Finite Group Databasebeta

← #27183834

Subgroups — #27183834

Order 12928.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265886C211yesmaximal, characteristic, complemented
2#20265847C211yesmaximal, characteristic, complemented
3#20265541C211yesmaximal, characteristic, complemented
4#20265540C211yesmaximal, characteristic, complemented
5#20265522C211yesmaximal, characteristic, complemented
6#20265478C211yesmaximal, characteristic, complemented
7#20265475C211yesmaximal, characteristic, complemented
8#20265465C211yesmaximal, characteristic, complemented
9#20265370C211yesmaximal, characteristic, complemented
10#20265370C211yesmaximal, characteristic, complemented
11#20265306C211yesmaximal, characteristic, complemented
12#20265289C211yesmaximal, characteristic, complemented
13#20265287C211yesmaximal, characteristic, complemented
14D8×Dic101C211yesmaximal, characteristic, complemented
15#459×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108059C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692251C2211yescharacteristic
20#15692251C2211yescharacteristic
21#15692247C2211yescharacteristic
22#15692246C2211yescharacteristic
23#15692234—21no—
24#15692232—21no—
25#15692232—21no—
26#15692227—21no—
27#15692227—21no—
28#15692226—21no—
29C4×Dic202C2211yescharacteristic
30C4×D404C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33D4×Dic101C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35C2×#13108053—21no—
36#15692189C2211yescharacteristic
37#15692187C2211yescharacteristic
38#15692186C2211yescharacteristic
39#15692184C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692177C2211yescharacteristic
42#15692177C2211yescharacteristic
43#15692176C2211yescharacteristic
44#15692176C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692166C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×D8×C101C2211yescharacteristic
53C2×D8×C101C2211yescharacteristic
54#127×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×Dic202D411yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C22×Dic101—21no—
68C22×Dic101—21no—
69C22×Dic101—21no—
70C22×Dic101—21no—
71C2×C4×D101—21no—
72#13108059—22no—
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C4⋊Dic101—21no—
76C4⋊Dic101D411yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053D411yescharacteristic
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90#13108051—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93C2×D4×C101—21no—
94C2×D4×C101—21no—
95C2×D4×C101C2311yescharacteristic
96C2×D4×C101C2311yescharacteristic
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101—21no—
99D8×C101—22no—
100D8×C101—22no—
101C2×C8×C101C2311yescharacteristic
102C2×C8×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C23×C101—21no—
108C23×C101—21no—
109C22×D101—21no—
110D4×C101—21no—
111D4×C101—21no—
112D4×C101—21no—
113D4×C101—21no—
114D4×C101—21no—
115D4×C101—42no—
116D4×C101—81no—
117D4×C101#4112yescomplemented
118C2×C4×C101—21no—
119C2×C4×C101—81no—
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2411yescharacteristic
123C22⋊D101—42no—
124C22⋊D101—42no—
125C22⋊D101—81no—
126C22⋊D101—81no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—42no—
131C2×Dic101—42no—
132C2×Dic101—42no—
133C2×Dic101—81no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138D404—21no—
139D404—21no—
140C4×D101—42no—
141Dic202—22no—
142Dic202—81no—
143C8×C101—21no—
144C8×C101—21no—
145#11717579—21no—
146#11717579—21no—
147C22×C101—22no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101—41no—
156C22×C101C22×D411yescharacteristic
157C2×D101—41no—
158C2×D101—41no—
159C2×D101—41no—
160C4×C101—21no—
161C4×C101—21no—
162C4×C101—41no—
163C4×C101C2×#4111yescharacteristic
164C4×C1012−1+411yescharacteristic
165Dic101—21no—
166Dic101—21no—
167Dic101—21no—
168Dic101—41no—
169Dic101—41no—
170Dic101—41no—
171Dic101AGL(1,ℤ/8)12yes—
172C2×C101—42no—
173C2×C101—81no—
174C2×C101—81no—
175C2×C101#54811yescharacteristic
176C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C101#57611yescharacteristic
178D101—81no—
179#3366—1011nomaximal, Sylow
180C101#336611yescharacteristic, complemented, Sylow
181#554—1011no—
182#550—1011no—
183#490—1011no—
184#490—1011no—
185#486—1011no—
186#478—1011no—
187#478—1011no—
188#476—1011no—
189#459—1011no—
190#459—1011no—
191#459D10111yescharacteristic, complemented
192#446—1011no—
193#442—1011no—
194#442—1011no—
195C4×D8—1011no—
196C2×#41—2021no—
197C2×D8—1011no—
198C2×D8C2×D10111yescharacteristic
199C2×D8C2×D10111yescharacteristic
200C4⋊Q8—1011no—
201#127—1011no—
202#127—1011no—
203#127C2×D10111yescharacteristic
204#125—1011no—
205#125—1011no—
206C22⋊Q8—2021no—
207C22⋊Q8—2021no—
208C22⋊Q8—2021no—
209#121—2021no—
210#121—2021no—
211#121—2021no—
212C4×Q8—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—1011no—
217C4×D4—1011no—
218C4×D4C2×D10111yescharacteristic
219C2×C4⋊C4—2021no—
220#107—1011no—
221#106—1011no—
222#106—1011no—
223C4⋊C8—1011no—
224C4⋊C8—1011no—
225C4⋊C8C2×D10111yescharacteristic
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102C2×D10111yescharacteristic
235#102C2×D10111yescharacteristic
236#97—1011no—
237#97—1011no—
238C4×C8—1011no—
239#41—4044no—
240C2×Q8—1011no—
241C2×Q8—2021no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1011no—
245C2×D4—2021no—
246C2×D4—2021no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4C22×D10111yescharacteristic
250C2×D4C22×D10111yescharacteristic
251C2×D4C22×D10111yescharacteristic
252C22×C4—21no—
253C22×C4—2021no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4—2021no—
258D8—22no—
259D8—22no—
260D8—2022no—
261C2×C8—1011no—
262C2×C8—1011no—
263C2×C8C22×D10111yescharacteristic
264C2×C8C22×D10111yescharacteristic
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—1011no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4—2021no—
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C42—1011no—
284C42—1011no—
285C42C22×D10111yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—2021no—
290Q8—8081no—
291Q8—2022no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—8081no—
300D4—8081no—
301D4—4042no—
302D4—4042no—
303D4—2021no—
304D4—2021no—
305D4#1310807312yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—8081no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—1011no—
312C2×C4—1011no—
313C2×C4—4042no—
314C2×C4—4042no—
315C2×C4—4042no—
316C2×C4—4042no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4D4×D10111yescharacteristic
321C2×C4D4×D10111yescharacteristic
322C2×C4C23×D10111yescharacteristic
323C8—21no—
324C8—21no—
325C8—2021no—
326C8—2021no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—4041no—
337C22—4041no—
338C22—4041no—
339C22C2×D4×D10111yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—1012no—
344C4—4041no—
345C4—4041no—
346C4—4041no—
347C4—2021no—
348C4—2021no—
349C4—2021no—
350C4C2×#1310807311yescharacteristic
351C4#1569230211yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—8081no—
356C2#2026583311yescharacteristic
357C2AGL(1,ℤ/8)×D10111yescharacteristic
358C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3366
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20265886C2semidirect
2. #20265847C2semidirect
3. #20265541C2semidirect
4. #20265540C2semidirect
5. #20265522C2semidirect
6. #20265478C2semidirect
7. #20265475C2semidirect
8. #20265465C2semidirect
9. #20265370C2semidirect
10. #20265370C2semidirect
11. #20265306C2semidirect
12. #20265289C2semidirect
13. #20265287C2semidirect
14. D8×Dic101C2semidirect
15. #459×C101C2semidirect
117. D4×C101#41semidirect
180. C101#3366semidirect
191. #459D101semidirect
305. D4#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658862
#202658472
#202655412
#202655402
#202655222
#202654782
#202654752
#202654652
#202653702
#202653062
#202652892
#202652872
D8×Dic1012
#459×C1012
#3366101