Finite Group Databasebeta

← #28374849

Subgroups — #28374849

Order 13952.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295734C211yesmaximal, characteristic, complemented
2D4×C22⋊D109C211yesmaximal, characteristic
3#20295371C211yesmaximal, characteristic, complemented
4#20295345C211yesmaximal, characteristic, complemented
5#20295343C211yesmaximal, characteristic, complemented
6#20295314C211yesmaximal, characteristic, complemented
7#20295257C211yesmaximal, characteristic, complemented
8#20295252C211yesmaximal, characteristic, complemented
9#20295224C211yesmaximal, characteristic, complemented
10#20295175C211yesmaximal, characteristic, complemented
11SD16×Dic109C211yesmaximal, characteristic, complemented
12#20295089C211yesmaximal, characteristic, complemented
13C2×#16812620C211yesmaximal, characteristic, complemented
14#102×D109C211yesmaximal, characteristic, complemented
15#450×C109C211yesmaximal, characteristic, complemented
16C22×C22⋊D109—21no—
17C2×D4×D109C2211yescharacteristic
18C2×#13182016C2211yescharacteristic, complemented
19C22×#11724784C2211yescharacteristic, complemented
20C2×C8×D109C2211yescharacteristic, complemented
21#16812698C2211yescharacteristic, complemented
22#16812695C2211yescharacteristic
23#16812689—21no—
24#16812679—21no—
25#16812676—21no—
26#16812675C2211yescharacteristic, complemented
27#16812673C2211yescharacteristic
28#16812671—21no—
29#16812670C2211yescharacteristic
30#16812668—21no—
31#16812667—21no—
32Q8×Dic109C2211yescharacteristic, complemented
33D4×Dic109C2211yescharacteristic
34C4×C22⋊D109C2211yescharacteristic
35#16812659—21no—
36C2×C4⋊Dic109C2211yescharacteristic, complemented
37C4⋊C4×D109C2211yescharacteristic, complemented
38#16812631C2211yescharacteristic, complemented
39#16812631C2211yescharacteristic, complemented
40#16812630C2211yescharacteristic, complemented
41#16812628C2211yescharacteristic, complemented
42#16812623C2211yescharacteristic, complemented
43#16812621C2211yescharacteristic, complemented
44#16812621C2211yescharacteristic, complemented
45#16812620C2211yescharacteristic
46#16812620C2211yescharacteristic
47#16812620C2211yescharacteristic, complemented
48#16812620C2211yescharacteristic, complemented
49#16812619C2211yescharacteristic, complemented
50#16812611C2211yescharacteristic, complemented
51C8×Dic109C2211yescharacteristic, complemented
52C22×D4×C109C2211yescharacteristic
53C2×SD16×C109C2211yescharacteristic
54C2×SD16×C109C2211yescharacteristic, complemented
55C22⋊Q8×C109C2211yescharacteristic, complemented
56#102×C109C2211yescharacteristic
57#102×C109C2211yescharacteristic, complemented
58C22⋊C8×C109C2211yescharacteristic, complemented
59C23×D109—21no—
60C2×Dic218—21no—
61C2×D436C2311yescharacteristic
62C2×C22⋊D109—21no—
63C2×C22⋊D109—21no—
64C2×C22⋊D109—21no—
65C2×C22⋊D109—21no—
66C2×C22⋊D109—21no—
67C2×C22⋊D109—44no—
68C2×C22⋊D109D411yescharacteristic
69D4×D109—22no—
70D4×D109—41no—
71C22×Dic109—21no—
72C22×Dic109—21no—
73C2×C4×D109—21no—
74C2×C4×D109C2311yescharacteristic
75#13182016—41no—
76C2×#11724784—41no—
77C2×#11724784C2311yescharacteristic
78C2×#11724784C2311yescharacteristic
79C8×D109—41no—
80C4⋊Dic109—21no—
81C4⋊Dic109—21no—
82C4⋊Dic109C2311yescharacteristic
83C4⋊Dic109C2311yescharacteristic
84C4⋊Dic109C2311yescharacteristic
85#13182008—21no—
86#13182008—21no—
87#13182008—21no—
88#13182008D411yescharacteristic
89C22⋊Dic109—21no—
90C22⋊Dic109—21no—
91C22⋊Dic109D411yescharacteristic
92#13182006—21no—
93#13182006—21no—
94#13182006D411yescharacteristic
95C4×Dic109—21no—
96C4×Dic109C2311yescharacteristic
97C24×C109—21no—
98C2×Q8×C109C2311yescharacteristic
99C2×D4×C109—22no—
100C2×D4×C109—21no—
101C2×D4×C109—41no—
102C2×D4×C109C2311yescharacteristic
103C2×D4×C109C2311yescharacteristic
104C22×C4×C109C2311yescharacteristic
105SD16×C109—22no—
106SD16×C109—41no—
107C2×C8×C109C2311yescharacteristic
108C2×C8×C109C2311yescharacteristic
109C4⋊C4×C109—21no—
110C4⋊C4×C109C2311yescharacteristic
111C22⋊C4×C109—21no—
112C23×C109—21no—
113C23×C109—21no—
114C23×C109—42no—
115C23×C109—42no—
116C23×C109—81no—
117C23×C109C2×D411yescharacteristic
118C22×D109—21no—
119C22×D109—42no—
120C22×D109—42no—
121C22×D109—81no—
122C22×D109C2×D411yescharacteristic
123Q8×C109—21no—
124Q8×C109—21no—
125D4×C109—21no—
126D4×C109—21no—
127D4×C109—21no—
128D4×C109—42no—
129D4×C109#4112yes—
130C2×C4×C109—21no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109—21no—
134C2×C4×C109C2×D411yescharacteristic
135C2×C4×C109C2411yescharacteristic
136C22⋊D109—42no—
137C22⋊D109—42no—
138C22⋊D109—42no—
139C22⋊D109—42no—
140C22⋊D109—81no—
141C22⋊D109SD1614yescomplemented
142C2×Dic109—21no—
143C2×Dic109—21no—
144C2×Dic109—21no—
145C2×Dic109—42no—
146C2×Dic109—81no—
147C2×Dic109C2×D411yescharacteristic
148C2×Dic109C2×D411yescharacteristic
149C2×Dic109C2×D411yescharacteristic
150D436—21no—
151D436—21no—
152C4×D109—21no—
153C4×D109—21no—
154C4×D109—81no—
155Dic218—81no—
156C8×C109—21no—
157C8×C109—21no—
158#11724784—21no—
159#11724784—21no—
160C22×C109—21no—
161C22×C109—21no—
162C22×C109—22no—
163C22×C109—41no—
164C22×C109—41no—
165C22×C109—41no—
166C22×C109—41no—
167C22×C109—41no—
168C22×C109—84no—
169C22×C109C2×SD1612yes—
170C22×C109C22×D411yescharacteristic
171C2×D109—21no—
172C2×D109—21no—
173C2×D109—41no—
174C2×D109—41no—
175C2×D109—41no—
176C2×D109—84no—
177C2×D109C2×SD1612yes—
178C4×C109—21no—
179C4×C109—41no—
180C4×C109—41no—
181C4×C109C2×#4111yescharacteristic
182C4×C109H(5,𝔽2)11yescharacteristic
183Dic109—21no—
184Dic109—21no—
185Dic109—41no—
186Dic109—41no—
187Dic109—41no—
188Dic109C2×SD1612yes—
189C2×C109—22no—
190C2×C109—42no—
191C2×C109—81no—
192C2×C109#54611yescharacteristic
193C2×C109C22×SD1611yescharacteristic
194C2×C109#57611yescharacteristic
195D109—22no—
196D109—81no—
197#3295—1091nomaximal, Sylow
198C109#329511yescharacteristic, complemented, Sylow
199#554—1091no—
200D42—1091no—
201#486—1091no—
202#481—1091no—
203#481—1091no—
204#476—1091no—
205#465—1091no—
206#465—1091no—
207#460—1091no—
208#450—1091no—
209#450D10911yescharacteristic, complemented
210C4×SD16—1091no—
211C8×D4—1091no—
212C2×#102—1091no—
213C2×#102—1091no—
214C22×D4—1091no—
215C22×D4—2181no—
216C22×D4C2×D10911yescharacteristic
217C2×SD16—1091no—
218C2×SD16C2×D10911yescharacteristic
219C2×SD16C2×D10911yescharacteristic, complemented
220C22×C8—1091no—
221C22×C8—1091no—
222C4⋊Q8—1091no—
223#127—1091no—
224#125—2181no—
225C22⋊Q8—1091no—
226C22⋊Q8—2181no—
227C22⋊Q8—2181no—
228C22⋊Q8C2×D10911yescharacteristic, complemented
229#121—1091no—
230#121—1091no—
231#121—2181no—
232#120—2181no—
233#120—2181no—
234C4×Q8—1091no—
235C4×D4—1091no—
236C4×D4—1091no—
237C4×D4—2181no—
238C2×C4⋊C4—1091no—
239C2×C4⋊C4—1091no—
240#106—1091no—
241#106—1091no—
242#106—1091no—
243C4⋊C8—1091no—
244#103—1091no—
245#102—1091no—
246#102—1091no—
247#102—1091no—
248#102—1091no—
249#102—1091no—
250#102—1091no—
251#102—1091no—
252#102C2×D10911yescharacteristic
253#102C2×D10911yescharacteristic, complemented
254C22⋊C8—1091no—
255C22⋊C8C2×D10911yescharacteristic, complemented
256C4×C8—1091no—
257C24—21no—
258C24—2181no—
259C2×Q8—2181no—
260C2×Q8C22×D10911yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1091no—
265C2×D4—1091no—
266C2×D4—4364no—
267C2×D4—4361no—
268C2×D4—2181no—
269C2×D4—2181no—
270C2×D4—2181no—
271C2×D4—2181no—
272C2×D4—2181no—
273C2×D4—2182no—
274C2×D4C22×D10911yescharacteristic
275C2×D4C22×D10911yescharacteristic
276C22×C4—1091no—
277C22×C4—2181no—
278C22×C4—2181no—
279C22×C4—2181no—
280C22×C4C22×D10911yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—4361no—
284C2×C8—1091no—
285C2×C8—1091no—
286C2×C8—4361no—
287C2×C8—4361no—
288C2×C8C22×D10911yescharacteristic
289C2×C8C22×D10911yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1091no—
292C4⋊C4—1091no—
293C4⋊C4—1091no—
294C4⋊C4—1091no—
295C4⋊C4—2181no—
296C4⋊C4—2181no—
297C4⋊C4—2181no—
298C4⋊C4—2181no—
299C4⋊C4—2181no—
300C4⋊C4C22×D10911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—1091no—
303C22⋊C4—1091no—
304C22⋊C4—2181no—
305C22⋊C4—2181no—
306C22⋊C4—2181no—
307C22⋊C4—2181no—
308C42—1091no—
309C42—2181no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—8721no—
316C23—1091no—
317C23—4362no—
318C23—4362no—
319C23—2181no—
320C23D4×D10911yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—8721no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—8721no—
329D4—1094no—
330D4—4362no—
331D4—4362no—
332D4—4362no—
333D4—4362no—
334D4—2181no—
335D4—2181no—
336D4#1318202812yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—8721no—
342C2×C4—8721no—
343C2×C4—1091no—
344C2×C4—1091no—
345C2×C4—1091no—
346C2×C4—4362no—
347C2×C4—2181no—
348C2×C4—2181no—
349C2×C4—2181no—
350C2×C4—2181no—
351C2×C4—2181no—
352C2×C4D4×D10911yescharacteristic
353C2×C4C23×D10911yescharacteristic
354C8—21no—
355C8—21no—
356C8—2181no—
357C8—2181no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—8724no—
368C22—1092no—
369C22—4361no—
370C22—4361no—
371C22—4361no—
372C22—2181no—
373C22—2181no—
374C22SD16×D10912yes—
375C22C2×D4×D10911yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—1092no—
380C4—4361no—
381C4—4361no—
382C4—4361no—
383C4—2181no—
384C4—2181no—
385C4C2×#1318202811yescharacteristic
386C4#1681274411yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—8721no—
391C2—2182no—
392C2#2029566311yescharacteristic
393C2C2×SD16×D10911yescharacteristic
394C2#2029583011yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#450×C109C2
socleC22×C109C22×D4
prime radicalD4×C22⋊D109C2
hypercenter#450D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20295734C2semidirect
3. #20295371C2semidirect
4. #20295345C2semidirect
5. #20295343C2semidirect
6. #20295314C2semidirect
7. #20295257C2semidirect
8. #20295252C2semidirect
9. #20295224C2semidirect
10. #20295175C2semidirect
11. SD16×Dic109C2semidirect
12. #20295089C2semidirect
13. C2×#16812620C2semidirect
14. #102×D109C2semidirect
15. #450×C109C2semidirect
18. C2×#13182016C22semidirect
19. C22×#11724784C22semidirect
20. C2×C8×D109C22semidirect
21. #16812698C22semidirect
26. #16812675C22semidirect
32. Q8×Dic109C22semidirect
36. C2×C4⋊Dic109C22semidirect
37. C4⋊C4×D109C22semidirect
38. #16812631C22semidirect
39. #16812631C22semidirect
40. #16812630C22semidirect
41. #16812628C22semidirect
42. #16812623C22semidirect
43. #16812621C22semidirect
44. #16812621C22semidirect
47. #16812620C22semidirect
48. #16812620C22semidirect
49. #16812619C22semidirect
50. #16812611C22semidirect
51. C8×Dic109C22semidirect
54. C2×SD16×C109C22semidirect
55. C22⋊Q8×C109C22semidirect
57. #102×C109C22semidirect
58. C22⋊C8×C109C22semidirect
141. C22⋊D109SD16semidirect
198. C109#3295semidirect
209. #450D109semidirect
219. C2×SD16C2×D109semidirect
228. C22⋊Q8C2×D109semidirect
253. #102C2×D109semidirect
255. C22⋊C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957342
D4×C22⋊D1092
#202953712
#202953452
#202953432
#202953142
#202952572
#202952522
#202952242
#202951752
SD16×Dic1092
#202950892
C2×#168126202
#102×D1092
#450×C1092
#3295109