Finite Group Databasebeta

← #16857580

Subgroups — #16857580

Order 3968.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521757C211yesmaximal, characteristic
2Q8×C22⋊D31C211yesmaximal, characteristic
3#14521398C211yesmaximal, characteristic, complemented
4#14521385C211yesmaximal, characteristic, complemented
5#14521383C211yesmaximal, characteristic, complemented
6#14521342C211yesmaximal, characteristic, complemented
7#14521288C211yesmaximal, characteristic, complemented
8#14521287C211yesmaximal, characteristic, complemented
9#14521257C211yesmaximal, characteristic, complemented
10#14521203C211yesmaximal, characteristic, complemented
11Q16×Dic31C211yesmaximal, characteristic, complemented
12(C22⋊D31)⋊C8C211yesmaximal, characteristic
13C2×#11758423C211yesmaximal, characteristic, complemented
14#103×D31C211yesmaximal, characteristic, complemented
15C31×#451C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic
19C2×Q16×C31C2211yescharacteristic
20C31×C22⋊Q8C2211yescharacteristic
21#11758523—21no—
22#11758523C2211yescharacteristic
23#11758523C2211yescharacteristic
24Q8×Dic31C2211yescharacteristic
25Q8×Dic31C2211yescharacteristic
26#11758521—21no—
27#11758521C2211yescharacteristic
28C2×#92482C2211yescharacteristic
29C4×C22⋊D31—21no—
30C4×C22⋊D31C2211yescharacteristic
31C2×C4⋊Dic31C2211yescharacteristic
32#11758501—21no—
33#11758501C2211yescharacteristic
34C22×C31⋊C8C2211yescharacteristic
35C2×C8×D31C2211yescharacteristic
36#11758474—21no—
37C4⋊C4×D31C2211yescharacteristic
38#11758466—21no—
39#11758465C2211yescharacteristic
40(C22⋊D31)⋊C4—21no—
41#11758455—21no—
42C31×#103C2211yescharacteristic
43C31×#103C2211yescharacteristic
44C31×C22⋊C8C2211yescharacteristic
45#11758423C2211yescharacteristic
46#11758423C2211yescharacteristic
47#11758423C2211yescharacteristic
48#11758423C2211yescharacteristic
49#11758423C2211yescharacteristic
50(C2×D31)⋊C8C2211yescharacteristic
51#11758406C2211yescharacteristic
52#11758406C2211yescharacteristic
53#11758404C2211yescharacteristic
54#11758404C2211yescharacteristic
55Dic31⋊C8C2211yescharacteristic
56C8×Dic31C2211yescharacteristic
57#11758398C2211yescharacteristic
58#11758395C2211yescharacteristic
59C2×Q8×C31—21no—
60C2×Q8×C31—22no—
61C2×Q8×C31—41no—
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×Q8×C31C2311yescharacteristic
65C22×C4×C31—21no—
66C22×C4×C31C2311yescharacteristic
67C2×C22⋊D31D411yescharacteristic
68C22×Dic31—21no—
69Q8×D31—22no—
70Q8×D31—41no—
71C2×C4×D31—21no—
72C2×C4×D31—21no—
73C2×C4×D31C2311yescharacteristic
74C2×Dic62—21no—
75C2×Dic62—21no—
76C2×Dic62C2311yescharacteristic
77Q16×C31—22no—
78Q16×C31—41no—
79C2×C8×C31C2311yescharacteristic
80C2×C8×C31C2311yescharacteristic
81C4⋊C4×C31—21no—
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85C22⋊Dic31D411yescharacteristic
86#92482—41no—
87(C2×D31)⋊C4—21no—
88(C2×D31)⋊C4—21no—
89(C2×D31)⋊C4D411yescharacteristic
90C4⋊Dic31—21no—
91C4⋊Dic31—21no—
92C4⋊Dic31—21no—
93C4⋊Dic31C2311yescharacteristic
94C4⋊Dic31C2311yescharacteristic
95C4⋊Dic31C2311yescharacteristic
96Dic31⋊C4—21no—
97Dic31⋊C4—21no—
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4—21no—
101Dic31⋊C4—21no—
102Dic31⋊C4—21no—
103Dic31⋊C4D411yescharacteristic
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31C2311yescharacteristic
107C2×C31⋊C8—41no—
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C8×D31—41no—
111C23×C31C2×D411yescharacteristic
112C22×D31C2×D411yescharacteristic
113Q8×C31—21no—
114Q8×C31—21no—
115Q8×C31—21no—
116Q8×C31—21no—
117Q8×C31—21no—
118Q8×C31—42no—
119Q8×C31#4112yes—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—42no—
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2411yescharacteristic
129C22⋊D31Q1614yescomplemented
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—81no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137C2×Dic31C2×D411yescharacteristic
138C4×D31—21no—
139C4×D31—21no—
140C4×D31—42no—
141C4×D31—81no—
142Dic62—21no—
143Dic62—21no—
144Dic62—42no—
145Dic62—81no—
146C8×C31—21no—
147C8×C31—21no—
148C31⋊C8—21no—
149C31⋊C8—21no—
150#3299—311nomaximal, Sylow
151C22×C31—21no—
152C22×C31—21no—
153C22×C31C2×Q1612yes—
154C22×C31C22×D411yescharacteristic
155C2×D31—21no—
156C2×D31—21no—
157C2×D31C2×Q1612yes—
158C4×C31—22no—
159C4×C31—21no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31—41no—
163C4×C31C2×#4111yescharacteristic
164C4×C31H(5,𝔽2)11yescharacteristic
165Dic31—21no—
166Dic31—21no—
167Dic31—41no—
168Dic31—41no—
169Dic31—41no—
170Dic31—41no—
171Dic31C2×Q1612yes—
172#554—311no—
173D4×Q8—311no—
174#487—311no—
175#484—311no—
176#484—311no—
177#477—311no—
178#467—311no—
179#467—311no—
180#462—311no—
181#451—311no—
182#451D3111yescharacteristic, complemented
183C4×Q16—311no—
184C8×D4—311no—
185C2×#103—311no—
186C2×#103—311no—
187C2×C31—22no—
188C2×C31#54611yescharacteristic
189C2×C31C22×Q1611yescharacteristic
190C2×C31#57711yescharacteristic
191D31—22no—
192C22×Q8—311no—
193C22×Q8C2×D3111yescharacteristic
194C2×Q16—311no—
195C2×Q16C2×D3111yescharacteristic
196C2×Q16C2×D3111yescharacteristic
197C22×C8—311no—
198C22×C8—311no—
199C4⋊Q8—311no—
200C4⋊Q8—311no—
201C4⋊Q8—621no—
202#125—621no—
203C22⋊Q8—311no—
204C22⋊Q8—311no—
205C22⋊Q8—311no—
206C22⋊Q8—621no—
207C22⋊Q8—621no—
208C22⋊Q8—621no—
209C22⋊Q8—621no—
210C22⋊Q8C2×D3111yescharacteristic
211C4×Q8—311no—
212C4×Q8—311no—
213C4×D4—311no—
214C4×D4—621no—
215C4×D4—621no—
216C2×C4⋊C4—311no—
217C2×C4⋊C4—311no—
218#107—311no—
219#107—311no—
220#107—311no—
221C4⋊C8—311no—
222#103—311no—
223#103—311no—
224#103—311no—
225#103—311no—
226#103—311no—
227#103—311no—
228#103—311no—
229#103—311no—
230#103C2×D3111yescharacteristic
231#103C2×D3111yescharacteristic
232C22⋊C8—311no—
233C22⋊C8C2×D3111yescharacteristic
234C4×C8—311no—
235C31#329911yescharacteristic, complemented, Sylow
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—311no—
240C2×Q8—621no—
241C2×Q8—621no—
242C2×Q8—622no—
243C2×Q8—1241no—
244C2×Q8C22×D3111yescharacteristic
245C2×Q8C22×D3111yescharacteristic
246C2×Q8C22×D3111yescharacteristic
247C2×D4—311no—
248C22×C4—21no—
249C22×C4—311no—
250C22×C4—621no—
251C22×C4—621no—
252C22×C4—621no—
253C22×C4C22×D3111yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—1241no—
257C2×C8—311no—
258C2×C8—311no—
259C2×C8—1241no—
260C2×C8—1241no—
261C2×C8C22×D3111yescharacteristic
262C2×C8C22×D3111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—311no—
265C4⋊C4—311no—
266C4⋊C4—311no—
267C4⋊C4—311no—
268C4⋊C4—621no—
269C4⋊C4—621no—
270C4⋊C4—621no—
271C4⋊C4—621no—
272C4⋊C4—621no—
273C4⋊C4—621no—
274C4⋊C4—621no—
275C4⋊C4—621no—
276C4⋊C4—621no—
277C4⋊C4—621no—
278C4⋊C4C22×D3111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—311no—
281C22⋊C4—311no—
282C22⋊C4—621no—
283C22⋊C4—621no—
284C22⋊C4—621no—
285C42—311no—
286C42—621no—
287C42—621no—
288C23—311no—
289C23D4×D3111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—621no—
297Q8—621no—
298Q8—1242no—
299Q8—2481no—
300Q8#9249712yes—
301D4—314no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—311no—
310C2×C4—311no—
311C2×C4—311no—
312C2×C4—621no—
313C2×C4—621no—
314C2×C4—621no—
315C2×C4—621no—
316C2×C4—621no—
317C2×C4—621no—
318C2×C4—1242no—
319C2×C4—2481no—
320C2×C4—2481no—
321C2×C4D4×D3111yescharacteristic
322C2×C4C23×D3111yescharacteristic
323C8—21no—
324C8—21no—
325C8—621no—
326C8—621no—
327C22—21no—
328C22—21no—
329C22—312no—
330C22—621no—
331C22—621no—
332C22Q16×D3112yes—
333C22C2×D4×D3111yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—312no—
340C4—621no—
341C4—621no—
342C4—1241no—
343C4—1241no—
344C4—1241no—
345C4—1241no—
346C4C2×#9249711yescharacteristic
347C4#1175856111yescharacteristic
348C2—22no—
349C2—622no—
350C2#1452168611yescharacteristic
351C2C2×Q16×D3111yescharacteristic
352C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radicalC2×C22⋊D31D4
hypercenter#451D31
hypocenterC31#3299
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #14521398C2semidirect
4. #14521385C2semidirect
5. #14521383C2semidirect
6. #14521342C2semidirect
7. #14521288C2semidirect
8. #14521287C2semidirect
9. #14521257C2semidirect
10. #14521203C2semidirect
11. Q16×Dic31C2semidirect
13. C2×#11758423C2semidirect
14. #103×D31C2semidirect
15. C31×#451C2semidirect
129. C22⋊D31Q16semidirect
182. #451D31semidirect
235. C31#3299semidirect

Maximal subgroups

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

subgroupindex
#145217572
Q8×C22⋊D312
#145213982
#145213852
#145213832
#145213422
#145212882
#145212872
#145212572
#145212032
Q16×Dic312
(C22⋊D31)⋊C82
C2×#117584232
#103×D312
C31×#4512
#329931