Finite Group Databasebeta

← #16857702

Subgroups — #16857702

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#14521703C211yesmaximal, characteristic, complemented
2#14521408C211yesmaximal, characteristic, complemented
3#14521382C211yesmaximal, characteristic, complemented
4#14521360C211yesmaximal, characteristic, complemented
5#14521344C211yesmaximal, characteristic, complemented
6#14521297C211yesmaximal, characteristic, complemented
7#14521283C211yesmaximal, characteristic, complemented
8#14521261C211yesmaximal, characteristic, complemented
9#14521194C211yesmaximal, characteristic, complemented
10#14521188C211yesmaximal, characteristic, complemented
11#14521147C211yesmaximal, characteristic, complemented
12#14521121C211yesmaximal, characteristic, complemented
13#14521050C211yesmaximal, characteristic, complemented
14C2×#11758423C211yesmaximal, characteristic, complemented
15C31×#548C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic
18C2×#92497—21no—
19C31×#124—21no—
20C31×#124C2211yescharacteristic
21C31×C22⋊Q8—21no—
22C31×C22⋊Q8C2211yescharacteristic
23C31×#121—21no—
24C31×#121C2211yescharacteristic
25C4×Q8×C31C2211yescharacteristic
26C4×D4×C31—21no—
27C4×D4×C31C2211yescharacteristic
28C2×#92481C2211yescharacteristic
29C2×#92481C2211yescharacteristic
30C2×#92479C2211yescharacteristic
31#11758507C2211yescharacteristic
32#11758507C2211yescharacteristic
33C2×C4⋊Dic31C2211yescharacteristic
34C2×C4⋊Dic31C2211yescharacteristic
35C2×#92474C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37#11758474—21no—
38#11758463—21no—
39#11758463—21no—
40C4×D124C2211yescharacteristic
41#11758446C2211yescharacteristic
42#11758423C2211yescharacteristic
43#11758423C2211yescharacteristic
44#11758423C2211yescharacteristic
45#11758423C2211yescharacteristic
46#11758423C2211yescharacteristic
47#11758423C2211yescharacteristic
48#11758420C2211yescharacteristic
49#11758420C2211yescharacteristic
50#11758420C2211yescharacteristic
51C22⋊(C31⋊C8)C2211yescharacteristic
52C22⋊(C31⋊C8)C2211yescharacteristic
53#11758397C2211yescharacteristic
54#11758395C2211yescharacteristic
55#11758395C2211yescharacteristic
56#11758395C2211yescharacteristic
57C4⋊(C31⋊C8)C2211yescharacteristic
58#11758391C2211yescharacteristic
59#41×C31—22no—
60#41×C31—41no—
61C2×Q8×C31—21no—
62C2×Q8×C31—22no—
63C2×Q8×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×Q8×C31C2311yescharacteristic
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31D411yescharacteristic
69C2×D4×C31C2311yescharacteristic
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31C2311yescharacteristic
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C22×Dic31—21no—
77C22×Dic31—21no—
78#92497—44no—
79C2×D124C2311yescharacteristic
80C2×C4×D31—21no—
81C2×Dic62—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31D411yescharacteristic
85C4⋊C4×C31C2311yescharacteristic
86C22⋊C4×C31—21no—
87C22⋊C4×C31—21no—
88C22⋊C4×C31—21no—
89C22⋊C4×C31—21no—
90C22⋊C4×C31D411yescharacteristic
91C22⋊C4×C31D411yescharacteristic
92C42×C31—21no—
93C42×C31C2311yescharacteristic
94#92481—22no—
95#92481—41no—
96#92479—41no—
97(C2×D31)⋊C4—21no—
98(C2×D31)⋊C4—21no—
99(C2×D31)⋊C4—21no—
100C4⋊Dic31—21no—
101C4⋊Dic31—21no—
102C4⋊Dic31—21no—
103C4⋊Dic31C2311yescharacteristic
104C4⋊Dic31C2311yescharacteristic
105C4⋊Dic31C2311yescharacteristic
106#92474—41no—
107C2×C31⋊C8—41no—
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C2×C31⋊C8C2311yescharacteristic
111C2×C31⋊C8C2311yescharacteristic
112C23×C31—21no—
113C23×C31C2×D411yescharacteristic
114C23×C31C2×D411yescharacteristic
115C22×D31—21no—
116Q8×C31—21no—
117Q8×C31—21no—
118Q8×C31—21no—
119Q8×C31—42no—
120Q8×C31#4112yescomplemented
121D4×C31—22no—
122D4×C31—21no—
123D4×C31—21no—
124D4×C31—42no—
125D4×C31—42no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31C2×D411yescharacteristic
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2411yescharacteristic
140C22⋊D31—42no—
141C22⋊D31—42no—
142C2×Dic31—21no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145C2×Dic31—42no—
146C2×Dic31—42no—
147D124—21no—
148D124—21no—
149C4×D31—42no—
150Dic62—42no—
151C31⋊C8—21no—
152C31⋊C8—21no—
153C31⋊C8—21no—
154C31⋊C8—21no—
155#3308—311nomaximal, Sylow
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31AGL(1,ℤ/8)12yes—
165C22×C31C22×D411yescharacteristic
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—41no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—22no—
172C4×C31—21no—
173C4×C31—21no—
174C4×C31—41no—
175C4×C31—41no—
176C4×C31C2×#4111yescharacteristic
177C4×C31H(5,𝔽2)11yescharacteristic
178Dic31—41no—
179Dic31—41no—
180Dic31—41no—
181#548D3111yescharacteristic, complemented
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#435—311no—
194#417—311no—
195C2×#103—311no—
196C2×C31—22no—
197C2×C31—41no—
198C2×C31—81no—
199C2×C31#54611yescharacteristic
200C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
201C2×C31#57811yescharacteristic
202D31—81no—
203C2×#41—621no—
204C2×#41C2×D3111yescharacteristic
205C22×Q8C2×D3111yescharacteristic
206C2×SD16—311no—
207C2×SD16—311no—
208C2×D8—311no—
209C2×M16—311no—
210C22×C8—311no—
211C4⋊Q8—311no—
212#124—21no—
213#124C2×D3111yescharacteristic
214#123—621no—
215#123—621no—
216C22⋊Q8—21no—
217C22⋊Q8—621no—
218C22⋊Q8C2×D3111yescharacteristic
219#121—21no—
220#121—311no—
221#121—311no—
222#121C2×D3111yescharacteristic
223C4×Q8C2×D3111yescharacteristic
224C4×D4—21no—
225C4×D4—311no—
226C4×D4C2×D3111yescharacteristic
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#103—311no—
238#103—311no—
239#102—311no—
240#102—311no—
241#102—311no—
242#102—311no—
243C22⋊C8—311no—
244C22⋊C8—311no—
245#97—311no—
246C31#330811yescharacteristic, complemented, Sylow
247#41—22no—
248#41—41no—
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—21no—
258C2×D4—311no—
259C2×D4—621no—
260C2×D4—621no—
261C2×D4C22⋊D3111yescharacteristic
262C2×D4C22×D3111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4—621no—
268C22×C4C22×D3111yescharacteristic
269C22×C4C22×D3111yescharacteristic
270SD16—622no—
271SD16—1241no—
272D8—1241no—
273M16—1241no—
274C2×C8—311no—
275C2×C8—311no—
276C2×C8—311no—
277C2×C8—311no—
278C2×C8—1241no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—311no—
282C4⋊C4—311no—
283C4⋊C4—311no—
284C4⋊C4—621no—
285C4⋊C4—621no—
286C4⋊C4—621no—
287C4⋊C4C22⋊D3111yescharacteristic
288C4⋊C4C22×D3111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4C22⋊D3111yescharacteristic
297C22⋊C4C22⋊D3111yescharacteristic
298C42—21no—
299C42C22×D3111yescharacteristic
300C23—21no—
301C23—621no—
302C23C2×C22⋊D3111yescharacteristic
303C23C2×C22⋊D3111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—1242no—
309Q8#9249712yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—621no—
316D4—621no—
317D4—1242no—
318D4—1242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—1242no—
332C2×C4—1242no—
333C2×C4—1242no—
334C2×C4C2×C22⋊D3111yescharacteristic
335C2×C4C2×C22⋊D3111yescharacteristic
336C2×C4C2×C22⋊D3111yescharacteristic
337C2×C4C2×C22⋊D3111yescharacteristic
338C2×C4C23×D3111yescharacteristic
339C8—621no—
340C8—621no—
341C8—621no—
342C8—621no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—1241no—
352C22—1241no—
353C22—1241no—
354C22#1175852612yes—
355C22C22×C22⋊D3111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—1241no—
364C4—1241no—
365C4—1241no—
366C4C2×#9249711yescharacteristic
367C4#1175856111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—2481no—
372C2#1452169311yescharacteristic
373C2C2×#1175852611yescharacteristic
374C2#1452186411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#548C2
socleC22×C31C22×D4
hypercenter#548D31
hypocenterC31#3308
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #14521703C2semidirect
2. #14521408C2semidirect
3. #14521382C2semidirect
4. #14521360C2semidirect
5. #14521344C2semidirect
6. #14521297C2semidirect
7. #14521283C2semidirect
8. #14521261C2semidirect
9. #14521194C2semidirect
10. #14521188C2semidirect
11. #14521147C2semidirect
12. #14521121C2semidirect
13. #14521050C2semidirect
14. C2×#11758423C2semidirect
15. C31×#548C2semidirect
120. Q8×C31#41semidirect
181. #548D31semidirect
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
#145217032
#145214082
#145213822
#145213602
#145213442
#145212972
#145212832
#145212612
#145211942
#145211882
#145211472
#145211212
#145210502
C2×#117584232
C31×#5482
#330831