Finite Group Databasebeta

← #27183023

Subgroups — #27183023

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265869C211yesmaximal, characteristic, complemented
2#20265838C211yesmaximal, characteristic
3#20265533C211yesmaximal, characteristic, complemented
4#20265517C211yesmaximal, characteristic, complemented
5#20265491C211yesmaximal, characteristic, complemented
6#20265486C211yesmaximal, characteristic, complemented
7#20265415C211yesmaximal, characteristic, complemented
8#20265408C211yesmaximal, characteristic, complemented
9#20265401C211yesmaximal, characteristic, complemented
10#20265338C211yesmaximal, characteristic, complemented
11Q16×Dic101C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13#20265172C211yesmaximal, characteristic, complemented
14C2×#15692179C211yesmaximal, characteristic, complemented
15#451×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C22×#11717579C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692250—21no—
21#15692246—21no—
22#15692246C2211yescharacteristic
23#15692244—21no—
24#15692244C2211yescharacteristic
25#15692239—21no—
26#15692231—21no—
27#15692231C2211yescharacteristic
28#15692231C2211yescharacteristic
29#15692229—21no—
30#15692229C2211yescharacteristic
31Q8×Dic101C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33C4×C22⋊D101—21no—
34C4×C22⋊D101C2211yescharacteristic
35#15692213C2211yescharacteristic
36#15692211—21no—
37C2×C4⋊Dic101C2211yescharacteristic
38#15692189C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692177C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic
54C2×Q16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108075—22no—
60#13108075—41no—
61C2×D404—21no—
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101D411yescharacteristic
65C22×Dic101—21no—
66C2×C4×D101—21no—
67C2×C4×D101C2311yescharacteristic
68#13108061—41no—
69C2×#11717579—41no—
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C8×D101—41no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84C22⋊Dic101D411yescharacteristic
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101—21no—
97C2×Q8×C101—22no—
98C2×Q8×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×Q8×C101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104Q16×C101—22no—
105Q16×C101—41no—
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101—21no—
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101—21no—
113C22×D101C2×D411yescharacteristic
114Q8×C101—21no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—21no—
119Q8×C101—42no—
120Q8×C101#4112yes—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
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—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—42no—
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139D404—21no—
140D404—21no—
141D404—42no—
142C4×D101—21no—
143C4×D101—21no—
144C4×D101—42no—
145C8×C101—21no—
146C8×C101—21no—
147#11717579—21no—
148#11717579—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101#13512yes—
152C22×C101C22×D411yescharacteristic
153C2×D101—21no—
154C2×D101—21no—
155C2×D101—21no—
156C2×D101—41no—
157C2×D101—41no—
158C2×D101—41no—
159C4×C101—22no—
160C4×C101—21no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101—41no—
164C4×C101C2×#4111yescharacteristic
165C4×C101H(5,𝔽2)11yescharacteristic
166Dic101—21no—
167Dic101—21no—
168Dic101—21no—
169Dic101—41no—
170Dic101—41no—
171Dic101—41no—
172C2×C101—22no—
173C2×C101#54611yescharacteristic
174C2×C101C2×#13511yescharacteristic
175C2×C101#57811yescharacteristic
176D101—41no—
177D101—81no—
178#3300—1011nomaximal, Sylow
179C101#330011yescharacteristic, complemented, Sylow
180#552—1011no—
181#548—1011no—
182#488—1011no—
183#485—1011no—
184#480—1011no—
185#479—1011no—
186#467—1011no—
187#465—1011no—
188#464—1011no—
189#452—1011no—
190#451D10111yescharacteristic, complemented
191C4×Q16—1011no—
192C8×D4—1011no—
193#416—1011no—
194C2×#103—1011no—
195C2×#41—1011no—
196C22×Q8C2×D10111yescharacteristic
197C2×Q16C2×D10111yescharacteristic
198C2×Q16C2×D10111yescharacteristic
199C2×SD16—1011no—
200C22×C8—1011no—
201C22×C8—1011no—
202#126—2021no—
203#125—1011no—
204#125—2021no—
205#124—1011no—
206#124—2021no—
207#123—2021no—
208C22⋊Q8—1011no—
209C22⋊Q8—1011no—
210C22⋊Q8—2021no—
211C22⋊Q8C2×D10111yescharacteristic
212#121—1011no—
213#121—2021no—
214C4×Q8—1011no—
215C4×Q8—1011no—
216C4×D4—1011no—
217C4×D4—2021no—
218#117—1011no—
219#117—2021no—
220C2×C4⋊C4—1011no—
221#107—1011no—
222#107—1011no—
223#106—1011no—
224C4⋊C8—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103—1011no—
230#103C2×D10111yescharacteristic
231#103C2×D10111yescharacteristic
232#102—1011no—
233#102—1011no—
234#102—1011no—
235C22⋊C8—1011no—
236C22⋊C8C2×D10111yescharacteristic
237C4×C8—1011no—
238#41—4041no—
239#41—2022no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D10111yescharacteristic
244C2×Q8C22×D10111yescharacteristic
245C2×Q8C22×D10111yescharacteristic
246C2×D4—1011no—
247C2×D4—1011no—
248C2×D4—2021no—
249C2×D4—2021no—
250C22×C4—21no—
251C22×C4—1011no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4C22×D10111yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—4041no—
258C2×C8—1011no—
259C2×C8—1011no—
260C2×C8—4041no—
261C2×C8—4041no—
262C2×C8C22×D10111yescharacteristic
263C2×C8C22×D10111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4—2021no—
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1011no—
279C22⋊C4—1011no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—1011no—
287C42—2021no—
288C42—2021no—
289C42—2021no—
290C23—1011no—
291C23—2021no—
292C23D4×D10111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1310807312yes—
300D4—4042no—
301D4—4042no—
302D4—2022no—
303D4—2021no—
304D4—2021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—1011no—
313C2×C4—1011no—
314C2×C4—1011no—
315C2×C4—4042no—
316C2×C4—4042no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4D4×D10111yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C8—21no—
325C8—21no—
326C8—2021no—
327C8—2021no—
328C22—21no—
329C22—21no—
330C22—4041no—
331C22—4041no—
332C22—4041no—
333C22—2021no—
334C22—2021no—
335C22—2021no—
336C22#1569227812yes—
337C22C2×D4×D10111yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4041no—
344C4—4041no—
345C4—4041no—
346C4—2021no—
347C4—2021no—
348C4—2021no—
349C4C2×#1310807311yescharacteristic
350C4#1569230011yescharacteristic
351C2—22no—
352C2—8081no—
353C2—4041no—
354C2#2026581411yescharacteristic
355C2C2×#1569227811yescharacteristic
356C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#451×C101C2
socleC22×C101C22×D4
prime radical#20265838C2
hypercenter#451D101
hypocenterC101#3300
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #20265869C2semidirect
3. #20265533C2semidirect
4. #20265517C2semidirect
5. #20265491C2semidirect
6. #20265486C2semidirect
7. #20265415C2semidirect
8. #20265408C2semidirect
9. #20265401C2semidirect
10. #20265338C2semidirect
11. Q16×Dic101C2semidirect
12. #20265240C2semidirect
13. #20265172C2semidirect
14. C2×#15692179C2semidirect
15. #451×C101C2semidirect
179. C101#3300semidirect
190. #451D101semidirect

Maximal subgroups

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

subgroupindex
#202658692
#202658382
#202655332
#202655172
#202654912
#202654862
#202654152
#202654082
#202654012
#202653382
Q16×Dic1012
#202652402
#202651722
C2×#156921792
#451×C1012
#3300101