Finite Group Databasebeta

← #27181722

Subgroups — #27181722

Order 12928.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265722C211yesmaximal, characteristic, complemented
2#20265708C211yesmaximal, characteristic, complemented
3#20265584C211yesmaximal, characteristic, complemented
4#20265565C211yesmaximal, characteristic, complemented
5#20265540C211yesmaximal, characteristic, complemented
6#20265535C211yesmaximal, characteristic, complemented
7#20265428C211yesmaximal, characteristic, complemented
8#20265423C211yesmaximal, characteristic, complemented
9#20265421C211yesmaximal, characteristic, complemented
10#20265268C211yesmaximal, characteristic, complemented
11#20265264C211yesmaximal, characteristic, complemented
12D8×Dic101C211yesmaximal, characteristic, complemented
13#20265257C211yesmaximal, characteristic, complemented
14M16×Dic101C211yesmaximal, characteristic, complemented
15#469×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108063C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×#13108058C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692251C2211yescharacteristic
23#15692248—21no—
24#15692246C2211yescharacteristic
25#15692242C2211yescharacteristic
26#15692241—21no—
27#15692236—21no—
28#15692234C2211yescharacteristic
29#15692229C2211yescharacteristic
30#15692227—21no—
31#15692220C2211yescharacteristic
32D4×Dic101—21no—
33D4×Dic101C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692217C2211yescharacteristic
36#15692215—21no—
37#15692211—21no—
38C2×C4×Dic101C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692165C2211yescharacteristic
49#15692165C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×D8×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54#121×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×Dic202C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C22×Dic101—21no—
67C22×Dic101—21no—
68C22×Dic101D411yescharacteristic
69C22×Dic101D411yescharacteristic
70C2×C4×D101—21no—
71#13108063—22no—
72#13108062—22no—
73#13108059—22no—
74#13108058—22no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89#13108051—21no—
90#13108051—21no—
91C4×Dic101—21no—
92C4×Dic101D411yescharacteristic
93C4×Dic101D411yescharacteristic
94C4×Dic101C2311yescharacteristic
95C4×Dic101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C2×D4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101D8×C101—22no—
102M16×C101—22no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C22⋊C4×C101—21no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101—21no—
113D4×C101—21no—
114D4×C101—21no—
115D4×C101—21no—
116D4×C101—21no—
117D4×C101—42no—
118D4×C101—42no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2411yescharacteristic
125C22⋊D101—42no—
126C22⋊D101—42no—
127C22⋊D101—42no—
128C2×Dic101—22no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—41no—
133C2×Dic101—42no—
134C2×Dic101—42no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139D404—21no—
140D404—21no—
141C4×D101—42no—
142Dic202—21no—
143Dic202—21no—
144Dic202—42no—
145C8×C101—21no—
146C8×C101#4112yescomplemented
147#11717579—21no—
148#11717579#4112yescomplemented
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101C22×D411yescharacteristic
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—41no—
162C4×C101—21no—
163C4×C101—41no—
164C4×C101—41no—
165C4×C101C2×#4111yescharacteristic
166C4×C101C2×#4111yescharacteristic
167Dic101—21no—
168Dic101—21no—
169Dic101—21no—
170Dic101—41no—
171Dic101—41no—
172Dic101—41no—
173Dic101AGL(1,ℤ/8)12yes—
174C2×C101—41no—
175C2×C101—81no—
176C2×C101—81no—
177C2×C101#53211yescharacteristic
178C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
179C2×C101#57511yescharacteristic
180D101—81no—
181#3171—1011nomaximal, Sylow
182C101#317111yescharacteristic, complemented, Sylow
183#532—1011no—
184#529—1011no—
185#501—1011no—
186#496—1011no—
187#490—1011no—
188#489—1011no—
189#469—1011no—
190#469D10111yescharacteristic, complemented
191#468—1011no—
192#468—1011no—
193C4×SD16—1011no—
194C4×SD16—1011no—
195C4×D8—1011no—
196C4×D8—1011no—
197C4×M16—1011no—
198C2×#41—2021no—
199C2×SD16—1011no—
200C2×SD16—1011no—
201C2×D8—1011no—
202C2×D8C2×D10111yescharacteristic
203C2×M16—1011no—
204C2×M16C2×D10111yescharacteristic
205C4⋊Q8—1011no—
206#127—1011no—
207#126—2021no—
208#125—1011no—
209#124—1011no—
210#124—2021no—
211#123—2021no—
212C22⋊Q8—1011no—
213#121—1011no—
214#121—2021no—
215#121C2×D10111yescharacteristic
216#121C2×D10111yescharacteristic
217C4×Q8—1011no—
218C4×D4—1011no—
219C4×D4—1011no—
220C4×D4—1011no—
221C4×D4—2021no—
222C4×D4—2021no—
223#117—2021no—
224C2×C42—1011no—
225#107—1011no—
226#107—1011no—
227#106—1011no—
228#106C2×D10111yescharacteristic
229#103—1011no—
230#103—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102C2×D10111yescharacteristic
236#102C2×D10111yescharacteristic
237#97—1011no—
238#97—1011no—
239C4×C8—1011no—
240C4×C8—1011no—
241#41—4044no—
242C2×Q8—1011no—
243C2×Q8—2021no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—1011no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4—2021no—
250C2×D4C22×D10111yescharacteristic
251C2×D4C22×D10111yescharacteristic
252C22×C4—1011no—
253C22×C4—1011no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4C22×D10111yescharacteristic
258SD16—2022no—
259SD16—2022no—
260D8—22no—
261D8—2022no—
262M16—22no—
263M16—2022no—
264C2×C8—1011no—
265C2×C8—1011no—
266C2×C8C22×D10111yescharacteristic
267C2×C8C22×D10111yescharacteristic
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—1011no—
287C42—1011no—
288C42—1011no—
289C42—1011no—
290C42—2021no—
291C23—21no—
292C23—21no—
293C23—2021no—
294C23D4×D10111yescharacteristic
295Q8—4042no—
296Q8—2021no—
297Q8—2021no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—42no—
304D4—4042no—
305D4—4042no—
306D4—4042no—
307D4—2021no—
308D4—2021no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—1011no—
314C2×C4—1011no—
315C2×C4—1011no—
316C2×C4—1011no—
317C2×C4—4042no—
318C2×C4—4042no—
319C2×C4—4042no—
320C2×C4—4041no—
321C2×C4—2022no—
322C2×C4—2021no—
323C2×C4—2021no—
324C2×C4—2021no—
325C2×C4D4×D10111yescharacteristic
326C2×C4C23×D10111yescharacteristic
327C8—21no—
328C8—1012no—
329C8—2021no—
330C8#1310807312yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—4041no—
341C22—4041no—
342C22—4041no—
343C22C2×D4×D10111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—1012no—
348C4—4041no—
349C4—4041no—
350C4—4041no—
351C4—2021no—
352C4—2021no—
353C4—2021no—
354C4#41×D10111yescharacteristic
355C4C2×#1310807311yescharacteristic
356C2—41no—
357C2—81no—
358C2—81no—
359C2—8081no—
360C2#2026572211yescharacteristic
361C2AGL(1,ℤ/8)×D10111yescharacteristic
362C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3171
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. #20265722C2semidirect
2. #20265708C2semidirect
3. #20265584C2semidirect
4. #20265565C2semidirect
5. #20265540C2semidirect
6. #20265535C2semidirect
7. #20265428C2semidirect
8. #20265423C2semidirect
9. #20265421C2semidirect
10. #20265268C2semidirect
11. #20265264C2semidirect
12. D8×Dic101C2semidirect
13. #20265257C2semidirect
14. M16×Dic101C2semidirect
15. #469×C101C2semidirect
146. C8×C101#41semidirect
148. #11717579#41semidirect
182. C101#3171semidirect
190. #469D101semidirect
330. C8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202657222
#202657082
#202655842
#202655652
#202655402
#202655352
#202654282
#202654232
#202654212
#202652682
#202652642
D8×Dic1012
#202652572
M16×Dic1012
#469×C1012
#3171101