Finite Group Databasebeta

← #27183127

Subgroups — #27183127

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265838C211yesmaximal, characteristic, complemented
2#20265827C211yesmaximal, characteristic, complemented
3#20265538C211yesmaximal, characteristic, complemented
4#20265511C211yesmaximal, characteristic, complemented
5#20265491C211yesmaximal, characteristic, complemented
6#20265470C211yesmaximal, characteristic, complemented
7#20265420C211yesmaximal, characteristic, complemented
8#20265411C211yesmaximal, characteristic, complemented
9#20265394C211yesmaximal, characteristic, complemented
10#20265321C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12#20265245C211yesmaximal, characteristic, complemented
13#20265177C211yesmaximal, characteristic, complemented
14C2×#15692179C211yesmaximal, characteristic, complemented
15#448×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073—21no—
18C2×#13108059C2211yescharacteristic
19C2×#13108057C2211yescharacteristic
20C22×#11717579C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692244—21no—
23#15692244C2211yescharacteristic
24#15692237—21no—
25#15692236—21no—
26#15692232—21no—
27#15692231—21no—
28#15692231C2211yescharacteristic
29#15692229—21no—
30#15692229C2211yescharacteristic
31#15692226C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33D4×Dic101C2211yescharacteristic
34C4×C22⋊D101—21no—
35C4×C22⋊D101C2211yescharacteristic
36C2×C4⋊Dic101C2211yescharacteristic
37C4⋊C4×D101C2211yescharacteristic
38#15692189C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692165C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108075—22no—
60#13108075—41no—
61#13108073—44no—
62C2×Dic202—21no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C2×C22⋊D101D411yescharacteristic
68C22×Dic101—21no—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73#13108059—41no—
74#13108057—41no—
75C2×#11717579—41no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C4⋊Dic101—21no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101D411yescharacteristic
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051D411yescharacteristic
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97C2×Q8×C101—21no—
98C2×Q8×C101—22no—
99C2×Q8×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C2×D4×C101—21no—
103C2×D4×C101C2311yescharacteristic
104C22×C4×C101—21no—
105C22×C4×C101C2311yescharacteristic
106SD16×C101—22no—
107SD16×C101—41no—
108C2×C8×C101C2311yescharacteristic
109C2×C8×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115C22×D101C2×D411yescharacteristic
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—21no—
119Q8×C101—42no—
120Q8×C101#4112yescomplemented
121D4×C101—21no—
122D4×C101—21no—
123D4×C101—42no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—42no—
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—22no—
133C22⋊D101—42no—
134C22⋊D101—42no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—42no—
139C2×Dic101—42no—
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143D404—21no—
144D404—21no—
145D404—42no—
146C4×D101—21no—
147C4×D101—21no—
148C4×D101—42no—
149C4×D101—42no—
150Dic202—42no—
151C8×C101—21no—
152C8×C101—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101AGL(1,ℤ/8)12yes—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—21no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C4×C101—22no—
169C4×C101—21no—
170C4×C101—41no—
171C4×C101—41no—
172C4×C101C2×#4111yescharacteristic
173C4×C101H(5,𝔽2)11yescharacteristic
174Dic101—21no—
175Dic101—21no—
176Dic101—21no—
177Dic101—41no—
178Dic101—41no—
179Dic101—41no—
180C2×C101—22no—
181C2×C101—81no—
182C2×C101#54611yescharacteristic
183C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C101#57811yescharacteristic
185D101—41no—
186D101—81no—
187#3308—1011nomaximal, Sylow
188C101#330811yescharacteristic, complemented, Sylow
189#548—1011no—
190#547—1011no—
191#489—1011no—
192#484—1011no—
193#480—1011no—
194#477—1011no—
195#468—1011no—
196#466—1011no—
197#463—1011no—
198#449—1011no—
199#448D10111yescharacteristic, complemented
200#440—1011no—
201#435—1011no—
202#417—1011no—
203C2×#103—1011no—
204C2×#41—1011no—
205C2×#41—2021no—
206C22×Q8C2×D10111yescharacteristic
207C2×SD16C2×D10111yescharacteristic
208C2×SD16C2×D10111yescharacteristic
209C2×D8—1011no—
210C2×M16—1011no—
211C22×C8—1011no—
212C4⋊Q8—1011no—
213#124—1011no—
214#124—2021no—
215#123—2021no—
216#123—2021no—
217C22⋊Q8—1011no—
218C22⋊Q8—2021no—
219C22⋊Q8—2021no—
220#121—1011no—
221#121—1011no—
222#121—2021no—
223#121C2×D10111yescharacteristic
224C4×Q8—1011no—
225C4×D4—1011no—
226C4×D4—1011no—
227C4×D4—2021no—
228C2×C4⋊C4—1011no—
229C2×C4⋊C4—1011no—
230#107—1011no—
231#107—1011no—
232#107—1011no—
233C4⋊C8—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#103—1011no—
238#103C2×D10111yescharacteristic
239#103C2×D10111yescharacteristic
240#102—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244C22⋊C8—1011no—
245C22⋊C8C2×D10111yescharacteristic
246#97—1011no—
247#41—4044no—
248#41—4041no—
249#41—2022no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—2021no—
254C2×Q8C22×D10111yescharacteristic
255C2×Q8C22×D10111yescharacteristic
256C2×D4—21no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4—2021no—
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—1011no—
265C22×C4—2021no—
266C22×C4—2021no—
267C22×C4—2021no—
268C22×C4—2021no—
269C22×C4C22×D10111yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—4041no—
273M16—4041no—
274C2×C8—1011no—
275C2×C8—1011no—
276C2×C8—4041no—
277C2×C8C22×D10111yescharacteristic
278C2×C8C22×D10111yescharacteristic
279C4⋊C4—1011no—
280C4⋊C4—1011no—
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—2021no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4—2021no—
288C4⋊C4C22×D10111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1011no—
291C22⋊C4—1011no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C42—1011no—
299C42—2021no—
300C23—21no—
301C23—1011no—
302C23—2021no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—4042no—
309Q8#1310807312yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—4042no—
314D4—4042no—
315D4—4042no—
316D4—2022no—
317D4—2021no—
318D4—2021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—1011no—
326C2×C4—1011no—
327C2×C4—1011no—
328C2×C4—4042no—
329C2×C4—4042no—
330C2×C4—4042no—
331C2×C4—4042no—
332C2×C4—2021no—
333C2×C4—2021no—
334C2×C4—2021no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4D4×D10111yescharacteristic
338C2×C4C23×D10111yescharacteristic
339C8—21no—
340C8—21no—
341C8—2021no—
342C8—2021no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—4041no—
349C22—4041no—
350C22—4041no—
351C22—2021no—
352C22—2021no—
353C22—2021no—
354C22#1569228212yes—
355C22C2×D4×D10111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—4041no—
361C4—4041no—
362C4—4041no—
363C4—2021no—
364C4—2021no—
365C4—2021no—
366C4C2×#1310807311yescharacteristic
367C4#1569230011yescharacteristic
368C2—22no—
369C2—81no—
370C2—8081no—
371C2—4041no—
372C2#2026581411yescharacteristic
373C2C2×#1569228211yescharacteristic
374C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#448×C101C2
socleC22×C101C22×D4
hypercenter#448D101
hypocenterC101#3308
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #20265838C2semidirect
2. #20265827C2semidirect
3. #20265538C2semidirect
4. #20265511C2semidirect
5. #20265491C2semidirect
6. #20265470C2semidirect
7. #20265420C2semidirect
8. #20265411C2semidirect
9. #20265394C2semidirect
10. #20265321C2semidirect
11. #20265278C2semidirect
12. #20265245C2semidirect
13. #20265177C2semidirect
14. C2×#15692179C2semidirect
15. #448×C101C2semidirect
120. Q8×C101#41semidirect
188. C101#3308semidirect
199. #448D101semidirect
309. Q8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658382
#202658272
#202655382
#202655112
#202654912
#202654702
#202654202
#202654112
#202653942
#202653212
#202652782
#202652452
#202651772
C2×#156921792
#448×C1012
#3308101