Finite Group Databasebeta

← #27180809

Subgroups — #27180809

Order 12928.

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×#15692279C211yesmaximal, characteristic, complemented
2C2×#15692263C211yesmaximal, characteristic, complemented
3#20265756C211yesmaximal, characteristic, complemented
4#20265743C211yesmaximal, characteristic, complemented
5#20265429C211yesmaximal, characteristic, complemented
6#20265423C211yesmaximal, characteristic, complemented
7#20265422C211yesmaximal, characteristic, complemented
8#20265411C211yesmaximal, characteristic, complemented
9#20265410C211yesmaximal, characteristic, complemented
10#20265407C211yesmaximal, characteristic, complemented
11#20265404C211yesmaximal, characteristic, complemented
12#20265216C211yesmaximal, characteristic, complemented
13#20265170C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#469×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692279—24no—
20C2×#13108063C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108059C2211yescharacteristic
23#15692263—24no—
24C2×#13108057C2211yescharacteristic
25C2×#13108057C2211yescharacteristic, complemented
26C22×#11717579C2211yescharacteristic, complemented
27C2×C8×D101C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic, complemented
29#15692251—21no—
30#15692241—21no—
31#15692235C2211yescharacteristic, complemented
32#15692234C2211yescharacteristic, complemented
33#15692232—21no—
34#15692230C2211yescharacteristic
35#15692229C2211yescharacteristic
36#15692228—21no—
37#15692227—21no—
38#15692226—21no—
39#15692226C2211yescharacteristic
40#15692226C2211yescharacteristic, complemented
41#15692225—21no—
42#15692223—21no—
43#15692213C2211yescharacteristic
44C4⋊C4×D101C2211yescharacteristic
45#15692189C2211yescharacteristic, complemented
46#15692188C2211yescharacteristic, complemented
47#15692186C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic
51#15692176C2211yescharacteristic
52#15692176C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic
54C2×D8×C101C2211yescharacteristic
55C2×M16×C101C2211yescharacteristic, complemented
56#121×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic, complemented
58#106×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—22no—
64#13108073—22no—
65#13108073—22no—
66C2×Dic202D411yescharacteristic
67C2×Dic202C2311yescharacteristic
68C2×D404—21no—
69C2×D404D411yescharacteristic
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101D411yescharacteristic
77C2×C22⋊D101D411yescharacteristic
78D4×D101—22no—
79D4×D101—22no—
80C22×Dic101—21no—
81C2×C4×D101—21no—
82C2×C4×D101C2311yescharacteristic
83C2×C4×D101C2311yescharacteristic
84#13108063—22no—
85#13108062—22no—
86#13108059—22no—
87#13108057—22no—
88#13108057D414yescomplemented
89C2×#11717579—22no—
90C2×#11717579C2311yescharacteristic
91C2×#11717579C2311yescharacteristic
92C8×D101—22no—
93C8×D101—22no—
94C4⋊Dic101—21no—
95C4⋊Dic101C2311yescharacteristic
96C4⋊Dic101C2311yescharacteristic
97#13108053—21no—
98#13108053—21no—
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106C4×Dic101—21no—
107C2×D4×C101—21no—
108C2×D4×C101—21no—
109C2×D4×C101C2311yescharacteristic
110C2×D4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112D8×C101—22no—
113M16×C101—22no—
114C2×C8×C101C2311yescharacteristic
115C2×C8×C101C2311yescharacteristic
116C4⋊C4×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101C2×D411yescharacteristic
123C22×D101—21no—
124C22×D101—42no—
125C22×D101—81no—
126C22×D101—81no—
127C22×D101C2×D411yescharacteristic
128C22×D101C2×D411yescharacteristic
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—21no—
133D4×C101—81no—
134D4×C101—81no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101C2×D411yescharacteristic
140C2×C4×C101C2411yescharacteristic
141C22⋊D101—22no—
142C22⋊D101—22no—
143C22⋊D101—42no—
144C22⋊D101—42no—
145C22⋊D101—81no—
146C22⋊D101—81no—
147C22⋊D101—81no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101—42no—
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154D404—22no—
155D404—21no—
156D404—21no—
157D404—81no—
158C4×D101—21no—
159C4×D101—21no—
160C4×D101—81no—
161C4×D101C2×D412yes—
162C4×D101#4112yes—
163Dic202—22no—
164Dic202—21no—
165Dic202—21no—
166C8×C101—21no—
167C8×C101C2×D412yes—
168#11717579—21no—
169#11717579C2×D412yes—
170C22×C101—21no—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—41no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101C22×D411yescharacteristic
180C2×D101—21no—
181C2×D101—21no—
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—84no—
189C2×D101#12112yes—
190C4×C101—21no—
191C4×C101—41no—
192C4×C101—41no—
193C4×C101C22×D411yescharacteristic
194C4×C101C2×#4111yescharacteristic
195Dic101—21no—
196Dic101—41no—
197Dic101—41no—
198Dic101—41no—
199Dic101#12112yes—
200C2×C101—41no—
201C2×C101—81no—
202C2×C101—81no—
203C2×C101C2×#12111yescharacteristic
204C2×C101#57611yescharacteristic
205C2×C101#57711yescharacteristic
206D101—22no—
207D101—41no—
208D101—81no—
209#3062—1011nomaximal, Sylow
210C101#306211yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—1011no—
212C2×#131—1011no—
213#537—1011no—
214#535—1011no—
215#469—1011no—
216#469D10111yescharacteristic, complemented
217#468—1011no—
218#468—1011no—
219#466—1011no—
220#466—1011no—
221#465—1011no—
222#465—1011no—
223#428—1011no—
224#416—1011no—
225C2×#102—1011no—
226C2×#41—1011no—
227C2×#41—1011no—
228C22×D4—1011no—
229AGL(1,ℤ/8)—2024no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×D8—1011no—
233C2×D8C2×D10111yescharacteristic
234#131—2024no—
235C2×M16—1011no—
236C2×M16—1011no—
237C2×M16C2×D10111yescharacteristic, complemented
238C22×C8—1011no—
239C22×C8—1011no—
240C22×C8—1011no—
241#127—2021no—
242#124—2021no—
243C22⋊Q8—1011no—
244C22⋊Q8—1011no—
245C22⋊Q8—2021no—
246#121—1011no—
247#121—1011no—
248#121—1011no—
249#121—1011no—
250#121—2021no—
251#121—2021no—
252#121—2021no—
253#121C2×D10111yescharacteristic
254#121C2×D10111yescharacteristic, complemented
255#120—2021no—
256#120—2021no—
257#117—1011no—
258C2×C4⋊C4—1011no—
259#107—1011no—
260#107—1011no—
261#106—1011no—
262#106C2×D10111yescharacteristic, complemented
263#103—1011no—
264#103—1011no—
265#102—1011no—
266#102—1011no—
267#102—1011no—
268#102—1011no—
269#102C2×D10111yescharacteristic
270#102C2×D10111yescharacteristic, complemented
271C24—2021no—
272#41—2022no—
273#41—2022no—
274#41—2022no—
275#41—2022no—
276C2×Q8—1011no—
277C2×Q8—1011no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—1011no—
281C2×D4—1011no—
282C2×D4—1011no—
283C2×D4—1011no—
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—2021no—
297C22×C4—2021no—
298C22×C4C22×D10111yescharacteristic
299SD16—2022no—
300SD16—2022no—
301D8—22no—
302D8—2022no—
303M16—22no—
304M16—1014no—
305M16—2022no—
306C2×C8—1011no—
307C2×C8—1011no—
308C2×C8—2022no—
309C2×C8—2022no—
310C2×C8—2022no—
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—1011no—
314C4⋊C4—1011no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4C22×D10111yescharacteristic
319C4⋊C4C22×D10111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C42—2021no—
330C23—21no—
331C23—21no—
332C23—8081no—
333C23—8081no—
334C23—1011no—
335C23—1011no—
336C23—4042no—
337C23—2021no—
338C23D4×D10111yescharacteristic
339Q8—2022no—
340Q8—2021no—
341Q8—2021no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—8081no—
349D4—8081no—
350D4—8081no—
351D4—8081no—
352D4—4042no—
353D4—4042no—
354D4—2022no—
355D4—2022no—
356D4—2022no—
357D4—2021no—
358D4—2021no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—8081no—
364C2×C4—1011no—
365C2×C4—1012no—
366C2×C4—1012no—
367C2×C4—1011no—
368C2×C4—4042no—
369C2×C4—2021no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4D4×D10111yescharacteristic
375C2×C4C23×D10111yescharacteristic
376C8—21no—
377C8—1012no—
378C8—2021no—
379C8D4×D10112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—8084no—
390C22—1012no—
391C22—4041no—
392C22—4041no—
393C22—4041no—
394C22—2021no—
395C22—2021no—
396C22—2021no—
397C22—2021no—
398C22—2021no—
399C22C2×D4×D10111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—1012no—
404C4—4041no—
405C4—4041no—
406C4—4041no—
407C4—2021no—
408C4C2×D4×D10111yescharacteristic
409C4#41×D10111yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—8081no—
414C2—4041no—
415C2—2022no—
416C2#121×D10111yescharacteristic
417C2#2026598111yescharacteristic
418C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3062
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×#15692263C2semidirect
3. #20265756C2semidirect
4. #20265743C2semidirect
5. #20265429C2semidirect
6. #20265423C2semidirect
7. #20265422C2semidirect
8. #20265411C2semidirect
9. #20265410C2semidirect
10. #20265407C2semidirect
11. #20265404C2semidirect
12. #20265216C2semidirect
13. #20265170C2semidirect
14. #102×D101C2semidirect
15. #469×C101C2semidirect
25. C2×#13108057C22semidirect
26. C22×#11717579C22semidirect
28. C2×C8×D101C22semidirect
31. #15692235C22semidirect
32. #15692234C22semidirect
40. #15692226C22semidirect
45. #15692189C22semidirect
46. #15692188C22semidirect
47. #15692186C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
52. #15692176C22semidirect
55. C2×M16×C101C22semidirect
57. #121×C101C22semidirect
58. #106×C101C22semidirect
60. #102×C101C22semidirect
88. #13108057D4semidirect
210. C101#3062semidirect
216. #469D101semidirect
237. C2×M16C2×D101semidirect
254. #121C2×D101semidirect
262. #106C2×D101semidirect
270. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×#156922632
#202657562
#202657432
#202654292
#202654232
#202654222
#202654112
#202654102
#202654072
#202654042
#202652162
#202651702
#102×D1012
#469×C1012
#3062101