Finite Group Databasebeta

← #16855957

Subgroups — #16855957

Order 3968.

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#14521659C211yesmaximal, characteristic, complemented
2#14521524C211yesmaximal, characteristic
3#14521391C211yesmaximal, characteristic, complemented
4#14521387C211yesmaximal, characteristic, complemented
5#14521351C211yesmaximal, characteristic, complemented
6#14521347C211yesmaximal, characteristic, complemented
7#14521252C211yesmaximal, characteristic, complemented
8#14521252C211yesmaximal, characteristic, complemented
9#14521200C211yesmaximal, characteristic
10#14521150C211yesmaximal, characteristic, complemented
11#14521146C211yesmaximal, characteristic, complemented
12#14521140C211yesmaximal, characteristic, complemented
13SD16×Dic31C211yesmaximal, characteristic, complemented
14#14521107C211yesmaximal, characteristic, complemented
15C31×#450C211yesmaximal, characteristic
16C22×D4×C31C2211yescharacteristic
17C2×SD16×C31C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic
19C31×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Dic31—21no—
21C2×C22⋊Dic31—21no—
22Q8×Dic31C2211yescharacteristic, complemented
23#11758521C2211yescharacteristic, complemented
24D4×Dic31—21no—
25D4×Dic31—21no—
26D4×Dic31C2211yescharacteristic
27D4×Dic31C2211yescharacteristic
28D4×Dic31C2211yescharacteristic
29D4×Dic31C2211yescharacteristic
30C2×#92480C2211yescharacteristic
31C2×#92480C2211yescharacteristic
32#11758503C2211yescharacteristic
33#11758503C2211yescharacteristic
34#11758501C2211yescharacteristic
35#11758467—21no—
36#11758467C2211yescharacteristic, complemented
37#11758466—21no—
38#11758464C2211yescharacteristic, complemented
39#11758456—21no—
40#11758456—21no—
41C31×#102C2211yescharacteristic
42C31×#102C2211yescharacteristic
43C31×C22⋊C8C2211yescharacteristic
44#11758423C2211yescharacteristic, complemented
45#11758423C2211yescharacteristic, complemented
46#11758420C2211yescharacteristic
47#11758420C2211yescharacteristic
48C22⋊(C31⋊C8)C2211yescharacteristic
49#11758406C2211yescharacteristic, complemented
50#11758405C2211yescharacteristic, complemented
51#11758404C2211yescharacteristic, complemented
52#11758404C2211yescharacteristic, complemented
53#11758403C2211yescharacteristic, complemented
54Dic31⋊C8C2211yescharacteristic, complemented
55Dic31⋊C8C2211yescharacteristic, complemented
56C8×Dic31C2211yescharacteristic, complemented
57#11758396C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C24×C31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×D4×C31—22no—
62C2×D4×C31—22no—
63C2×D4×C31—21no—
64C2×D4×C31C2311yescharacteristic
65C2×D4×C31C2311yescharacteristic
66C22×C4×C31C2311yescharacteristic
67C22×Dic31—21no—
68C22×Dic31—21no—
69C22×Dic31—21no—
70C22×Dic31—21no—
71C2×Dic62C2311yescharacteristic
72SD16×C31—22no—
73SD16×C31—22no—
74C2×C8×C31C2311yescharacteristic
75C2×C8×C31C2311yescharacteristic
76C4⋊C4×C31—21no—
77C4⋊C4×C31C2311yescharacteristic
78C22⋊C4×C31—21no—
79C22⋊Dic31—21no—
80C22⋊Dic31—21no—
81C22⋊Dic31—21no—
82C22⋊Dic31—21no—
83C22⋊Dic31—21no—
84C22⋊Dic31—21no—
85C22⋊Dic31D411yescharacteristic
86C22⋊Dic31D411yescharacteristic
87#92480—22no—
88#92480—22no—
89C4⋊Dic31—21no—
90C4⋊Dic31—21no—
91C4⋊Dic31D411yescharacteristic
92C4⋊Dic31D411yescharacteristic
93C4⋊Dic31C2311yescharacteristic
94C4⋊Dic31C2311yescharacteristic
95C4⋊Dic31C2311yescharacteristic
96Dic31⋊C4—21no—
97Dic31⋊C4—21no—
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4—21no—
101C4×Dic31—21no—
102C4×Dic31—21no—
103C4×Dic31—21no—
104C4×Dic31C2311yescharacteristic
105C4×Dic31C2311yescharacteristic
106C2×C31⋊C8C2311yescharacteristic
107C2×C31⋊C8C2311yescharacteristic
108C23×C31—24no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31—81no—
112C23×C31—81no—
113C23×C31C2×D411yescharacteristic
114Q8×C31—21no—
115Q8×C31—21no—
116D4×C31—24no—
117D4×C31—21no—
118D4×C31—21no—
119D4×C31#4112yes—
120D4×C31#4112yes—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2411yescharacteristic
127C2×Dic31—21no—
128C2×Dic31—21no—
129C2×Dic31—21no—
130C2×Dic31—21no—
131C2×Dic31—81no—
132C2×Dic31—81no—
133C2×Dic31—81no—
134C2×Dic31—81no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137C2×Dic31C2×D411yescharacteristic
138C2×Dic31C2×D411yescharacteristic
139Dic62—21no—
140Dic62—21no—
141C8×C31—21no—
142C8×C31—21no—
143C31⋊C8—21no—
144C31⋊C8—21no—
145#3123—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—22no—
148C22×C31—22no—
149C22×C31—21no—
150C22×C31—21no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—84no—
156C22×C31C22×D411yescharacteristic
157C4×C31—21no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31C2×#4111yescharacteristic
161C4×C31C2×#4111yescharacteristic
162Dic31—21no—
163Dic31—21no—
164Dic31—21no—
165Dic31—21no—
166Dic31—41no—
167Dic31—41no—
168Dic31—41no—
169Dic31—41no—
170#541—311no—
171#518—311no—
172#485—311no—
173#485—311no—
174#478—311no—
175#478—311no—
176#461—311no—
177#461—311no—
178#450—311no—
179#450D3111yescharacteristic
180#440—311no—
181#440—311no—
182C4×SD16—311no—
183C4×SD16—311no—
184#433—311no—
185C2×C31—41no—
186C2×C31—42no—
187C2×C31—42no—
188C2×C31#52511yescharacteristic
189C2×C31#57711yescharacteristic
190C2×C31#57711yescharacteristic
191C22×D4C2×D3111yescharacteristic
192C2×SD16—311no—
193C2×SD16—311no—
194C2×SD16C2×D3111yescharacteristic
195C2×SD16C2×D3111yescharacteristic
196C4⋊Q8—311no—
197#126—621no—
198#126—621no—
199#125—311no—
200#125—621no—
201#125—621no—
202C22⋊Q8—311no—
203C22⋊Q8C2×D3111yescharacteristic
204C4×Q8—311no—
205C4×Q8—311no—
206C4×D4—311no—
207C4×D4—311no—
208C4×D4—311no—
209C4×D4—311no—
210C4×D4—621no—
211C4×D4—621no—
212#117—311no—
213#117—311no—
214C2×C22⋊C4—621no—
215C2×C22⋊C4—621no—
216#107—311no—
217#107—311no—
218#106—311no—
219#106—311no—
220C4⋊C8—311no—
221C4⋊C8—311no—
222#103—311no—
223#103—311no—
224#103—311no—
225#103—311no—
226#102—311no—
227#102—311no—
228#102C2×D3111yescharacteristic
229#102C2×D3111yescharacteristic
230C22⋊C8—311no—
231C22⋊C8C2×D3111yescharacteristic
232#97—311no—
233C4×C8—311no—
234C31#312311yescharacteristic, complemented, Sylow
235C24—21no—
236C2×Q8—311no—
237C2×Q8C22×D3111yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D3111yescharacteristic
242C2×D4C22×D3111yescharacteristic
243C22×C4—621no—
244C22×C4—621no—
245C22×C4—621no—
246C22×C4—621no—
247C22×C4C22×D3111yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—622no—
251SD16—622no—
252C2×C8—311no—
253C2×C8—311no—
254C2×C8C22×D3111yescharacteristic
255C2×C8C22×D3111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—311no—
258C4⋊C4—311no—
259C4⋊C4—311no—
260C4⋊C4—311no—
261C4⋊C4—311no—
262C4⋊C4—621no—
263C4⋊C4—621no—
264C4⋊C4—621no—
265C4⋊C4—621no—
266C4⋊C4—621no—
267C4⋊C4—621no—
268C4⋊C4—621no—
269C4⋊C4C22×D3111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—311no—
272C22⋊C4—311no—
273C22⋊C4—621no—
274C22⋊C4—621no—
275C22⋊C4—621no—
276C22⋊C4—621no—
277C22⋊C4—621no—
278C22⋊C4—621no—
279C42—311no—
280C42—311no—
281C42—621no—
282C42—621no—
283C42—621no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D3111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—621no—
293Q8—621no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#9249712yes—
298D4#9249712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—311no—
304C2×C4—311no—
305C2×C4—311no—
306C2×C4—311no—
307C2×C4—621no—
308C2×C4—621no—
309C2×C4—621no—
310C2×C4—621no—
311C2×C4—2481no—
312C2×C4—2481no—
313C2×C4—2481no—
314C2×C4—2481no—
315C2×C4D4×D3111yescharacteristic
316C2×C4C23×D3111yescharacteristic
317C8—21no—
318C8—21no—
319C8—621no—
320C8—621no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D3111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—621no—
336C4—621no—
337C4—621no—
338C4—621no—
339C4—1241no—
340C4—1241no—
341C4—1241no—
342C4—1241no—
343C4C2×#9249711yescharacteristic
344C4C2×#9249711yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#1452155511yescharacteristic
349C2#1452185511yescharacteristic
350C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
prime radicalC22×D4×C31C22
hypercenter#450D31
hypocenterC31#3123
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #14521659C2semidirect
3. #14521391C2semidirect
4. #14521387C2semidirect
5. #14521351C2semidirect
6. #14521347C2semidirect
7. #14521252C2semidirect
8. #14521252C2semidirect
10. #14521150C2semidirect
11. #14521146C2semidirect
12. #14521140C2semidirect
13. SD16×Dic31C2semidirect
14. #14521107C2semidirect
22. Q8×Dic31C22semidirect
23. #11758521C22semidirect
36. #11758467C22semidirect
38. #11758464C22semidirect
44. #11758423C22semidirect
45. #11758423C22semidirect
49. #11758406C22semidirect
50. #11758405C22semidirect
51. #11758404C22semidirect
52. #11758404C22semidirect
53. #11758403C22semidirect
54. Dic31⋊C8C22semidirect
55. Dic31⋊C8C22semidirect
56. C8×Dic31C22semidirect
57. #11758396C22semidirect
58. #11758395C22semidirect
234. C31#3123semidirect

Maximal subgroups

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

subgroupindex
#145216592
#145215242
#145213912
#145213872
#145213512
#145213472
#145212522
#145212002
#145211502
#145211462
#145211402
SD16×Dic312
#145211072
C31×#4502
#312331