Finite Group Databasebeta

← #27179678

Subgroups — #27179678

Order 12928.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20266013C211yesmaximal, characteristic
2#20265943C211yesmaximal, characteristic, complemented
3#20265645C211yesmaximal, characteristic, complemented
4#20265195C211yesmaximal, characteristic, complemented
5#20265192C214yesmaximal, complemented
6C2×#15692182C211yesmaximal, characteristic, complemented
7C2×#15692181C211yesmaximal, characteristic, complemented
8#104×D101C214yesmaximal, complemented
9#421×C101C211yesmaximal, characteristic, complemented
10#15692303C412yescomplemented
11#15692302—21no—
12#15692302C412yescomplemented
13#15692301—21no—
14#15692301C412yescomplemented
15#15692300—22no—
16#15692300C412yescomplemented
17C2×#13108074—22no—
18C2×#13108074C2211yescharacteristic
19C2×#13108074C2211yescharacteristic
20#41×D101—22no—
21#41×D101C2214yes—
22#15692263—22no—
23#15692261—22no—
24C2×#13108058C2211yescharacteristic, complemented
25M16×D101C2214yescomplemented
26#15692220—21no—
27#15692217—21no—
28#15692215—22no—
29#15692212C2212yescomplemented
30C2×C4×Dic101C2211yescharacteristic, complemented
31C42×D101C2212yescomplemented
32#15692183C2214yescomplemented
33#15692182C2212yescomplemented
34#15692182C2212yescomplemented
35#15692181C2212yescomplemented
36#15692181C2212yescomplemented
37C2×#41×C101C2211yescharacteristic
38C2×M16×C101C2211yescharacteristic, complemented
39#117×C101C2211yescharacteristic, complemented
40#104×C101C2214yescomplemented
41#13108075—22no—
42#13108075C2×C414yescomplemented
43#13108074—21no—
44#13108074—21no—
45#13108074—24no—
46#13108074—22no—
47#13108074—24no—
48#13108074—22no—
49#13108074C2×C412yes—
50#13108074C2×C412yes—
51#13108074C2×C412yescomplemented
52#13108074D412yes—
53#13108074D412yes—
54#13108074C2312yes—
55#13108073—24no—
56#13108073—24no—
57#13108073—22no—
58#13108073C2×C414yescomplemented
59C2×Dic202—22no—
60C2×Dic202C2×C411yescharacteristic, complemented
61C2×Dic202D411yescharacteristic
62Q8×D101—22no—
63Q8×D101C2×C414yescomplemented
64C2×D404—22no—
65C2×D404C2×C411yescharacteristic, complemented
66C2×D404D411yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—24no—
69C2×C22⋊D101D412yes—
70D4×D101—24no—
71D4×D101—24no—
72D4×D101—22no—
73D4×D101C2×C414yescomplemented
74C22×Dic101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—24no—
77C2×C4×D101—22no—
78C2×C4×D101C2312yes—
79#13108058—21no—
80#13108058C2312yes—
81#13108057—24no—
82C2×#11717579—21no—
83C8×D101—24no—
84#13108053—22no—
85#13108051—22no—
86C4×Dic101—21no—
87C4×Dic101—22no—
88C4×Dic101C2311yescharacteristic
89C4×Dic101C2311yescharacteristic
90#41×C101—21no—
91#41×C101—21no—
92#41×C101C2×C412yescomplemented
93#41×C101C2312yes—
94C2×Q8×C101C2×C411yescharacteristic, complemented
95C2×D4×C101—21no—
96C2×D4×C101C2×C411yescharacteristic, complemented
97C22×C4×C101—21no—
98C22×C4×C101C2311yescharacteristic
99M16×C101—21no—
100M16×C101C2312yes—
101C2×C8×C101—21no—
102C4⋊C4×C101—21no—
103C22⋊C4×C101—21no—
104C42×C101C2312yes—
105C23×C101—21no—
106C23×C101C2×D411yescharacteristic
107C22×D101—21no—
108C22×D101—24no—
109C22×D101C2×D412yes—
110Q8×C101—21no—
111Q8×C101C22×C412yes—
112D4×C101—22no—
113D4×C101—22no—
114D4×C101—21no—
115D4×C101C22×C412yes—
116C2×C4×C101—21no—
117C2×C4×C101—22no—
118C2×C4×C101—22no—
119C2×C4×C101—22no—
120C2×C4×C101C22×C411yescharacteristic
121C2×C4×C101C22×C411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C22⋊D101—22no—
127C22⋊D101—22no—
128C22⋊D101—24no—
129C22⋊D101—24no—
130C22⋊D101C22⋊C414yes—
131C22⋊D101C22⋊C414yes—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—24no—
135C2×Dic101—21no—
136C2×Dic101—42no—
137C2×Dic101C2×D412yes—
138D404—24no—
139D404—21no—
140D404C22⋊C412yes—
141D404C22⋊C412yes—
142D404C22×C412yes—
143C4×D101—24no—
144C4×D101—22no—
145C4×D101—22no—
146C4×D101—24no—
147C4×D101—44no—
148C4×D101C22×C414yes—
149C4×D101C2×D414yes—
150Dic202—21no—
151Dic202—24no—
152Dic202C22⋊C412yes—
153Dic202C22⋊C412yes—
154Dic202C22×C412yes—
155C8×C101—22no—
156#11717579—22no—
157C22×C101—22no—
158C22×C101—41no—
159C22×C101—81no—
160C22×C101C2×C22⋊C411yescharacteristic
161C22×C101C2×C22⋊C411yescharacteristic
162C22×C101C22×D411yescharacteristic
163C2×D101—22no—
164C2×D101—42no—
165C2×D101—84no—
166C2×D101—81no—
167C2×D101C2×C22⋊C414yes—
168C4×C101—22no—
169C4×C101—42no—
170C4×C101C2×C22⋊C411yescharacteristic
171C4×C101C2×C22⋊C411yescharacteristic
172C4×C101C23×C411yescharacteristic
173C4×C101C22×D411yescharacteristic
174Dic101—22no—
175Dic101—22no—
176Dic101—41no—
177Dic101C2×C22⋊C414yes—
178C2×C101—21no—
179C2×C101—21no—
180C2×C101—21no—
181C2×C101—42no—
182C2×C101C22×C22⋊C411yescharacteristic
183D101—24no—
184D101—42no—
185#2901—1011nomaximal, Sylow
186C101#290111yescharacteristic, complemented, Sylow
187#585—1011no—
188#568—1011no—
189C4×#41—1011no—
190#421—1011no—
191#421—1014no—
192#421D10111yescharacteristic, complemented
193C2×#104—1011no—
194C2×#104—1011no—
195C2×#104—1014no—
1962−1+4—1012no—
1972−1+4—1012no—
1982−1+4—2021no—
199H(5,𝔽2)—1012no—
200H(5,𝔽2)—1012no—
201H(5,𝔽2)—2022no—
202H(5,𝔽2)—2021no—
203C2×#41—1011no—
204C2×#41—1014no—
205C2×#41—1011no—
206C2×#41—2022no—
207C2×#41—2022no—
208C2×#41C2×D10111yescharacteristic
209#131—2022no—
210#131—2022no—
211C2×M16—1011no—
212C2×M16—1014no—
213C2×M16C2×D10111yescharacteristic, complemented
214C4×Q8—2021no—
215C4×D4—2022no—
216C4×D4—2021no—
217#117—1012no—
218#117C2×D10111yescharacteristic, complemented
219C2×C42—1011no—
220C2×C42—1012no—
221#104—1012no—
222#104—1012no—
223#104—1012no—
224#104—1012no—
225#104—1014no—
226#104C2×D10114yescomplemented
227#41—21no—
228#41—21no—
229#41—1012no—
230#41—1012no—
231#41—1012no—
232#41—1012no—
233#41—1014no—
234#41—1014no—
235#41—1012no—
236#41—1012no—
237#41—2024no—
238#41—2024no—
239#41—2022no—
240#41—2022no—
241#41—2024no—
242#41—2024no—
243#41—2022no—
244#41—2022no—
245#41—2021no—
246#41—2021no—
247#41C4×D10112yescomplemented
248#41C22×D10112yes—
249C2×Q8—1011no—
250C2×Q8—1014no—
251C2×Q8—1011no—
252C2×Q8—2022no—
253C2×Q8—2022no—
254C2×Q8C4×D10111yescharacteristic, complemented
255C2×D4—21no—
256C2×D4—1012no—
257C2×D4—1014no—
258C2×D4—1011no—
259C2×D4—1011no—
260C2×D4—2022no—
261C2×D4—2024no—
262C2×D4—2022no—
263C2×D4—2024no—
264C2×D4—2024no—
265C2×D4—2021no—
266C2×D4C4×D10111yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—1012no—
269C22×C4—2024no—
270C22×C4—2022no—
271C22×C4—2021no—
272C22×C4—2021no—
273C22×C4C22×D10111yescharacteristic
274M16—21no—
275M16—1012no—
276M16—2024no—
277M16—2021no—
278M16C22×D10112yes—
279C2×C8—21no—
280C2×C8—2024no—
281C2×C8—2021no—
282C4⋊C4—21no—
283C4⋊C4—2022no—
284C22⋊C4—21no—
285C22⋊C4—2022no—
286C42—1011no—
287C42—1011no—
288C42—2022no—
289C42—2021no—
290C42C22×D10112yes—
291C23—21no—
292C23—1012no—
293C23—2024no—
294C23—2021no—
295C23D4×D10111yescharacteristic
296Q8—21no—
297Q8—1012no—
298Q8—1012no—
299Q8—1012no—
300Q8—2024no—
301Q8—2021no—
302Q8C2×C4×D10112yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—1014no—
307D4—1014no—
308D4—1012no—
309D4—1012no—
310D4—1012no—
311D4—2024no—
312D4—2024no—
313D4—2024no—
314D4—2022no—
315D4—2022no—
316D4—2021no—
317D4C2×C4×D10112yes—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—1012no—
323C2×C4—1014no—
324C2×C4—1014no—
325C2×C4—4044no—
326C2×C4—4042no—
327C2×C4—2024no—
328C2×C4—2024no—
329C2×C4—2024no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4—2022no—
334C2×C4—2022no—
335C2×C4C2×C4×D10111yescharacteristic
336C2×C4C2×C4×D10111yescharacteristic
337C2×C4D4×D10111yescharacteristic
338C2×C4D4×D10111yescharacteristic
339C2×C4D4×D10111yescharacteristic
340C2×C4C23×D10111yescharacteristic
341C8—22no—
342C8—2022no—
343C22—22no—
344C22—41no—
345C22—81no—
346C22—8081no—
347C22—8084no—
348C22—1014no—
349C22—4042no—
350C22—2022no—
351C22C22⋊C4×D10111yescharacteristic
352C22C22⋊C4×D10111yescharacteristic
353C22C2×D4×D10111yescharacteristic
354C4—22no—
355C4—42no—
356C4—1014no—
357C4—4041no—
358C4—2022no—
359C4—2022no—
360C4C22⋊C4×D10111yescharacteristic
361C4C22⋊C4×D10111yescharacteristic
362C4C22×C4×D10111yescharacteristic
363C4C2×D4×D10111yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—4042no—
369C2—2024no—
370C2C2×C22⋊C4×D10111yescharacteristic
subgroupG/H
centerC4C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#421×C101C2
socleC2×C101C22×C22⋊C4
prime radical#20266013C2
cosocleC2×C4×C101C24
hypercenter#421D101
hypocenterC101#2901
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20265943C2semidirect
3. #20265645C2semidirect
4. #20265195C2semidirect
5. #20265192C2semidirect
6. C2×#15692182C2semidirect
7. C2×#15692181C2semidirect
8. #104×D101C2semidirect
9. #421×C101C2semidirect
10. #15692303C4semidirect
12. #15692302C4semidirect
14. #15692301C4semidirect
16. #15692300C4semidirect
24. C2×#13108058C22semidirect
25. M16×D101C22semidirect
29. #15692212C22semidirect
30. C2×C4×Dic101C22semidirect
31. C42×D101C22semidirect
32. #15692183C22semidirect
33. #15692182C22semidirect
34. #15692182C22semidirect
35. #15692181C22semidirect
36. #15692181C22semidirect
38. C2×M16×C101C22semidirect
39. #117×C101C22semidirect
40. #104×C101C22semidirect
42. #13108075C2×C4semidirect
51. #13108074C2×C4semidirect
58. #13108073C2×C4semidirect
60. C2×Dic202C2×C4semidirect
63. Q8×D101C2×C4semidirect
65. C2×D404C2×C4semidirect
73. D4×D101C2×C4semidirect
92. #41×C101C2×C4semidirect
94. C2×Q8×C101C2×C4semidirect
96. C2×D4×C101C2×C4semidirect
186. C101#2901semidirect
192. #421D101semidirect
213. C2×M16C2×D101semidirect
218. #117C2×D101semidirect
226. #104C2×D101semidirect
247. #41C4×D101semidirect
254. C2×Q8C4×D101semidirect
266. C2×D4C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
#202659432
#202656452
#202651952
#202651922
C2×#156921822
C2×#156921812
#104×D1012
#421×C1012
#2901101