Finite Group Databasebeta

← #27182987

Subgroups — #27182987

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265877C211yesmaximal, characteristic, complemented
2#20265822C211yesmaximal, characteristic, complemented
3#20265532C211yesmaximal, characteristic, complemented
4#20265514C211yesmaximal, characteristic, complemented
5#20265504C211yesmaximal, characteristic, complemented
6#20265477C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic, complemented
8#20265405C211yesmaximal, characteristic, complemented
9#20265392C211yesmaximal, characteristic, complemented
10#20265319C211yesmaximal, characteristic, complemented
11#20265309C211yesmaximal, characteristic, complemented
12C8×D404C211yesmaximal, characteristic, complemented
13#20265170C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15C4×D8×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C2×#13108062C2211yescharacteristic, complemented
19C2×#13108059C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21C2×C8×D101C2211yescharacteristic, complemented
22#15692248—21no—
23#15692247C2211yescharacteristic, complemented
24#15692243C2211yescharacteristic
25#15692239—21no—
26#15692238—21no—
27#15692235C2211yescharacteristic
28#15692230C2211yescharacteristic
29#15692229—21no—
30#15692228—21no—
31#15692226C2211yescharacteristic
32#15692226C2211yescharacteristic, complemented
33#15692225—21no—
34C4×D404C2211yescharacteristic
35C4×C22⋊D101—21no—
36#15692213C2211yescharacteristic, complemented
37C4⋊C4×D101C2211yescharacteristic
38C2×#13108051—21no—
39#15692188C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41C4⋊#11717579C2211yescharacteristic, complemented
42#15692180C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692178C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692167C2211yescharacteristic, complemented
52C2×D8×C101C2211yescharacteristic
53C4×D4×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic, complemented
55#107×C101C2211yescharacteristic, complemented
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic, complemented
58C4×C8×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108073—22no—
61#13108073—41no—
62C2×Dic202C2311yescharacteristic
63C2×D404—21no—
64C2×D404D411yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70D4×D101—22no—
71D4×D101—41no—
72C22×Dic101—21no—
73C2×C4×D101—21no—
74C2×C4×D101C2311yescharacteristic
75C2×C4×D101C2311yescharacteristic
76#13108062—41no—
77#13108059—41no—
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C8×D101—41no—
81C8×D101—41no—
82C4⋊Dic101—21no—
83C4⋊Dic101D411yescharacteristic
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051D411yescharacteristic
97#13108051D411yescharacteristic
98C4×Dic101—21no—
99C2×D4×C101C2311yescharacteristic
100C2×D4×C101C2311yescharacteristic
101C22×C4×C101—21no—
102C22×C4×C101—21no—
103D8×C101—22no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C22⋊C4×C101—21no—
109C22⋊C4×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C23×C101—21no—
113C22×D101—21no—
114C22×D101—42no—
115C22×D101—42no—
116C22×D101—81no—
117C22×D101C2×D411yescharacteristic
118C22×D101C2×D411yescharacteristic
119D4×C101—21no—
120D4×C101—21no—
121D4×C101—21no—
122D4×C101#4112yes—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—81no—
126C2×C4×C101—81no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—42no—
131C22⋊D101—42no—
132C22⋊D101—42no—
133C22⋊D101—81no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—42no—
138C2×Dic101C2×D411yescharacteristic
139C2×Dic101C2×D411yescharacteristic
140D404—22no—
141D404—21no—
142D404—21no—
143D404—81no—
144C4×D101—21no—
145C4×D101—21no—
146C4×D101—21no—
147C4×D101—21no—
148C4×D101—81no—
149Dic202—21no—
150Dic202—21no—
151C8×C101—21no—
152C8×C101—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—22no—
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—
165C2×D101—21no—
166C2×D101—21no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—84no—
171C2×D101#13512yes—
172C4×C101—21no—
173C4×C101—21no—
174C4×C101—41no—
175C4×C101—41no—
176C4×C101C2×#4111yescharacteristic
177C4×C101H(5,𝔽2)11yescharacteristic
178Dic101—21no—
179Dic101—21no—
180Dic101—41no—
181Dic101—41no—
182Dic101—41no—
183C2×C101—42no—
184C2×C101—81no—
185C2×C101#54611yescharacteristic
186C2×C101C2×#13511yescharacteristic
187C2×C101#57611yescharacteristic
188D101—22no—
189D101—41no—
190D101—81no—
191#3297—1011nomaximal, Sylow
192C101#329711yescharacteristic, complemented, Sylow
193#553—1011no—
194#546—1011no—
195#488—1011no—
196#484—1011no—
197#482—1011no—
198#478—1011no—
199#466—1011no—
200#465—1011no—
201#463—1011no—
202#449—1011no—
203#447—1011no—
204C4×D8D10111yescharacteristic, complemented
205C8×D4—1011no—
206#416—1011no—
207C2×#102—1011no—
208C2×#41—1011no—
209C22×D4—1011no—
210C2×SD16—1011no—
211C2×D8—1011no—
212C2×D8C2×D10111yescharacteristic
213C22×C8—1011no—
214C22×C8—1011no—
215#126—2021no—
216#125—1011no—
217#124—1011no—
218#123—2021no—
219#123—2021no—
220C22⋊Q8—1011no—
221#121—1011no—
222#121—1011no—
223#121—1011no—
224#121—2021no—
225#121—2021no—
226#120—2021no—
227C4×D4—1011no—
228C4×D4—2021no—
229C4×D4C2×D10111yescharacteristic
230C4×D4C2×D10111yescharacteristic, complemented
231#117—1011no—
232C2×C4⋊C4—1011no—
233C2×C22⋊C4—2021no—
234#107—1011no—
235#107C2×D10111yescharacteristic, complemented
236#106—1011no—
237C4⋊C8—1011no—
238#103—1011no—
239#103—1011no—
240#103—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102C2×D10111yescharacteristic
247#102C2×D10111yescharacteristic, complemented
248C22⋊C8—1011no—
249C22⋊C8—1011no—
250C4×C8C2×D10111yescharacteristic, complemented
251C24—2021no—
252#41—4041no—
253#41—2022no—
254C2×Q8—1011no—
255C2×D4—1011no—
256C2×D4—1011no—
257C2×D4—4041no—
258C2×D4—2021no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4—2021no—
262C2×D4—2021no—
263C2×D4—2022no—
264C2×D4C22×D10111yescharacteristic
265C2×D4C22×D10111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1011no—
269C22×C4—1011no—
270C22×C4—2021no—
271C22×C4—2021no—
272SD16—4041no—
273D8—22no—
274D8—4041no—
275C2×C8—1011no—
276C2×C8—1011no—
277C2×C8—4041no—
278C2×C8—4041no—
279C2×C8C22×D10111yescharacteristic
280C2×C8C22×D10111yescharacteristic
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—1011no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4—2021no—
288C4⋊C4C22×D10111yescharacteristic
289C4⋊C4C22×D10111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1011no—
293C22⋊C4—1011no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C22⋊C4—2021no—
300C22⋊C4—2021no—
301C42—2021no—
302C42C22×D10111yescharacteristic
303C23—21no—
304C23—21no—
305C23—8081no—
306C23—1011no—
307C23—1011no—
308C23—4042no—
309C23—4042no—
310C23—2021no—
311Q8—2021no—
312Q8—2021no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—8081no—
317D4—8081no—
318D4—4042no—
319D4—4042no—
320D4—4042no—
321D4—2022no—
322D4—2021no—
323D4—2021no—
324D4#1310807412yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—8081no—
330C2×C4—1011no—
331C2×C4—1011no—
332C2×C4—4042no—
333C2×C4—2021no—
334C2×C4—2021no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4D4×D10111yescharacteristic
341C2×C4D4×D10111yescharacteristic
342C2×C4C23×D10111yescharacteristic
343C8—21no—
344C8—21no—
345C8—2021no—
346C8—2021no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—8084no—
354C22—1012no—
355C22—4041no—
356C22—4041no—
357C22—4041no—
358C22—2021no—
359C22—2021no—
360C22—2021no—
361C22—2021no—
362C22—2021no—
363C22C2×D4×D10111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—4041no—
369C4—4041no—
370C4—4041no—
371C4—2021no—
372C4—2021no—
373C4C2×#1310807411yescharacteristic
374C4#1569230111yescharacteristic
375C2—42no—
376C2—81no—
377C2—8081no—
378C2—4041no—
379C2—2022no—
380C2#2026582311yescharacteristic
381C2#135×D10111yescharacteristic
382C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×D8×C101C2
socleC22×C101C22×D4
hypercenterC4×D8D101
hypocenterC101#3297
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20265877C2semidirect
2. #20265822C2semidirect
3. #20265532C2semidirect
4. #20265514C2semidirect
5. #20265504C2semidirect
6. #20265477C2semidirect
7. #20265412C2semidirect
8. #20265405C2semidirect
9. #20265392C2semidirect
10. #20265319C2semidirect
11. #20265309C2semidirect
12. C8×D404C2semidirect
13. #20265170C2semidirect
14. #102×D101C2semidirect
15. C4×D8×C101C2semidirect
18. C2×#13108062C22semidirect
21. C2×C8×D101C22semidirect
23. #15692247C22semidirect
32. #15692226C22semidirect
36. #15692213C22semidirect
39. #15692188C22semidirect
40. #15692186C22semidirect
41. C4⋊#11717579C22semidirect
42. #15692180C22semidirect
43. #15692180C22semidirect
44. #15692178C22semidirect
51. #15692167C22semidirect
54. C4×D4×C101C22semidirect
55. #107×C101C22semidirect
57. #102×C101C22semidirect
58. C4×C8×C101C22semidirect
192. C101#3297semidirect
204. C4×D8D101semidirect
230. C4×D4C2×D101semidirect
235. #107C2×D101semidirect
247. #102C2×D101semidirect
250. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658772
#202658222
#202655322
#202655142
#202655042
#202654772
#202654122
#202654052
#202653922
#202653192
#202653092
C8×D4042
#202651702
#102×D1012
C4×D8×C1012
#3297101