Finite Group Databasebeta

← #16857706

Subgroups — #16857706

Order 3968.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521710C211yesmaximal, characteristic, complemented
2#14521699C211yesmaximal, characteristic, complemented
3#14521410C211yesmaximal, characteristic, complemented
4#14521383C211yesmaximal, characteristic, complemented
5#14521363C211yesmaximal, characteristic, complemented
6#14521342C211yesmaximal, characteristic, complemented
7#14521292C211yesmaximal, characteristic, complemented
8#14521283C211yesmaximal, characteristic, complemented
9#14521266C211yesmaximal, characteristic, complemented
10#14521193C211yesmaximal, characteristic, complemented
11#14521150C211yesmaximal, characteristic, complemented
12#14521117C211yesmaximal, characteristic, complemented
13#14521049C211yesmaximal, characteristic, complemented
14C2×#11758423C211yesmaximal, characteristic, complemented
15C31×#448C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C2×#92499C2211yescharacteristic
18C2×#92497—21no—
19C2×SD16×C31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C31×#121C2211yescharacteristic
22#11758524—21no—
23#11758524C2211yescharacteristic
24#11758523—21no—
25#11758523C2211yescharacteristic
26Q8×Dic31C2211yescharacteristic
27#11758515C2211yescharacteristic
28#11758512—21no—
29D4×Dic31C2211yescharacteristic
30C2×#92479C2211yescharacteristic
31#11758507—21no—
32#11758507C2211yescharacteristic
33C4×C22⋊D31—21no—
34C4×C22⋊D31C2211yescharacteristic
35C2×C4⋊Dic31C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37C2×#92469C2211yescharacteristic
38C4⋊C4×D31C2211yescharacteristic
39#11758465C2211yescharacteristic
40#11758460—21no—
41#11758455—21no—
42C31×#103C2211yescharacteristic
43C31×#103C2211yescharacteristic
44C31×C22⋊C8C2211yescharacteristic
45#11758423C2211yescharacteristic
46#11758423C2211yescharacteristic
47#11758423C2211yescharacteristic
48#11758423C2211yescharacteristic
49#11758420C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758409C2211yescharacteristic
52(C2×D31)⋊C8C2211yescharacteristic
53#11758406C2211yescharacteristic
54#11758406C2211yescharacteristic
55#11758403C2211yescharacteristic
56Dic31⋊C8C2211yescharacteristic
57#11758397C2211yescharacteristic
58#11758395C2211yescharacteristic
59C2×Q8×C31—21no—
60C2×Q8×C31—22no—
61C2×Q8×C31—41no—
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—21no—
65C2×D4×C31C2311yescharacteristic
66C22×C4×C31—21no—
67C22×C4×C31C2311yescharacteristic
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31D411yescharacteristic
71C22×Dic31—21no—
72C22×Dic31—21no—
73#92499—22no—
74#92499—41no—
75#92497—44no—
76C2×D124—21no—
77C2×D124C2311yescharacteristic
78C2×C4×D31—21no—
79C2×C4×D31—21no—
80C2×C4×D31C2311yescharacteristic
81C2×Dic62—21no—
82SD16×C31—22no—
83SD16×C31—41no—
84C2×C8×C31C2311yescharacteristic
85C2×C8×C31C2311yescharacteristic
86C4⋊C4×C31C2311yescharacteristic
87C22⋊C4×C31—21no—
88C22⋊Dic31—21no—
89C22⋊Dic31—21no—
90C22⋊Dic31D411yescharacteristic
91#92479—41no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4D411yescharacteristic
97C4⋊Dic31—21no—
98C4⋊Dic31—21no—
99C4⋊Dic31C2311yescharacteristic
100C4⋊Dic31C2311yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105Dic31⋊C4D411yescharacteristic
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8—41no—
109C2×C31⋊C8C2311yescharacteristic
110C2×C31⋊C8C2311yescharacteristic
111#92469—41no—
112C23×C31—21no—
113C23×C31C2×D411yescharacteristic
114C22×D31—21no—
115C22×D31C2×D411yescharacteristic
116Q8×C31—21no—
117Q8×C31—21no—
118Q8×C31—21no—
119Q8×C31—42no—
120Q8×C31#4112yescomplemented
121D4×C31—21no—
122D4×C31—21no—
123D4×C31—42no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—42no—
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2411yescharacteristic
132C22⋊D31—22no—
133C22⋊D31—42no—
134C22⋊D31—42no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31—42no—
139C2×Dic31—42no—
140C2×Dic31C2×D411yescharacteristic
141C2×Dic31C2×D411yescharacteristic
142C2×Dic31C2×D411yescharacteristic
143D124—21no—
144D124—21no—
145D124—42no—
146C4×D31—21no—
147C4×D31—21no—
148C4×D31—42no—
149C4×D31—42no—
150Dic62—42no—
151C8×C31—21no—
152C8×C31—21no—
153C31⋊C8—21no—
154C31⋊C8—21no—
155#3308—311nomaximal, Sylow
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31AGL(1,ℤ/8)12yes—
162C22×C31C22×D411yescharacteristic
163C2×D31—21no—
164C2×D31—21no—
165C2×D31—21no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—41no—
169C4×C31—22no—
170C4×C31—21no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31C2×#4111yescharacteristic
174C4×C31H(5,𝔽2)11yescharacteristic
175Dic31—21no—
176Dic31—21no—
177Dic31—21no—
178Dic31—41no—
179Dic31—41no—
180Dic31—41no—
181#548—311no—
182#547—311no—
183#489—311no—
184#484—311no—
185#480—311no—
186#477—311no—
187#468—311no—
188#466—311no—
189#463—311no—
190#449—311no—
191#448D3111yescharacteristic, complemented
192#440—311no—
193#435—311no—
194#417—311no—
195C2×#103—311no—
196C2×C31—22no—
197C2×C31—81no—
198C2×C31#54611yescharacteristic
199C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C31#57811yescharacteristic
201D31—41no—
202D31—81no—
203C2×#41—311no—
204C2×#41—621no—
205C22×Q8C2×D3111yescharacteristic
206C2×SD16C2×D3111yescharacteristic
207C2×SD16C2×D3111yescharacteristic
208C2×D8—311no—
209C2×M16—311no—
210C22×C8—311no—
211C4⋊Q8—311no—
212#124—311no—
213#124—621no—
214#123—621no—
215#123—621no—
216C22⋊Q8—311no—
217C22⋊Q8—621no—
218C22⋊Q8—621no—
219#121—311no—
220#121—311no—
221#121—621no—
222#121C2×D3111yescharacteristic
223C4×Q8—311no—
224C4×D4—311no—
225C4×D4—311no—
226C4×D4—621no—
227C2×C4⋊C4—311no—
228C2×C4⋊C4—311no—
229#107—311no—
230#107—311no—
231#107—311no—
232C4⋊C8—311no—
233#103—311no—
234#103—311no—
235#103—311no—
236#103—311no—
237#103C2×D3111yescharacteristic
238#103C2×D3111yescharacteristic
239#102—311no—
240#102—311no—
241#102—311no—
242#102—311no—
243C22⋊C8—311no—
244C22⋊C8C2×D3111yescharacteristic
245#97—311no—
246C31#330811yescharacteristic, complemented, Sylow
247#41—622no—
248#41—1241no—
249#41—1244no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—621no—
254C2×Q8C22×D3111yescharacteristic
255C2×Q8C22×D3111yescharacteristic
256C2×D4—21no—
257C2×D4—311no—
258C2×D4—311no—
259C2×D4—621no—
260C2×D4—621no—
261C2×D4—621no—
262C2×D4C22×D3111yescharacteristic
263C22×C4—21no—
264C22×C4—311no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4—621no—
268C22×C4—621no—
269C22×C4C22×D3111yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—1241no—
273M16—1241no—
274C2×C8—311no—
275C2×C8—311no—
276C2×C8—1241no—
277C2×C8C22×D3111yescharacteristic
278C2×C8C22×D3111yescharacteristic
279C4⋊C4—311no—
280C4⋊C4—311no—
281C4⋊C4—311no—
282C4⋊C4—311no—
283C4⋊C4—621no—
284C4⋊C4—621no—
285C4⋊C4—621no—
286C4⋊C4—621no—
287C4⋊C4—621no—
288C4⋊C4C22×D3111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—311no—
291C22⋊C4—311no—
292C22⋊C4—621no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4—621no—
297C22⋊C4—621no—
298C42—311no—
299C42—621no—
300C23—21no—
301C23—311no—
302C23—621no—
303C23D4×D3111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—1242no—
309Q8#9249712yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—622no—
314D4—621no—
315D4—621no—
316D4—1242no—
317D4—1242no—
318D4—1242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—311no—
326C2×C4—311no—
327C2×C4—311no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—621no—
332C2×C4—621no—
333C2×C4—1242no—
334C2×C4—1242no—
335C2×C4—1242no—
336C2×C4—1242no—
337C2×C4D4×D3111yescharacteristic
338C2×C4C23×D3111yescharacteristic
339C8—21no—
340C8—21no—
341C8—621no—
342C8—621no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—621no—
349C22—621no—
350C22—621no—
351C22—1241no—
352C22—1241no—
353C22—1241no—
354C22#1175849112yes—
355C22C2×D4×D3111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—621no—
361C4—621no—
362C4—621no—
363C4—1241no—
364C4—1241no—
365C4—1241no—
366C4C2×#9249711yescharacteristic
367C4#1175856111yescharacteristic
368C2—22no—
369C2—81no—
370C2—1241no—
371C2—2481no—
372C2#1452168611yescharacteristic
373C2C2×#1175849111yescharacteristic
374C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
hypercenter#448D31
hypocenterC31#3308
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #14521710C2semidirect
2. #14521699C2semidirect
3. #14521410C2semidirect
4. #14521383C2semidirect
5. #14521363C2semidirect
6. #14521342C2semidirect
7. #14521292C2semidirect
8. #14521283C2semidirect
9. #14521266C2semidirect
10. #14521193C2semidirect
11. #14521150C2semidirect
12. #14521117C2semidirect
13. #14521049C2semidirect
14. C2×#11758423C2semidirect
15. C31×#448C2semidirect
120. Q8×C31#41semidirect
191. #448D31semidirect
246. C31#3308semidirect
309. Q8#92497semidirect

Maximal subgroups

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

subgroupindex
#145217102
#145216992
#145214102
#145213832
#145213632
#145213422
#145212922
#145212832
#145212662
#145211932
#145211502
#145211172
#145210492
C2×#117584232
C31×#4482
#330831