Finite Group Databasebeta

← #27181378

Subgroups — #27181378

Order 12928.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265787C211yesmaximal, characteristic, complemented
2#20265652C211yesmaximal, characteristic
3#20265519C211yesmaximal, characteristic, complemented
4#20265515C211yesmaximal, characteristic, complemented
5#20265479C211yesmaximal, characteristic, complemented
6#20265475C211yesmaximal, characteristic, complemented
7#20265380C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265328C211yesmaximal, characteristic
10#20265278C211yesmaximal, characteristic, complemented
11#20265274C211yesmaximal, characteristic, complemented
12#20265268C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic
16C2×#13108062C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18#15692253C2211yescharacteristic, complemented
19#15692248—21no—
20#15692248—21no—
21#15692246—21no—
22#15692246C2211yescharacteristic, complemented
23#15692245—21no—
24#15692234C2211yescharacteristic
25Q8×Dic101C2211yescharacteristic, complemented
26#15692220C2211yescharacteristic, complemented
27D4×Dic101—21no—
28D4×Dic101—21no—
29D4×Dic101C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32D4×Dic101C2211yescharacteristic
33#15692214C2211yescharacteristic
34#15692214C2211yescharacteristic
35C2×C22⋊Dic101—21no—
36C2×C22⋊Dic101—21no—
37#15692189C2211yescharacteristic, complemented
38#15692188C2211yescharacteristic, complemented
39#15692187C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692184C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C22×D4×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C22×Dic101—21no—
61C22×Dic101—21no—
62C22×Dic101—21no—
63C22×Dic101—21no—
64#13108062—22no—
65#13108062—22no—
66C2×#11717579C2311yescharacteristic
67C2×#11717579C2311yescharacteristic
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101D411yescharacteristic
71C4⋊Dic101D411yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101D411yescharacteristic
87C22⋊Dic101D411yescharacteristic
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93C24×C101—21no—
94C2×Q8×C101C2311yescharacteristic
95C2×D4×C101—22no—
96C2×D4×C101—22no—
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C2×D4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101SD16×C101—22no—
102SD16×C101—22no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C23×C101—24no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101—81no—
112C23×C101—81no—
113C23×C101C2×D411yescharacteristic
114Q8×C101—21no—
115Q8×C101—21no—
116D4×C101—24no—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101#4112yes—
120D4×C101#4112yes—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—81no—
132C2×Dic101—81no—
133C2×Dic101—81no—
134C2×Dic101—81no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139Dic202—21no—
140Dic202—21no—
141C8×C101—21no—
142C8×C101—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—21no—
146C22×C101—22no—
147C22×C101—22no—
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—84no—
155C22×C101C22×D411yescharacteristic
156C4×C101—21no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C101C2×#4111yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—21no—
164Dic101—21no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168Dic101—41no—
169C2×C101—41no—
170C2×C101—42no—
171C2×C101—42no—
172C2×C101#52511yescharacteristic
173C2×C101#57711yescharacteristic
174C2×C101#57711yescharacteristic
175#3123—1011nomaximal, Sylow
176C101#312311yescharacteristic, complemented, Sylow
177#541—1011no—
178#518—1011no—
179#485—1011no—
180#485—1011no—
181#478—1011no—
182#478—1011no—
183#461—1011no—
184#461—1011no—
185#450—1011no—
186#450D10111yescharacteristic
187#440—1011no—
188#440—1011no—
189C4×SD16—1011no—
190C4×SD16—1011no—
191#433—1011no—
192C22×D4C2×D10111yescharacteristic
193C2×SD16—1011no—
194C2×SD16—1011no—
195C2×SD16C2×D10111yescharacteristic
196C2×SD16C2×D10111yescharacteristic
197C4⋊Q8—1011no—
198#126—2021no—
199#126—2021no—
200#125—1011no—
201#125—2021no—
202#125—2021no—
203C22⋊Q8—1011no—
204C22⋊Q8C2×D10111yescharacteristic
205C4×Q8—1011no—
206C4×Q8—1011no—
207C4×D4—1011no—
208C4×D4—1011no—
209C4×D4—1011no—
210C4×D4—1011no—
211C4×D4—2021no—
212C4×D4—2021no—
213#117—1011no—
214#117—1011no—
215C2×C22⋊C4—2021no—
216C2×C22⋊C4—2021no—
217#107—1011no—
218#107—1011no—
219#106—1011no—
220#106—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#102—1011no—
228#102—1011no—
229#102C2×D10111yescharacteristic
230#102C2×D10111yescharacteristic
231C22⋊C8—1011no—
232C22⋊C8C2×D10111yescharacteristic
233#97—1011no—
234C4×C8—1011no—
235C24—21no—
236C2×Q8—1011no—
237C2×Q8C22×D10111yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D10111yescharacteristic
242C2×D4C22×D10111yescharacteristic
243C22×C4—2021no—
244C22×C4—2021no—
245C22×C4—2021no—
246C22×C4—2021no—
247C22×C4C22×D10111yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—2022no—
251SD16—2022no—
252C2×C8—1011no—
253C2×C8—1011no—
254C2×C8C22×D10111yescharacteristic
255C2×C8C22×D10111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—1011no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4C22×D10111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1011no—
272C22⋊C4—1011no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C42—1011no—
280C42—1011no—
281C42—2021no—
282C42—2021no—
283C42—2021no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D10111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2021no—
293Q8—2021no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1310807312yes—
298D4#1310807312yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—8081no—
304C2×C4—8081no—
305C2×C4—8081no—
306C2×C4—8081no—
307C2×C4—1011no—
308C2×C4—1011no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4D4×D10111yescharacteristic
316C2×C4C23×D10111yescharacteristic
317C8—21no—
318C8—21no—
319C8—2021no—
320C8—2021no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D10111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—4041no—
336C4—4041no—
337C4—4041no—
338C4—4041no—
339C4—2021no—
340C4—2021no—
341C4—2021no—
342C4—2021no—
343C4C2×#1310807311yescharacteristic
344C4C2×#1310807311yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2026568311yescharacteristic
349C2#2026598311yescharacteristic
350C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×C101C22
hypercenter#450D101
hypocenterC101#3123
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20265787C2semidirect
3. #20265519C2semidirect
4. #20265515C2semidirect
5. #20265479C2semidirect
6. #20265475C2semidirect
7. #20265380C2semidirect
8. #20265380C2semidirect
10. #20265278C2semidirect
11. #20265274C2semidirect
12. #20265268C2semidirect
13. SD16×Dic101C2semidirect
14. #20265235C2semidirect
18. #15692253C22semidirect
22. #15692246C22semidirect
25. Q8×Dic101C22semidirect
26. #15692220C22semidirect
37. #15692189C22semidirect
38. #15692188C22semidirect
39. #15692187C22semidirect
40. #15692186C22semidirect
41. #15692184C22semidirect
42. #15692184C22semidirect
43. #15692180C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692179C22semidirect
50. #15692165C22semidirect
51. C8×Dic101C22semidirect
176. C101#3123semidirect

Maximal subgroups

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

subgroupindex
#202657872
#202656522
#202655192
#202655152
#202654792
#202654752
#202653802
#202653282
#202652782
#202652742
#202652682
SD16×Dic1012
#202652352
#450×C1012
#3123101