Finite Group Databasebeta

← #27183053

Subgroups — #27183053

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#20265891C211yesmaximal, characteristic, complemented
2#20265826C211yesmaximal, characteristic, complemented
3#20265526C211yesmaximal, characteristic, complemented
4#20265519C211yesmaximal, characteristic, complemented
5#20265517C211yesmaximal, characteristic, complemented
6#20265465C211yesmaximal, characteristic, complemented
7#20265404C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265380C211yesmaximal, characteristic, complemented
10#20265319C211yesmaximal, characteristic, complemented
11D8×Dic101C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13#20265173C211yesmaximal, characteristic, complemented
14#20265170C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C22×#11717579C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692254C2211yescharacteristic
21#15692253C2211yescharacteristic
22#15692242—21no—
23#15692241—21no—
24#15692238—21no—
25#15692236—21no—
26#15692234C2211yescharacteristic
27#15692231C2211yescharacteristic
28#15692227—21no—
29#15692227—21no—
30#15692226C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33C4×C22⋊D101C2211yescharacteristic
34#15692214C2211yescharacteristic
35#15692213C2211yescharacteristic
36#15692211—21no—
37C2×#13108053—21no—
38#15692189C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×D8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108073—22no—
60#13108073—41no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101D411yescharacteristic
66C22×Dic101—21no—
67C22×Dic101—21no—
68C22×Dic101—21no—
69C2×C4×D101C2311yescharacteristic
70#13108062—41no—
71C2×#11717579—41no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C8×D101—41no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101D411yescharacteristic
90#13108051—21no—
91#13108051—21no—
92#13108051D411yescharacteristic
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96#41×C101—22no—
97#41×C101—41no—
98C2×Q8×C101C2311yescharacteristic
99C2×D4×C101—21no—
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104C22×C4×C101C2311yescharacteristic
105SD16×C101—41no—
106D8×C101—22no—
107C2×C8×C101C2311yescharacteristic
108C2×C8×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101—21no—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101—21no—
120D4×C101—42no—
121D4×C101—81no—
122D4×C101#4112yes—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—42no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—22no—
131C22⋊D101—42no—
132C22⋊D101—81no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—42no—
138C2×Dic101—42no—
139C2×Dic101—42no—
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143C4×D101—21no—
144C4×D101—21no—
145Dic202—21no—
146Dic202—21no—
147Dic202—81no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—22no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—21no—
165C4×C101—21no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101C2×#4111yescharacteristic
169C4×C101H(5,𝔽2)11yescharacteristic
170Dic101—21no—
171Dic101—21no—
172Dic101—41no—
173Dic101—41no—
174Dic101—41no—
175Dic101—41no—
176Dic101#13512yes—
177C2×C101—41no—
178C2×C101—42no—
179C2×C101—81no—
180C2×C101#54611yescharacteristic
181C2×C101C2×#13511yescharacteristic
182C2×C101#57711yescharacteristic
183D101—41no—
184#3303—1011nomaximal, Sylow
185C101#330311yescharacteristic, complemented, Sylow
186#555—1011no—
187#547—1011no—
188#487—1011no—
189#485—1011no—
190#485—1011no—
191#476—1011no—
192#465—1011no—
193#465—1011no—
194#461—1011no—
195#449—1011no—
196#449D10111yescharacteristic, complemented
197C4×D8—1011no—
198C8×D4—1011no—
199#416—1011no—
200#416—1011no—
201C2×#41—1011no—
202C2×#41C2×D10111yescharacteristic
203C2×SD16—1011no—
204C2×SD16C2×D10111yescharacteristic
205C2×D8C2×D10111yescharacteristic
206C22×C8—1011no—
207C22×C8—1011no—
208C4⋊Q8—1011no—
209C4⋊Q8—1011no—
210#124—2021no—
211#124—2021no—
212#123—2021no—
213#123—2021no—
214C22⋊Q8—1011no—
215C22⋊Q8—1011no—
216#121—1011no—
217#121—2021no—
218#121—2021no—
219#121C2×D10111yescharacteristic
220C4×D4—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223#117—1011no—
224#117—1011no—
225#117—2021no—
226C2×C4⋊C4—2021no—
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#103—1011no—
236#103C2×D10111yescharacteristic
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102C2×D10111yescharacteristic
241C22⋊C8—1011no—
242C22⋊C8C2×D10111yescharacteristic
243C4×C8—1011no—
244#41—22no—
245#41—41no—
246#41—4041no—
247#41—2022no—
248C2×Q8—1011no—
249C2×Q8—2021no—
250C2×Q8C22×D10111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1011no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4C22×D10111yescharacteristic
257C2×D4C22×D10111yescharacteristic
258C22×C4—21no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263C22×C4C22×D10111yescharacteristic
264SD16—41no—
265SD16—4041no—
266D8—22no—
267C2×C8—1011no—
268C2×C8—1011no—
269C2×C8—4041no—
270C2×C8—4041no—
271C2×C8C22×D10111yescharacteristic
272C2×C8C22×D10111yescharacteristic
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
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—1011no—
294C42—2021no—
295C42—2021no—
296C23—21no—
297C23—21no—
298C23—1011no—
299C23D4×D10111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—8081no—
303Q8—2021no—
304Q8—2021no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—8081no—
311D4—4042no—
312D4—2022no—
313D4#1310807312yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1011no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—4042no—
323C2×C4—4042no—
324C2×C4—4042no—
325C2×C4—2021no—
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—22no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2021no—
347C22—2021no—
348C22—2021no—
349C22C2×D4×D10111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—1012no—
354C4—4041no—
355C4—4041no—
356C4—4041no—
357C4—4041no—
358C4—2021no—
359C4—2021no—
360C4C2×#1310807311yescharacteristic
361C4#1569230011yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—4041no—
366C2#2026581411yescharacteristic
367C2#135×D10111yescharacteristic
368C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3303
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20265891C2semidirect
2. #20265826C2semidirect
3. #20265526C2semidirect
4. #20265519C2semidirect
5. #20265517C2semidirect
6. #20265465C2semidirect
7. #20265404C2semidirect
8. #20265403C2semidirect
9. #20265380C2semidirect
10. #20265319C2semidirect
11. D8×Dic101C2semidirect
12. #20265240C2semidirect
13. #20265173C2semidirect
14. #20265170C2semidirect
15. #449×C101C2semidirect
185. C101#3303semidirect
196. #449D101semidirect

Maximal subgroups

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

subgroupindex
#202658912
#202658262
#202655262
#202655192
#202655172
#202654652
#202654042
#202654032
#202653802
#202653192
D8×Dic1012
#202652402
#202651732
#202651702
#449×C1012
#3303101