—
18C2×D4×D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692279—24no—
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108059C2211yescharacteristic, complemented
24C2×#13108058C2211yescharacteristic, complemented
25C2×#13108057C2211yescharacteristic
26C2×#13108057C2211yescharacteristic
27M16×D101—24no—
28C2×C8×D101C2211yescharacteristic, complemented
29C2×C8×D101C2211yescharacteristic, complemented
30#15692237—21no—
31#15692235C2211yescharacteristic, complemented
32#15692234C2211yescharacteristic, complemented
33#15692230C2211yescharacteristic
34#15692229C2211yescharacteristic
35#15692228—21no—
36#15692227—21no—
37#15692226C2211yescharacteristic
38#15692226C2211yescharacteristic, complemented
39#15692223—21no—
40D4×Dic101—21no—
41#15692215—21no—
42#15692213C2211yescharacteristic
43C4⋊C4×D101C2211yescharacteristic
44C22⋊C4×D101—21no—
45#15692189C2211yescharacteristic, complemented
46#15692188C2211yescharacteristic, complemented
47#15692186C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692176C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic, complemented
56#121×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic, complemented
58#106×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101D411yescharacteristic
63#13108073—22no—
64#13108073—22no—
65C2×Dic202C2311yescharacteristic
66C2×D404—21no—
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73D4×D101—22no—
74D4×D101—22no—
75D4×D101—44no—
76C22×Dic101—21no—
77C22×Dic101D411yescharacteristic
78C2×C4×D101—22no—
79C2×C4×D101—22no—
80C2×C4×D101—21no—
81C2×C4×D101D411yescharacteristic
82C2×C4×D101D411yescharacteristic
83C2×C4×D101C2311yescharacteristic
84C2×C4×D101C2311yescharacteristic
85#13108063—22no—
86#13108062—22no—
87#13108059—22no—
88#13108058—22no—
89#13108057—22no—
90#13108057D414yescomplemented
91C2×#11717579C2311yescharacteristic
92C2×#11717579C2311yescharacteristic
93C8×D101—22no—
94C8×D101—22no—
95C4⋊Dic101—21no—
96C4⋊Dic101C2311yescharacteristic
97C4⋊Dic101C2311yescharacteristic
98#13108053—21no—
99#13108053—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107C4×Dic101—21no—
108C2×D4×C101—21no—
109C2×D4×C101—21no—
110C2×D4×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113D8×C101—22no—
114M16×C101—22no—
115C2×C8×C101C2311yescharacteristic
116C2×C8×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C4⋊C4×C101C2311yescharacteristic
119C22⋊C4×C101—21no—
120C22⋊C4×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101C2×D411yescharacteristic
124C22×D101—22no—
125C22×D101—22no—
126C22×D101—21no—
127C22×D101—41no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101C2×D411yescharacteristic
132C22×D101C2×D411yescharacteristic
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—42no—
138D4×C101—42no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101C2×D411yescharacteristic
144C2×C4×C101C2411yescharacteristic
145C22⋊D101—42no—
146C22⋊D101—42no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C2×Dic101—22no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—42no—
155C2×Dic101C2×D411yescharacteristic
156C2×Dic101C2×D411yescharacteristic
157D404—21no—
158D404—21no—
159D404—42no—
160C4×D101—22no—
161C4×D101—21no—
162C4×D101—21no—
163C4×D101—22no—
164C4×D101—42no—
165C4×D101C2×D412yes—
166C4×D101#4112yes—
167Dic202—21no—
168Dic202—21no—
169C8×C101—21no—
170C8×C101C2×D412yescomplemented
171#11717579—21no—
172#11717579C2×D412yescomplemented
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101C22×D411yescharacteristic
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—44no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—84no—
192C2×D101#12112yes—
193C2×D101AGL(1,ℤ/8)12yescomplemented
194C4×C101—21no—
195C4×C101—41no—
196C4×C101—41no—
197C4×C101C22×D411yescharacteristic
198C4×C101C2×#4111yescharacteristic
199Dic101—21no—
200Dic101—41no—
201Dic101—41no—
202Dic101—41no—
203Dic101#12112yes—
204C2×C101—41no—
205C2×C101—81no—
206C2×C101—81no—
207C2×C101C2×#12111yescharacteristic
208C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C101#57511yescharacteristic
210D101—22no—
211D101—22no—
212D101—81no—
213#3055—1011nomaximal, Sylow
214C101#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—1011no—
216C22×M16—1011no—
217#525—1011no—
218C2×#121—1011no—
219#469—1011no—
220#469D10111yescharacteristic, complemented
221#468—1011no—
222#468—1011no—
223#466—1011no—
224#466—1011no—
225#465—1011no—
226#465—1011no—
227#428—1011no—
228#418—1011no—
229#417—1011no—
230C2×#41—1011no—
231C22×D4—1011no—
232C22×D4—2021no—
233C23×C4—1011no—
234AGL(1,ℤ/8)—2024no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×D8—1011no—
238C2×D8C2×D10111yescharacteristic, complemented
239C2×M16—1011no—
240C2×M16—1011no—
241C2×M16—1011no—
242C2×M16—2024no—
243C2×M16C2×D10111yescharacteristic, complemented
244C22×C8—1011no—
245C22×C8—1011no—
246#123—2021no—
247C22⋊Q8—1011no—
248C22⋊Q8—1011no—
249#121—1011no—
250#121—1011no—
251#121—1011no—
252#121—1011no—
253#121—2021no—
254#121—2021no—
255#121C2×D10111yescharacteristic
256#121C2×D10111yescharacteristic, complemented
257#120—2021no—
258C4×D4—2021no—
259C4×D4—2021no—
260#117—1011no—
261C2×C4⋊C4—1011no—
262C2×C22⋊C4—2021no—
263#107—1011no—
264#107—1011no—
265#106—1011no—
266#106C2×D10111yescharacteristic, complemented
267#103—1011no—
268#103—1011no—
269#102—1011no—
270#102—1011no—
271#102—1011no—
272#102—1011no—
273#102C2×D10111yescharacteristic, complemented
274#102C2×D10111yescharacteristic, complemented
275C24—1011no—
276C24—2021no—
277#41—2022no—
278#41—2022no—
279C2×Q8—1011no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1011no—
283C2×D4—4044no—
284C2×D4—2021no—
285C2×D4—2022no—
286C2×D4—2022no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4—2021no—
291C2×D4—2021no—
292C2×D4C22×D10111yescharacteristic
293C2×D4C22×D10111yescharacteristic
294C22×C4—1011no—
295C22×C4—1011no—
296C22×C4—1011no—
297C22×C4—1011no—
298C22×C4—1011no—
299C22×C4—2022no—
300C22×C4—2022no—
301C22×C4—2021no—
302C22×C4—2021no—
303C22×C4C22×D10111yescharacteristic
304SD16—2022no—
305SD16—2022no—
306D8—22no—
307D8—2022no—
308M16—22no—
309M16—1014no—
310M16—2022no—
311M16—2022no—
312C2×C8—1011no—
313C2×C8—1011no—
314C2×C8—2022no—
315C2×C8—2022no—
316C2×C8C22×D10111yescharacteristic
317C2×C8C22×D10111yescharacteristic
318C4⋊C4—1011no—
319C4⋊C4—1011no—
320C4⋊C4—2021no—
321C4⋊C4—2021no—
322C4⋊C4—2021no—
323C4⋊C4C22×D10111yescharacteristic
324C4⋊C4C22×D10111yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C42—2021no—
335C23—21no—
336C23—21no—
337C23—1011no—
338C23—1011no—
339C23—4041no—
340C23—4042no—
341C23—4042no—
342C23—4042no—
343C23—2022no—
344C23—2022no—
345C23—2021no—
346C23D4×D10111yescharacteristic
347Q8—2021no—
348Q8—2021no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—42no—
355D4—4042no—
356D4—4042no—
357D4—4042no—
358D4—4042no—
359D4—4042no—
360D4—4042no—
361D4—2021no—
362D4—2021no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—1011no—
368C2×C4—1012no—
369C2×C4—1012no—
370C2×C4—1011no—
371C2×C4—4042no—
372C2×C4—4042no—
373C2×C4—2022no—
374C2×C4—2022no—
375C2×C4—2022no—
376C2×C4—2021no—
377C2×C4—2021no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4D4×D10111yescharacteristic
382C2×C4C23×D10111yescharacteristic
383C8—21no—
384C8—1012no—
385C8—2021no—
386C8D4×D10112yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—8084no—
397C22—1012no—
398C22—1012no—
399C22—4044no—
400C22—4041no—
401C22—4041no—
402C22—4041no—
403C22—2021no—
404C22—2021no—
405C22—2021no—
406C22—2021no—
407C22C2×D4×D10111yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—1012no—
412C4—4041no—
413C4—4041no—
414C4—4041no—
415C4—2021no—
416C4C2×D4×D10111yescharacteristic
417C4#41×D10111yescharacteristic
418C2—41no—
419C2—81no—
420C2—81no—
421C2—8081no—
422C2—2022no—
423C2—2022no—
424C2#121×D10111yescharacteristic
425C2AGL(1,ℤ/8)×D10111yescharacteristic
426C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3055
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×M16×D101C2semidirect
3. #20265687C2semidirect
4. #121×D101C2semidirect
5. #20265428C2semidirect
6. #20265423C2semidirect
7. #20265421C2semidirect
8. #20265412C2semidirect
9. #20265410C2semidirect
10. #20265407C2semidirect
11. #20265405C2semidirect
12. #20265216C2semidirect
13. #20265180C2semidirect
14. #20265175C2semidirect
15. #469×C101C2semidirect
21. C2×#13108063C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108058C22semidirect
28. C2×C8×D101C22semidirect
29. C2×C8×D101C22semidirect
31. #15692235C22semidirect
32. #15692234C22semidirect
38. #15692226C22semidirect
45. #15692189C22semidirect
46. #15692188C22semidirect
47. #15692186C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
50. #15692177C22semidirect
51. #15692176C22semidirect
52. #15692176C22semidirect
53. #15692175C22semidirect
54. C2×D8×C101C22semidirect
55. C2×M16×C101C22semidirect
57. #121×C101C22semidirect
58. #106×C101C22semidirect
59. #102×C101C22semidirect
60. #102×C101C22semidirect
90. #13108057D4semidirect
170. C8×C101C2×D4semidirect
172. #11717579C2×D4semidirect
193. C2×D101AGL(1,ℤ/8)semidirect
214. C101#3055semidirect
220. #469D101semidirect
238. C2×D8C2×D101semidirect
243. C2×M16C2×D101semidirect
256. #121C2×D101semidirect
266. #106C2×D101semidirect
273. #102C2×D101semidirect
274. #102C2×D101semidirect
386. C8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×M16×D1012
#202656872
#121×D1012
#202654282
#202654232
#202654212
#202654122
#202654102
#202654072
#202654052
#202652162
#202651802
#202651752
#469×C1012
#3055101