Finite Group Databasebeta

← #16857741

Subgroups — #16857741

Order 3968.

402 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#14521695C211yesmaximal, characteristic
2#14521411C211yesmaximal, characteristic, complemented
3#14521390C211yesmaximal, characteristic, complemented
4#14521373C211yesmaximal, characteristic, complemented
5#14521336C211yesmaximal, characteristic, complemented
6#14521305C211yesmaximal, characteristic, complemented
7#14521280C211yesmaximal, characteristic, complemented
8#14521271C211yesmaximal, characteristic, complemented
9#14521209C211yesmaximal, characteristic, complemented
10#14521199C211yesmaximal, characteristic, complemented
11#14521147C211yesmaximal, characteristic, complemented
12#14521121C211yesmaximal, characteristic, complemented
13#14521053C211yesmaximal, characteristic, complemented
14#14521043C211yesmaximal, characteristic, complemented
15C31×#556C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C31×C4⋊Q8C2211yescharacteristic, complemented
19C31×#126—21no—
20C31×#126—21no—
21C31×#124—21no—
22C31×C22⋊Q8C2211yescharacteristic
23C31×C22⋊Q8C2211yescharacteristic, complemented
24C4×Q8×C31—21no—
25C4×Q8×C31C2211yescharacteristic, complemented
26C4×D4×C31C2211yescharacteristic
27C31×#117C2211yescharacteristic
28C31×#117C2211yescharacteristic, complemented
29C2×#92482C2211yescharacteristic, complemented
30C2×#92481C2211yescharacteristic
31C2×#92481C2211yescharacteristic
32#11758514—21no—
33#11758507C2211yescharacteristic
34#11758501C2211yescharacteristic
35C2×#92474C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic, complemented
37#11758471—21no—
38#11758461—21no—
39C22⋊C4×D31—21no—
40#11758452C2211yescharacteristic
41C4×D124C2211yescharacteristic
42#11758423C2211yescharacteristic, complemented
43C22⋊(C31⋊C8)C2211yescharacteristic
44C22⋊(C31⋊C8)C2211yescharacteristic, complemented
45#11758398C2211yescharacteristic, complemented
46#11758398C2211yescharacteristic, complemented
47#11758398C2211yescharacteristic, complemented
48#11758397C2211yescharacteristic
49#11758397C2211yescharacteristic
50#11758397C2211yescharacteristic
51#11758397C2211yescharacteristic
52#11758397C2211yescharacteristic
53#11758397C2211yescharacteristic
54#11758396C2211yescharacteristic, complemented
55#11758396C2211yescharacteristic, complemented
56#11758395C2211yescharacteristic, complemented
57C4⋊(C31⋊C8)C2211yescharacteristic, complemented
58#11758391C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31—21no—
61C2×Q8×C31C2311yescharacteristic
62C2×D4×C31D411yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67#92495—22no—
68#92495—41no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124—41no—
72C2×D124C2311yescharacteristic
73C2×D124C2311yescharacteristic
74C2×C4×D31—21no—
75C2×C4×D31—21no—
76C2×Dic62C2311yescharacteristic
77C4⋊C4×C31—21no—
78C4⋊C4×C31—21no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31D411yescharacteristic
82C4⋊C4×C31C2311yescharacteristic
83C4⋊C4×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C22⋊C4×C31—21no—
86C22⋊C4×C31—21no—
87C22⋊C4×C31—21no—
88C22⋊C4×C31—21no—
89C22⋊C4×C31D411yescharacteristic
90C22⋊C4×C31D411yescharacteristic
91C42×C31—21no—
92C42×C31—21no—
93C42×C31—21no—
94C42×C31C2311yescharacteristic
95C22⋊Dic31—21no—
96#92482—41no—
97#92481—22no—
98#92481—41no—
99(C2×D31)⋊C4—21no—
100(C2×D31)⋊C4—21no—
101(C2×D31)⋊C4—21no—
102(C2×D31)⋊C4—21no—
103C4⋊Dic31C2311yescharacteristic
104Dic31⋊C4—21no—
105#92474—41no—
106C2×C31⋊C8—41no—
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C2×C31⋊C8C2311yescharacteristic
111C23×C31C2×D411yescharacteristic
112C23×C31C2×D411yescharacteristic
113C22×D31—21no—
114C22×D31—21no—
115C22×D31—42no—
116C22×D31—42no—
117C22×D31—81no—
118Q8×C31—21no—
119Q8×C31—21no—
120Q8×C31—42no—
121D4×C31—41no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2411yescharacteristic
135C22⋊D31—42no—
136C22⋊D31—81no—
137C2×Dic31—21no—
138C2×Dic31—21no—
139D124—21no—
140D124—21no—
141D124—21no—
142D124—42no—
143D124#4112yes—
144C4×D31—42no—
145C4×D31—81no—
146Dic62—21no—
147Dic62—21no—
148C31⋊C8—21no—
149C31⋊C8—21no—
150C31⋊C8—21no—
151C31⋊C8—21no—
152#3311—311nomaximal, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31C22×D411yescharacteristic
160C2×D31—22no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—84no—
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31C2×#4111yescharacteristic
176C4×C31H(5,𝔽2)11yescharacteristic
177Dic31—41no—
178Dic31—41no—
179#556D3111yescharacteristic, complemented
180#546—311no—
181#489—311no—
182#485—311no—
183#482—311no—
184#476—311no—
185#470—311no—
186#465—311no—
187#464—311no—
188#452—311no—
189#450—311no—
190#440—311no—
191#435—311no—
192#418—311no—
193#416—311no—
194C2×C31—41no—
195C2×C31—41no—
196C2×C31#54611yescharacteristic
197C2×C31#57511yescharacteristic
198C2×C31#57711yescharacteristic
199D31—42no—
200D31—81no—
201C2×#41—311no—
202C22×D4—311no—
203C2×Q16—311no—
204C2×SD16—311no—
205C2×SD16—311no—
206C2×M16—311no—
207C22×C8—311no—
208C4⋊Q8C2×D3111yescharacteristic, complemented
209#126—21no—
210#126—21no—
211#124—21no—
212#124—311no—
213#123—621no—
214C22⋊Q8—311no—
215C22⋊Q8C2×D3111yescharacteristic
216C22⋊Q8C2×D3111yescharacteristic, complemented
217#121—311no—
218#121—621no—
219#120—621no—
220C4×Q8—21no—
221C4×Q8C2×D3111yescharacteristic, complemented
222C4×D4—311no—
223C4×D4C2×D3111yescharacteristic
224#117C2×D3111yescharacteristic
225#117C2×D3111yescharacteristic, complemented
226C2×C22⋊C4—621no—
227#107—311no—
228#106—311no—
229#106—311no—
230C4⋊C8—311no—
231#103—311no—
232#103—311no—
233#103—311no—
234#103—311no—
235#102—311no—
236#102—311no—
237#102—311no—
238#102—311no—
239#102—311no—
240#102—311no—
241C22⋊C8—311no—
242C22⋊C8—311no—
243#97—311no—
244C31#331111yescharacteristic, complemented, Sylow
245C24—621no—
246#41—622no—
247#41—1241no—
248C2×Q8—21no—
249C2×Q8—311no—
250C2×Q8C22×D3111yescharacteristic
251C2×D4—311no—
252C2×D4—311no—
253C2×D4—622no—
254C2×D4—621no—
255C2×D4—621no—
256C2×D4—621no—
257C2×D4—1241no—
258C2×D4C22⋊D3111yescharacteristic
259C22×C4—621no—
260C22×C4—621no—
261C22×C4C22×D3111yescharacteristic
262C22×C4C22×D3111yescharacteristic
263Q16—1241no—
264SD16—622no—
265SD16—1241no—
266M16—1241no—
267C2×C8—311no—
268C2×C8—311no—
269C2×C8—311no—
270C2×C8—311no—
271C2×C8—1241no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—311no—
277C4⋊C4—621no—
278C4⋊C4C22⋊D3111yescharacteristic
279C4⋊C4C22×D3111yescharacteristic
280C4⋊C4C22×D3111yescharacteristic
281C4⋊C4C22×D3111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C22⋊C4—621no—
290C22⋊C4—621no—
291C22⋊C4C22⋊D3111yescharacteristic
292C22⋊C4C22⋊D3111yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D3111yescharacteristic
297C23—621no—
298C23—621no—
299C23—1242no—
300C23—1242no—
301C23—2481no—
302C23C2×C22⋊D3111yescharacteristic
303C23C2×C22⋊D3111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—621no—
308Q8—621no—
309D4—41no—
310D4—312no—
311D4—621no—
312D4—621no—
313D4—621no—
314D4—1242no—
315D4—1242no—
316D4—2481no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—621no—
326C2×C4—621no—
327C2×C4—1242no—
328C2×C4—2481no—
329C2×C4C2×C22⋊D3111yescharacteristic
330C2×C4C2×C22⋊D3111yescharacteristic
331C2×C4C2×C22⋊D3111yescharacteristic
332C2×C4C2×C22⋊D3111yescharacteristic
333C2×C4C23×D3111yescharacteristic
334C8—621no—
335C8—621no—
336C8—621no—
337C8—621no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—622no—
345C22—1241no—
346C22—1241no—
347C22—1241no—
348C22—1241no—
349C22—1241no—
350C22—2484no—
351C22C22×C22⋊D3111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—1241no—
361C4—1241no—
362C4#41×D3111yescharacteristic
363C4#1175856611yescharacteristic
364C2—41no—
365C2—41no—
366C2—1242no—
367C2—2481no—
368C2#1452168711yescharacteristic
369C2#1452184711yescharacteristic
370C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#556C2
socleC22×C31C22×D4
prime radical#14521695C2
hypercenter#556D31
hypocenterC31#3311
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #14521411C2semidirect
3. #14521390C2semidirect
4. #14521373C2semidirect
5. #14521336C2semidirect
6. #14521305C2semidirect
7. #14521280C2semidirect
8. #14521271C2semidirect
9. #14521209C2semidirect
10. #14521199C2semidirect
11. #14521147C2semidirect
12. #14521121C2semidirect
13. #14521053C2semidirect
14. #14521043C2semidirect
15. C31×#556C2semidirect
18. C31×C4⋊Q8C22semidirect
23. C31×C22⋊Q8C22semidirect
25. C4×Q8×C31C22semidirect
28. C31×#117C22semidirect
29. C2×#92482C22semidirect
36. C22×C31⋊C8C22semidirect
42. #11758423C22semidirect
44. C22⋊(C31⋊C8)C22semidirect
45. #11758398C22semidirect
46. #11758398C22semidirect
47. #11758398C22semidirect
54. #11758396C22semidirect
55. #11758396C22semidirect
56. #11758395C22semidirect
57. C4⋊(C31⋊C8)C22semidirect
58. #11758391C22semidirect
179. #556D31semidirect
208. C4⋊Q8C2×D31semidirect
216. C22⋊Q8C2×D31semidirect
221. C4×Q8C2×D31semidirect
225. #117C2×D31semidirect
244. C31#3311semidirect

Maximal subgroups

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

subgroupindex
#145216952
#145214112
#145213902
#145213732
#145213362
#145213052
#145212802
#145212712
#145212092
#145211992
#145211472
#145211212
#145210532
#145210432
C31×#5562
#331131