Finite Group Databasebeta

← #27183035

Subgroups — #27183035

Order 12928.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265871C211yesmaximal, characteristic, complemented
2#20265834C211yesmaximal, characteristic, complemented
3#20265530C211yesmaximal, characteristic, complemented
4#20265518C211yesmaximal, characteristic, complemented
5#20265494C211yesmaximal, characteristic, complemented
6#20265482C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic, complemented
8#20265406C211yesmaximal, characteristic, complemented
9#20265391C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265317C211yesmaximal, characteristic, complemented
12#20265262C211yesmaximal, characteristic, complemented
13C8×C22⋊D101C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#416×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×#13108063C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692249—21no—
22#15692246C2211yescharacteristic
23#15692242—21no—
24#15692242C2211yescharacteristic
25#15692237—21no—
26#15692234—21no—
27#15692234C2211yescharacteristic
28#15692230C2211yescharacteristic
29#15692229C2211yescharacteristic
30#15692228—21no—
31#15692226—21no—
32#15692226C2211yescharacteristic
33Q8×Dic101C2211yescharacteristic
34C4×D404—21no—
35#15692217C2211yescharacteristic
36C4×C22⋊D101—21no—
37C4×C22⋊D101C2211yescharacteristic
38C4⋊C4×D101C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54#117×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61Q8×D101—22no—
62Q8×D101—41no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C2×C22⋊D101D411yescharacteristic
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101C2311yescharacteristic
72#13108063—41no—
73#13108061—22no—
74#13108059—41no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C8×D101—41no—
78C4⋊Dic101—21no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101D411yescharacteristic
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051D411yescharacteristic
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104C2×C8×C101—41no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C42×C101—21no—
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101#4112yes—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101—42no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—42no—
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2411yescharacteristic
128C22⋊D101—22no—
129C22⋊D101—42no—
130C22⋊D101—81no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137D404—21no—
138D404—21no—
139D404—42no—
140C4×D101—21no—
141C4×D101—21no—
142C4×D101—42no—
143C4×D101—81no—
144C4×D101—81no—
145Dic202—21no—
146Dic202—21no—
147Dic202—42no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101C22×D411yescharacteristic
159C2×D101—21no—
160C2×D101—21no—
161C2×D101—41no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101#13512yes—
165C4×C101—22no—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C2×#4111yescharacteristic
170C4×C101H(5,𝔽2)11yescharacteristic
171Dic101—21no—
172Dic101—21no—
173Dic101—21no—
174Dic101—41no—
175Dic101—41no—
176Dic101—41no—
177C2×C101—41no—
178C2×C101—81no—
179C2×C101#54611yescharacteristic
180C2×C101C2×#13511yescharacteristic
181C2×C101#57711yescharacteristic
182D101—22no—
183D101—81no—
184#3301—1011nomaximal, Sylow
185C101#330111yescharacteristic, complemented, Sylow
186#553—1011no—
187#548—1011no—
188#488—1011no—
189#485—1011no—
190#481—1011no—
191#479—1011no—
192#466—1011no—
193#465—1011no—
194#463—1011no—
195#449—1011no—
196#448—1011no—
197C4×SD16—1011no—
198C8×D4—1011no—
199#416D10111yescharacteristic, complemented
200C2×#103—1011no—
201C2×#41C2×D10111yescharacteristic
202C22×Q8—1011no—
203C2×SD16—1011no—
204C2×SD16—1011no—
205C2×D8—1011no—
206C22×C8—1011no—
207C22×C8C2×D10111yescharacteristic
208#126—2021no—
209#125—1011no—
210#124—1011no—
211#124—2021no—
212#123—2021no—
213C22⋊Q8—1011no—
214C22⋊Q8—2021no—
215#121—1011no—
216#121—1011no—
217#121—1011no—
218#121—2021no—
219#121—2021no—
220C4×Q8—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223C4×D4—2021no—
224C4×D4—2021no—
225#117C2×D10111yescharacteristic
226C2×C4⋊C4—1011no—
227#107—1011no—
228#106—1011no—
229#106—1011no—
230C4⋊C8—1011no—
231#103—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103C2×D10111yescharacteristic
236#103C2×D10111yescharacteristic
237#102—1011no—
238#102—1011no—
239#102C2×D10111yescharacteristic
240#102C2×D10111yescharacteristic
241C22⋊C8—1011no—
242C22⋊C8—1011no—
243C4×C8—1011no—
244#41—22no—
245#41—41no—
246C2×Q8—1011no—
247C2×Q8—4041no—
248C2×Q8—2021no—
249C2×Q8—2022no—
250C2×Q8C22×D10111yescharacteristic
251C2×D4—21no—
252C2×D4—1011no—
253C2×D4—1011no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4—2021no—
257C2×D4C22×D10111yescharacteristic
258C22×C4—21no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263C22×C4C22×D10111yescharacteristic
264SD16—4041no—
265SD16—2022no—
266D8—4041no—
267C2×C8—41no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8—4041no—
271C2×C8C22×D10111yescharacteristic
272C2×C8C22×D10111yescharacteristic
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—2021no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4C22×D10111yescharacteristic
282C4⋊C4C22×D10111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1011no—
285C22⋊C4—1011no—
286C22⋊C4—2021no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C42—21no—
294C42—1011no—
295C42—2021no—
296C23—21no—
297C23—1011no—
298C23—2021no—
299C23D4×D10111yescharacteristic
300Q8—21no—
301Q8—4042no—
302Q8—2021no—
303Q8—2021no—
304Q8#1310807412yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—8081no—
309D4—4042no—
310D4—4042no—
311D4—2022no—
312D4—2021no—
313D4—2021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—8081no—
321C2×C4—8081no—
322C2×C4—1011no—
323C2×C4—1011no—
324C2×C4—1011no—
325C2×C4—4042no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4D4×D10111yescharacteristic
332C2×C4C23×D10111yescharacteristic
333C8—21no—
334C8—21no—
335C8—2021no—
336C8—2021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1012no—
344C22—4041no—
345C22—4041no—
346C22—4041no—
347C22—2021no—
348C22—2021no—
349C22C2×D4×D10111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—4041no—
355C4—4041no—
356C4—4041no—
357C4—2021no—
358C4—2021no—
359C4—2021no—
360C4C2×#1310807411yescharacteristic
361C4#1569230011yescharacteristic
362C2—41no—
363C2—81no—
364C2—8081no—
365C2—2022no—
366C2#2026581711yescharacteristic
367C2#135×D10111yescharacteristic
368C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
hypercenter#416D101
hypocenterC101#3301
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20265871C2semidirect
2. #20265834C2semidirect
3. #20265530C2semidirect
4. #20265518C2semidirect
5. #20265494C2semidirect
6. #20265482C2semidirect
7. #20265412C2semidirect
8. #20265406C2semidirect
9. #20265391C2semidirect
10. #20265322C2semidirect
11. #20265317C2semidirect
12. #20265262C2semidirect
13. C8×C22⋊D101C2semidirect
14. #103×D101C2semidirect
15. #416×C101C2semidirect
185. C101#3301semidirect
199. #416D101semidirect

Maximal subgroups

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

subgroupindex
#202658712
#202658342
#202655302
#202655182
#202654942
#202654822
#202654122
#202654062
#202653912
#202653222
#202653172
#202652622
C8×C22⋊D1012
#103×D1012
#416×C1012
#3301101