Finite Group Databasebeta

← #16857704

Subgroups — #16857704

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#14521708C211yesmaximal, characteristic, complemented
2#14521703C211yesmaximal, characteristic, complemented
3#14521409C211yesmaximal, characteristic, complemented
4#14521385C211yesmaximal, characteristic, complemented
5#14521363C211yesmaximal, characteristic, complemented
6#14521345C211yesmaximal, characteristic, complemented
7#14521295C211yesmaximal, characteristic, complemented
8#14521285C211yesmaximal, characteristic, complemented
9#14521265C211yesmaximal, characteristic, complemented
10#14521196C211yesmaximal, characteristic, complemented
11#14521190C211yesmaximal, characteristic, complemented
12#14521151C211yesmaximal, characteristic, complemented
13#14521051C211yesmaximal, characteristic, complemented
14C2×#11758404C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×#92497—21no—
17C2×#92495C2211yescharacteristic
18C22×Dic62C2211yescharacteristic
19C2×M16×C31C2211yescharacteristic
20C22×C8×C31C2211yescharacteristic
21C4×D4×C31C2211yescharacteristic
22#11758516—21no—
23#11758507C2211yescharacteristic
24#11758507C2211yescharacteristic
25#11758507C2211yescharacteristic
26C2×C4⋊Dic31C2211yescharacteristic
27C2×C4⋊Dic31C2211yescharacteristic
28#11758501C2211yescharacteristic
29C2×D248C2211yescharacteristic
30C2×#92470C2211yescharacteristic
31C2×#92470C2211yescharacteristic
32#11758474—21no—
33#11758473—21no—
34#11758463—21no—
35#11758463—21no—
36#11758462—21no—
37(C22⋊D31)⋊C4—21no—
38#11758452C2211yescharacteristic
39C4×D124C2211yescharacteristic
40#11758446C2211yescharacteristic
41C4×Dic62C2211yescharacteristic
42C31×C4⋊C8C2211yescharacteristic
43C31×C22⋊C8C2211yescharacteristic
44C31×C22⋊C8C2211yescharacteristic
45C31×#97C2211yescharacteristic
46#11758409C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758409C2211yescharacteristic
50#11758406C2211yescharacteristic
51#11758406C2211yescharacteristic
52#11758406C2211yescharacteristic
53#11758404C2211yescharacteristic
54#11758404C2211yescharacteristic
55#11758404C2211yescharacteristic
56#11758404C2211yescharacteristic
57#11758404C2211yescharacteristic
58#11758404C2211yescharacteristic
59C2×D4×C31D411yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C2×C22⋊D31—21no—
63C2×C22⋊D31—21no—
64C2×C22⋊D31—21no—
65C2×C22⋊D31—21no—
66C22×Dic31—21no—
67C22×Dic31—21no—
68C22×Dic31—21no—
69#92497—44no—
70#92495—22no—
71#92495—41no—
72C2×D124C2311yescharacteristic
73C2×D124C2311yescharacteristic
74C2×C4×D31—21no—
75C2×C4×D31—21no—
76C2×Dic62—22no—
77C2×Dic62—21no—
78C2×Dic62—21no—
79C2×Dic62—41no—
80C2×Dic62C2311yescharacteristic
81C2×Dic62C2311yescharacteristic
82M16×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31C2311yescharacteristic
85C2×C8×C31C2311yescharacteristic
86C2×C8×C31C2311yescharacteristic
87C2×C8×C31C2311yescharacteristic
88C4⋊C4×C31D411yescharacteristic
89C22⋊C4×C31D411yescharacteristic
90C22⋊C4×C31D411yescharacteristic
91C42×C31C2311yescharacteristic
92C22⋊Dic31—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4—21no—
97(C2×D31)⋊C4—21no—
98(C2×D31)⋊C4—21no—
99C4⋊Dic31—21no—
100C4⋊Dic31—21no—
101C4⋊Dic31—21no—
102C4⋊Dic31C2311yescharacteristic
103C4⋊Dic31C2311yescharacteristic
104C4⋊Dic31C2311yescharacteristic
105C4⋊Dic31C2311yescharacteristic
106Dic31⋊C4—21no—
107Dic31⋊C4—21no—
108C4×Dic31—21no—
109D248—41no—
110#92470—22no—
111#92470—41no—
112C23×C31C2×D411yescharacteristic
113C23×C31C2×D411yescharacteristic
114C22×D31—21no—
115C22×D31—21no—
116D4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31C2×D411yescharacteristic
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2411yescharacteristic
126C22⋊D31—42no—
127C22⋊D31—42no—
128C22⋊D31—42no—
129C22⋊D31—42no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—42no—
137C2×Dic31—42no—
138C2×Dic31—42no—
139D124—21no—
140D124—21no—
141D124—21no—
142D124—21no—
143C4×D31—42no—
144C4×D31—42no—
145Dic62—21no—
146Dic62—21no—
147Dic62—21no—
148Dic62—42no—
149Dic62—42no—
150Dic62#4112yescomplemented
151C8×C31—21no—
152C8×C31—21no—
153C8×C31—21no—
154C8×C31—21no—
155#3308—311nomaximal, Sylow
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—21no—
161C22×C31AGL(1,ℤ/8)12yes—
162C22×C31C22×D411yescharacteristic
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—41no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—21no—
172C4×C31—21no—
173C4×C31C2×#4111yescharacteristic
174C4×C31H(5,𝔽2)11yescharacteristic
175Dic31—22no—
176Dic31—41no—
177Dic31—41no—
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#448—311no—
192#440—311no—
193#435D3111yescharacteristic, complemented
194#417—311no—
195C2×#103—311no—
196C2×C31—22no—
197C2×C31—41no—
198C2×C31#54611yescharacteristic
199C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C31#57811yescharacteristic
201D31—81no—
202D31—81no—
203C2×#41—311no—
204C2×#41—621no—
205C22×Q8—311no—
206C2×SD16—311no—
207C2×SD16—311no—
208C2×D8—311no—
209C2×M16C2×D3111yescharacteristic
210C22×C8C2×D3111yescharacteristic
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—311no—
222#121—621no—
223C4×Q8—311no—
224C4×D4—311no—
225C4×D4—621no—
226C4×D4C2×D3111yescharacteristic
227C2×C4⋊C4—311no—
228C2×C4⋊C4—311no—
229#107—311no—
230#107—311no—
231#107—311no—
232C4⋊C8C2×D3111yescharacteristic
233#103—311no—
234#103—311no—
235#103—311no—
236#103—311no—
237#103—311no—
238#103—311no—
239#102—311no—
240#102—311no—
241#102—311no—
242#102—311no—
243C22⋊C8C2×D3111yescharacteristic
244C22⋊C8C2×D3111yescharacteristic
245#97C2×D3111yescharacteristic
246C31#330811yescharacteristic, complemented, Sylow
247#41—622no—
248#41—1241no—
249#41—1244no—
250C2×Q8—311no—
251C2×Q8—311no—
252C2×Q8—622no—
253C2×Q8—621no—
254C2×Q8—621no—
255C2×Q8—1241no—
256C2×D4—311no—
257C2×D4—311no—
258C2×D4—621no—
259C2×D4—621no—
260C2×D4—621no—
261C2×D4—621no—
262C2×D4D12411yescharacteristic
263C22×C4—621no—
264C22×C4—621no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4—621no—
268C22×C4C22×D3111yescharacteristic
269C22×C4C22×D3111yescharacteristic
270SD16—622no—
271SD16—1241no—
272D8—1241no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D3111yescharacteristic
276C2×C8C22×D3111yescharacteristic
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⋊C4D12411yescharacteristic
289C22⋊C4—621no—
290C22⋊C4—621no—
291C22⋊C4—621no—
292C22⋊C4—621no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4D12411yescharacteristic
297C22⋊C4D12411yescharacteristic
298C42—621no—
299C42C22×D3111yescharacteristic
300C23—621no—
301C23—621no—
302C23C2×D12411yescharacteristic
303C23C2×D12411yescharacteristic
304Q8—312no—
305Q8—621no—
306Q8—621no—
307Q8—621no—
308Q8—1242no—
309Q8—1242no—
310D4—22no—
311D4—621no—
312D4—621no—
313D4—621no—
314D4—621no—
315D4—1242no—
316D4—1242no—
317D4—1242no—
318D4—1242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—621no—
326C2×C4—621no—
327C2×C4—621no—
328C2×C4—621no—
329C2×C4—1242no—
330C2×C4—1242no—
331C2×C4—1242no—
332C2×C4—1242no—
333C2×C4—1242no—
334C2×C4C2×D12411yescharacteristic
335C2×C4C2×D12411yescharacteristic
336C2×C4C2×D12411yescharacteristic
337C2×C4C2×D12411yescharacteristic
338C2×C4C23×D3111yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—1241no—
349C22—1241no—
350C22—1241no—
351C22—1241no—
352C22—1241no—
353C22—1241no—
354C22#1175848512yes—
355C22C22×D12411yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—622no—
361C4—1241no—
362C4—1241no—
363C4—1241no—
364C4—1241no—
365C4—1241no—
366C4#1175856111yescharacteristic
367C4#41×D3111yescharacteristic
368C2—22no—
369C2—41no—
370C2—2481no—
371C2—2481no—
372C2#1452169411yescharacteristic
373C2C2×#1175848511yescharacteristic
374C2#1452186611yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
hypercenter#435D31
hypocenterC31#3308
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #14521708C2semidirect
2. #14521703C2semidirect
3. #14521409C2semidirect
4. #14521385C2semidirect
5. #14521363C2semidirect
6. #14521345C2semidirect
7. #14521295C2semidirect
8. #14521285C2semidirect
9. #14521265C2semidirect
10. #14521196C2semidirect
11. #14521190C2semidirect
12. #14521151C2semidirect
13. #14521051C2semidirect
14. C2×#11758404C2semidirect
15. C31×#435C2semidirect
150. Dic62#41semidirect
193. #435D31semidirect
246. C31#3308semidirect

Maximal subgroups

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

subgroupindex
#145217082
#145217032
#145214092
#145213852
#145213632
#145213452
#145212952
#145212852
#145212652
#145211962
#145211902
#145211512
#145210512
C2×#117584042
C31×#4352
#330831