Finite Group Databasebeta

← #27183123

Subgroups — #27183123

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#20265831C211yesmaximal, characteristic, complemented
2#20265536C211yesmaximal, characteristic, complemented
3#20265510C211yesmaximal, characteristic, complemented
4#20265488C211yesmaximal, characteristic, complemented
5#20265472C211yesmaximal, characteristic, complemented
6#20265425C211yesmaximal, characteristic, complemented
7#20265411C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265316C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12#20265249C211yesmaximal, characteristic, complemented
13#20265178C211yesmaximal, characteristic, complemented
14C2×#15692179C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108061C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×#13108058C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22#15692255C2211yescharacteristic
23#15692239—21no—
24#15692239—21no—
25#15692235—21no—
26#15692229C2211yescharacteristic
27#15692229C2211yescharacteristic
28C4×D404C2211yescharacteristic
29C2×C4⋊Dic101C2211yescharacteristic
30C2×C4⋊Dic101C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692188C2211yescharacteristic
33#15692188C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35#15692179C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692179C2211yescharacteristic
39#15692179C2211yescharacteristic
40#15692179C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692176C2211yescharacteristic
43#15692176C2211yescharacteristic
44#15692175C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic
47#15692166C2211yescharacteristic
48C2×#41×C101C2211yescharacteristic
49C22×Q8×C101C2211yescharacteristic
50#124×C101—21no—
51#124×C101C2211yescharacteristic
52C22⋊Q8×C101—21no—
53C22⋊Q8×C101C2211yescharacteristic
54#121×C101—21no—
55#121×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic
57C4×D4×C101—21no—
58C4×D4×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C22×Dic101—21no—
65C22×Dic101—21no—
66C2×C4×D101—21no—
67#13108061—22no—
68#13108061—41no—
69#13108059—41no—
70#13108058—41no—
71C2×#11717579—41no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101—21no—
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108051—21no—
83#13108051—21no—
84#13108051—21no—
85#41×C101—22no—
86#41×C101—41no—
87C2×Q8×C101—21no—
88C2×Q8×C101—22no—
89C2×Q8×C101—41no—
90C2×Q8×C101C2311yescharacteristic
91C2×Q8×C101C2311yescharacteristic
92C2×D4×C101—21no—
93C2×D4×C101—21no—
94C2×D4×C101D411yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97C22×C4×C101—21no—
98C22×C4×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101—21no—
102C4⋊C4×C101D411yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C22⋊C4×C101—21no—
108C22⋊C4×C101D411yescharacteristic
109C22⋊C4×C101D411yescharacteristic
110C42×C101—21no—
111C42×C101C2311yescharacteristic
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C23×C101C2×D411yescharacteristic
115C22×D101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—21no—
119Q8×C101—42no—
120Q8×C101#4112yescomplemented
121D4×C101—22no—
122D4×C101—21no—
123D4×C101—21no—
124D4×C101—42no—
125D4×C101—42no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—42no—
134C2×C4×C101—42no—
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—42no—
141C22⋊D101—42no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—42no—
146C2×Dic101—42no—
147D404—21no—
148D404—21no—
149C4×D101—42no—
150Dic202—42no—
151#11717579—21no—
152#11717579—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—21no—
158C22×C101—21no—
159C22×C101—21no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101—41no—
163C22×C101AGL(1,ℤ/8)12yes—
164C22×C101C22×D411yescharacteristic
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C4×C101—21no—
169C4×C101—21no—
170C4×C101—22no—
171C4×C101—21no—
172C4×C101—21no—
173C4×C101—41no—
174C4×C101—41no—
175C4×C101C2×#4111yescharacteristic
176C4×C101H(5,𝔽2)11yescharacteristic
177Dic101—41no—
178Dic101—41no—
179Dic101—41no—
180C2×C101—22no—
181C2×C101—41no—
182C2×C101—81no—
183C2×C101#54611yescharacteristic
184C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C101#57811yescharacteristic
186D101—81no—
187#3308—1011nomaximal, Sylow
188C101#330811yescharacteristic, complemented, Sylow
189#548D10111yescharacteristic, complemented
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#448—1011no—
200#440—1011no—
201#435—1011no—
202#417—1011no—
203C2×#103—1011no—
204C2×#41—2021no—
205C2×#41C2×D10111yescharacteristic
206C22×Q8C2×D10111yescharacteristic
207C2×SD16—1011no—
208C2×SD16—1011no—
209C2×D8—1011no—
210C2×M16—1011no—
211C22×C8—1011no—
212C4⋊Q8—1011no—
213#124—21no—
214#124C2×D10111yescharacteristic
215#123—2021no—
216#123—2021no—
217C22⋊Q8—21no—
218C22⋊Q8—2021no—
219C22⋊Q8C2×D10111yescharacteristic
220#121—21no—
221#121—1011no—
222#121—1011no—
223#121C2×D10111yescharacteristic
224C4×Q8C2×D10111yescharacteristic
225C4×D4—21no—
226C4×D4—1011no—
227C4×D4C2×D10111yescharacteristic
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#103—1011no—
239#103—1011no—
240#102—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244C22⋊C8—1011no—
245C22⋊C8—1011no—
246#97—1011no—
247#41—22no—
248#41—41no—
249#41—4044no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—2021no—
254C2×Q8C22×D10111yescharacteristic
255C2×Q8C22×D10111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1011no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4C22⋊D10111yescharacteristic
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—2021no—
266C22×C4—2021no—
267C22×C4—2021no—
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270SD16—4041no—
271SD16—2022no—
272D8—4041no—
273M16—4041no—
274C2×C8—1011no—
275C2×C8—1011no—
276C2×C8—1011no—
277C2×C8—1011no—
278C2×C8—4041no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—1011no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4C22⋊D10111yescharacteristic
288C4⋊C4C22×D10111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4C22⋊D10111yescharacteristic
297C22⋊C4C22⋊D10111yescharacteristic
298C42—21no—
299C42C22×D10111yescharacteristic
300C23—21no—
301C23—2021no—
302C23C2×C22⋊D10111yescharacteristic
303C23C2×C22⋊D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—4042no—
309Q8#1310807312yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—4042no—
316D4—4042no—
317D4—2021no—
318D4—2021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—4042no—
329C2×C4—4042no—
330C2×C4—4042no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4—2021no—
334C2×C4C2×C22⋊D10111yescharacteristic
335C2×C4C2×C22⋊D10111yescharacteristic
336C2×C4C2×C22⋊D10111yescharacteristic
337C2×C4C2×C22⋊D10111yescharacteristic
338C2×C4C23×D10111yescharacteristic
339C8—2021no—
340C8—2021no—
341C8—2021no—
342C8—2021no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—4041no—
352C22—4041no—
353C22—4041no—
354C22#1569228012yes—
355C22C22×C22⋊D10111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—4041no—
364C4—4041no—
365C4—4041no—
366C4C2×#1310807311yescharacteristic
367C4#1569230011yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—8081no—
372C2#2026582111yescharacteristic
373C2C2×#1569228011yescharacteristic
374C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3308
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20265831C2semidirect
2. #20265536C2semidirect
3. #20265510C2semidirect
4. #20265488C2semidirect
5. #20265472C2semidirect
6. #20265425C2semidirect
7. #20265411C2semidirect
8. #20265389C2semidirect
9. #20265322C2semidirect
10. #20265316C2semidirect
11. #20265275C2semidirect
12. #20265249C2semidirect
13. #20265178C2semidirect
14. C2×#15692179C2semidirect
15. #548×C101C2semidirect
120. Q8×C101#41semidirect
188. C101#3308semidirect
189. #548D101semidirect
309. Q8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658312
#202655362
#202655102
#202654882
#202654722
#202654252
#202654112
#202653892
#202653222
#202653162
#202652752
#202652492
#202651782
C2×#156921792
#548×C1012
#3308101