Finite Group Databasebeta

← #27179972

Subgroups — #27179972

Order 12928.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic
2#20265656C211yesmaximal, characteristic
3#20265653C211yesmaximal, characteristic, complemented
4#20265285C211yesmaximal, characteristic, complemented
5#20265281C211yesmaximal, characteristic, complemented
6#20265274C211yesmaximal, characteristic, complemented
7#20265271C211yesmaximal, characteristic, complemented
8C4×#13108064C211yesmaximal, characteristic, complemented
9SD16×Dic101C211yesmaximal, characteristic, complemented
10C4×#13108062C211yesmaximal, characteristic, complemented
11#20265218C211yesmaximal, characteristic, complemented
12#20265170C211yesmaximal, characteristic
13#103×D101C211yesmaximal, characteristic, complemented
14#20265134C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18#15692284C418yescomplemented
19C2×Dic404C2211yescharacteristic
20C2×#13108064C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108057C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic
25C4×Dic202—21no—
26C4×Dic202C2211yescharacteristic
27C4×Dic202C2211yescharacteristic
28#15692220—21no—
29#15692220C2211yescharacteristic
30#15692220C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692213—21no—
35#15692213C2211yescharacteristic
36#15692212—21no—
37#15692212C2211yescharacteristic
38#15692211—21no—
39C2×#13108053—21no—
40C4⋊C4×D101C2211yescharacteristic
41#15692188C2211yescharacteristic
42#15692187C2211yescharacteristic
43#15692186C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692165C2211yescharacteristic
51C4×#11717579C2211yescharacteristic
52C8×Dic101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#97×C101C2211yescharacteristic
60#13108073—22no—
61#13108073C2×C414yes—
62C2×Dic202—21no—
63C2×Dic202C2311yescharacteristic
64C2×Dic202C2311yescharacteristic
65Q8×D101—22no—
66Q8×D101C2×C414yescomplemented
67C2×C22⋊D101—21no—
68C22×Dic101—21no—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73Dic404C2×C414yescomplemented
74#13108064C2×C414yescomplemented
75#13108062C2×C414yescomplemented
76#13108057C2×C414yescomplemented
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C8×D101—22no—
80C4⋊Dic101—21no—
81C4⋊Dic101C2311yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053—21no—
88#13108053D411yescharacteristic
89#13108053D411yescharacteristic
90C22⋊Dic101—21no—
91#13108051—21no—
92#13108051D411yescharacteristic
93#13108051D411yescharacteristic
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101—21no—
98C4×Dic101—21no—
99C4×Dic101—21no—
100C4×Dic101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104SD16×C101C2×C414yescomplemented
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101—21no—
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C42×C101—21no—
112C42×C101C2311yescharacteristic
113C23×C101—21no—
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101C22×C412yes—
117D4×C101—21no—
118D4×C101C22×C412yes—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—81no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C22⋊D101—24no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—24no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—81no—
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C4×D101—24no—
137C4×D101—81no—
138C4×D101C22×C412yes—
139C4×D101#4112yes—
140Dic202—21no—
141Dic202—21no—
142Dic202—24no—
143Dic202C22×C412yes—
144Dic202C22×C412yes—
145C8×C101—21no—
146C8×C101C22×C412yes—
147#11717579—21no—
148#11717579C22×C412yes—
149C22×C101—22no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101C22×D411yescharacteristic
153C2×D101—21no—
154C2×D101—21no—
155C2×D101C4×D412yes—
156C4×C101—22no—
157C4×C101—21no—
158C4×C101—21no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101C23×C411yescharacteristic
162C4×C101C2×#4111yescharacteristic
163Dic101—22no—
164Dic101—22no—
165Dic101—21no—
166Dic101—21no—
167Dic101—41no—
168Dic101—41no—
169Dic101C4×D412yes—
170C2×C101—42no—
171C2×C101C2×C4×D411yescharacteristic
172C2×C101#57711yescharacteristic
173C2×C101#57811yescharacteristic
174D101—22no—
175#2947—1011nomaximal, Sylow
176C101#294711yescharacteristic, complemented, Sylow
177C2×#137—1011no—
178#520—1011no—
179#519—1011no—
180#441—1011no—
181#441—1011no—
182#440—1011no—
183#440D10111yescharacteristic, complemented
184C4×Q16—1011no—
185C4×Q16—1011no—
186C4×SD16—1011no—
187C4×SD16—1011no—
188#428—1011no—
189#416—1011no—
190C2×#103—1011no—
191#405—1011no—
192C2×#41—1011no—
193C22×Q8—1011no—
194#137—1018no—
195C2×Q16—1011no—
196C2×Q16—1011no—
197C2×SD16—1011no—
198C2×SD16C2×D10111yescharacteristic
199C2×M16—1011no—
200C22×C8—1011no—
201C4×Q8—1011no—
202C4×Q8—1011no—
203C4×Q8—1011no—
204C4×Q8—1011no—
205C4×Q8—1011no—
206C4×Q8—2021no—
207C4×Q8—2021no—
208C4×Q8C2×D10111yescharacteristic
209C4×D4—1011no—
210C4×D4—2021no—
211C4×D4—2021no—
212C4×D4C2×D10111yescharacteristic
213#117—1011no—
214#117—1011no—
215#117—2021no—
216#117—2021no—
217#117—2021no—
218C2×C4⋊C4—1011no—
219C2×C4⋊C4—2021no—
220#107—1011no—
221#107C2×D10111yescharacteristic
222#106—1011no—
223#106—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103C2×D10111yescharacteristic
230#102—1011no—
231#102C2×D10111yescharacteristic
232#97—1011no—
233#97C2×D10111yescharacteristic
234C4×C8—1011no—
235C4×C8—1011no—
236#41—1014no—
237#41—2022no—
238C2×Q8—1014no—
239C2×Q8—1011no—
240C2×Q8—1011no—
241C2×Q8—2021no—
242C2×Q8—2022no—
243C2×Q8C22×D10111yescharacteristic
244C2×D4—2021no—
245C2×D4C22×D10111yescharacteristic
246C22×C4—21no—
247C22×C4—1011no—
248C22×C4—2021no—
249C22×C4—2021no—
250C22×C4—2021no—
251C22×C4—2021no—
252Q16—1014no—
253Q16—1014no—
254SD16—1014no—
255SD16C4×D10114yescomplemented
256M16—1014no—
257C2×C8—1011no—
258C2×C8—1011no—
259C2×C8—2022no—
260C2×C8C22×D10111yescharacteristic
261C2×C8C22×D10111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1011no—
264C4⋊C4—1011no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4C22×D10111yescharacteristic
274C4⋊C4C22×D10111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—1011no—
277C22⋊C4—1011no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C42—21no—
281C42—1011no—
282C42—2021no—
283C42—2021no—
284C42—2021no—
285C42—2021no—
286C42—2021no—
287C42—2021no—
288C42C22×D10111yescharacteristic
289C23—21no—
290C23—1011no—
291Q8—21no—
292Q8—1012no—
293Q8—1012no—
294Q8—2024no—
295Q8—2021no—
296Q8—2021no—
297Q8C2×C4×D10112yes—
298D4—21no—
299D4—2024no—
300D4C2×C4×D10112yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—8081no—
306C2×C4—8081no—
307C2×C4—1011no—
308C2×C4—1012no—
309C2×C4—1012no—
310C2×C4—1011no—
311C2×C4—1011no—
312C2×C4—2024no—
313C2×C4—2024no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4D4×D10111yescharacteristic
319C2×C4D4×D10111yescharacteristic
320C2×C4C23×D10111yescharacteristic
321C8—21no—
322C8—1012no—
323C8—2021no—
324C8C2×C4×D10112yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—1012no—
329C22—2021no—
330C22—2021no—
331C22C2×D4×D10111yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—1012no—
338C4—4041no—
339C4—4041no—
340C4—2021no—
341C4—2021no—
342C4—2022no—
343C4—2022no—
344C4C22×C4×D10111yescharacteristic
345C4#41×D10111yescharacteristic
346C2—42no—
347C2—2022no—
348C2C4×D4×D10111yescharacteristic
349C2#2026598711yescharacteristic
350C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108073C22
cosocleC2×C4×C101C24
hypercenter#440D101
hypocenterC101#2947
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. #20265653C2semidirect
4. #20265285C2semidirect
5. #20265281C2semidirect
6. #20265274C2semidirect
7. #20265271C2semidirect
8. C4×#13108064C2semidirect
9. SD16×Dic101C2semidirect
10. C4×#13108062C2semidirect
11. #20265218C2semidirect
13. #103×D101C2semidirect
14. #20265134C2semidirect
15. #440×C101C2semidirect
18. #15692284C4semidirect
66. Q8×D101C2×C4semidirect
73. Dic404C2×C4semidirect
74. #13108064C2×C4semidirect
75. #13108062C2×C4semidirect
76. #13108057C2×C4semidirect
104. SD16×C101C2×C4semidirect
176. C101#2947semidirect
183. #440D101semidirect
255. SD16C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
#202656562
#202656532
#202652852
#202652812
#202652742
#202652712
C4×#131080642
SD16×Dic1012
C4×#131080622
#202652182
#202651702
#103×D1012
#202651342
#440×C1012
#2947101