Finite Group Databasebeta

← #16857666

Subgroups — #16857666

Order 3968.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521695C211yesmaximal, characteristic, complemented
2#14521411C211yesmaximal, characteristic, complemented
3#14521379C211yesmaximal, characteristic, complemented
4#14521367C211yesmaximal, characteristic, complemented
5#14521336C211yesmaximal, characteristic, complemented
6#14521300C211yesmaximal, characteristic, complemented
7#14521276C211yesmaximal, characteristic, complemented
8#14521264C211yesmaximal, characteristic, complemented
9#14521192C211yesmaximal, characteristic, complemented
10#14521182C211yesmaximal, characteristic, complemented
11#14521159C211yesmaximal, characteristic, complemented
12#14521121C211yesmaximal, characteristic, complemented
13#14521048C211yesmaximal, characteristic, complemented
14C2×#11758397C211yesmaximal, characteristic, complemented
15C31×#547C211yesmaximal, characteristic, complemented
16C2×#41×C31—21no—
17C2×#92495C2211yescharacteristic
18C22×D124C2211yescharacteristic
19C31×C4⋊Q8C2211yescharacteristic, complemented
20C31×#123—21no—
21C31×#123—21no—
22C31×C22⋊Q8—21no—
23C31×#121C2211yescharacteristic
24C31×#121C2211yescharacteristic, complemented
25C4×D4×C31C2211yescharacteristic
26C4×D4×C31C2211yescharacteristic, complemented
27C2×C4⋊C4×C31C2211yescharacteristic
28C2×C4⋊C4×C31C2211yescharacteristic, complemented
29#11758514—21no—
30C2×#92480C2211yescharacteristic, complemented
31C2×#92479C2211yescharacteristic
32C2×#92479C2211yescharacteristic
33#11758507C2211yescharacteristic
34#11758501C2211yescharacteristic
35C2×#92474C2211yescharacteristic, complemented
36C22×C31⋊C8C2211yescharacteristic
37#11758471—21no—
38#11758461—21no—
39C22⋊C4×D31—21no—
40#11758452C2211yescharacteristic
41C4×D124C2211yescharacteristic
42#11758420C2211yescharacteristic, 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#11758396C2211yescharacteristic, complemented
57C4⋊(C31⋊C8)C2211yescharacteristic, complemented
58#11758391C2211yescharacteristic, complemented
59C23×D31—21no—
60#41×C31—44no—
61C2×Q8×C31—21no—
62C2×D4×C31—21no—
63C2×D4×C31—21no—
64C2×D4×C31D411yescharacteristic
65C2×D4×C31C2311yescharacteristic
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31C2311yescharacteristic
70C22×C4×C31C2311yescharacteristic
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73#92495—22no—
74#92495—41no—
75C2×D124—22no—
76C2×D124—21no—
77C2×D124—41no—
78C2×D124C2311yescharacteristic
79C2×D124C2311yescharacteristic
80C2×C4×D31—21no—
81C2×C4×D31—21no—
82C2×Dic62C2311yescharacteristic
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31D411yescharacteristic
87C4⋊C4×C31C2311yescharacteristic
88C4⋊C4×C31C2311yescharacteristic
89C4⋊C4×C31C2311yescharacteristic
90C22⋊C4×C31—21no—
91C22⋊C4×C31—21no—
92C22⋊C4×C31—21no—
93C22⋊C4×C31D411yescharacteristic
94C22⋊C4×C31D411yescharacteristic
95C42×C31C2311yescharacteristic
96C22⋊Dic31—21no—
97#92480—41no—
98#92479—22no—
99#92479—41no—
100(C2×D31)⋊C4—21no—
101(C2×D31)⋊C4—21no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—21no—
104C4⋊Dic31C2311yescharacteristic
105Dic31⋊C4—21no—
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—
116C22×D31—21no—
117C22×D31—42no—
118C22×D31—42no—
119C22×D31—81no—
120Q8×C31—42no—
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×C31—42no—
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2411yescharacteristic
141C22⋊D31—42no—
142C22⋊D31—81no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145D124—21no—
146D124—21no—
147D124—21no—
148D124—42no—
149D124#4112yescomplemented
150C4×D31—42no—
151C4×D31—81no—
152Dic62—21no—
153Dic62—21no—
154C31⋊C8—21no—
155C31⋊C8—21no—
156C31⋊C8—21no—
157C31⋊C8—21no—
158#3306—311nomaximal, Sylow
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31AGL(1,ℤ/8)12yes—
168C22×C31C22×D411yescharacteristic
169C2×D31—22no—
170C2×D31—41no—
171C2×D31—41no—
172C2×D31—41no—
173C2×D31—41no—
174C2×D31—41no—
175C2×D31—84no—
176C4×C31—21no—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—41no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31C2×#4111yescharacteristic
184C4×C31H(5,𝔽2)11yescharacteristic
185Dic31—41no—
186Dic31—41no—
187#547D3111yescharacteristic, complemented
188#546—311no—
189#489—311no—
190#483—311no—
191#481—311no—
192#476—311no—
193#469—311no—
194#465—311no—
195#463—311no—
196#449—311no—
197#447—311no—
198#442—311no—
199#435—311no—
200#417—311no—
201C2×#102—311no—
202C2×C31—22no—
203C2×C31—41no—
204C2×C31—81no—
205C2×C31#54611yescharacteristic
206C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C31#57711yescharacteristic
208D31—42no—
209D31—81no—
210C2×#41—21no—
211C2×#41—311no—
212C22×D4—311no—
213C2×SD16—311no—
214C2×D8—311no—
215C2×D8—311no—
216C2×M16—311no—
217C22×C8—311no—
218C4⋊Q8C2×D3111yescharacteristic, complemented
219#124—311no—
220#123—21no—
221#123—21no—
222#123—621no—
223C22⋊Q8—21no—
224C22⋊Q8—311no—
225#121—311no—
226#121—621no—
227#121C2×D3111yescharacteristic
228#121C2×D3111yescharacteristic, complemented
229#120—621no—
230C4×D4—311no—
231C4×D4C2×D3111yescharacteristic
232C4×D4C2×D3111yescharacteristic, complemented
233C2×C4⋊C4C2×D3111yescharacteristic
234C2×C4⋊C4C2×D3111yescharacteristic, complemented
235C2×C22⋊C4—621no—
236#106—311no—
237#106—311no—
238#106—311no—
239C4⋊C8—311no—
240#103—311no—
241#103—311no—
242#103—311no—
243#102—311no—
244#102—311no—
245#102—311no—
246#102—311no—
247#102—311no—
248#102—311no—
249#102—311no—
250C22⋊C8—311no—
251C22⋊C8—311no—
252#97—311no—
253C31#330611yescharacteristic, complemented, Sylow
254C24—621no—
255#41—44no—
256#41—622no—
257#41—1241no—
258C2×Q8—21no—
259C2×Q8—311no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—311no—
263C2×D4—311no—
264C2×D4—622no—
265C2×D4—621no—
266C2×D4—621no—
267C2×D4—621no—
268C2×D4—1241no—
269C2×D4C22⋊D3111yescharacteristic
270C2×D4C22×D3111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—621no—
275C22×C4—621no—
276C22×C4C22×D3111yescharacteristic
277C22×C4C22×D3111yescharacteristic
278SD16—1241no—
279D8—622no—
280D8—1241no—
281M16—1241no—
282C2×C8—311no—
283C2×C8—311no—
284C2×C8—311no—
285C2×C8—311no—
286C2×C8—1241no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—311no—
291C4⋊C4—621no—
292C4⋊C4C22⋊D3111yescharacteristic
293C4⋊C4C22×D3111yescharacteristic
294C4⋊C4C22×D3111yescharacteristic
295C4⋊C4C22×D3111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—621no—
300C22⋊C4—621no—
301C22⋊C4—621no—
302C22⋊C4—621no—
303C22⋊C4—621no—
304C22⋊C4C22⋊D3111yescharacteristic
305C22⋊C4C22⋊D3111yescharacteristic
306C42C22×D3111yescharacteristic
307C23—21no—
308C23—621no—
309C23—621no—
310C23—1242no—
311C23—1242no—
312C23—2481no—
313C23C2×C22⋊D3111yescharacteristic
314C23C2×C22⋊D3111yescharacteristic
315Q8—42no—
316Q8—621no—
317Q8—621no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—312no—
324D4—621no—
325D4—621no—
326D4—621no—
327D4—1242no—
328D4—1242no—
329D4—2481no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—621no—
341C2×C4—621no—
342C2×C4—1242no—
343C2×C4—2481no—
344C2×C4C2×C22⋊D3111yescharacteristic
345C2×C4C2×C22⋊D3111yescharacteristic
346C2×C4C2×C22⋊D3111yescharacteristic
347C2×C4C2×C22⋊D3111yescharacteristic
348C2×C4C23×D3111yescharacteristic
349C8—621no—
350C8—621no—
351C8—621no—
352C8—621no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—622no—
362C22—1241no—
363C22—1241no—
364C22—1241no—
365C22—1241no—
366C22—1241no—
367C22—2484no—
368C22#1175850912yes—
369C22C22×C22⋊D3111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—1241no—
378C4—1241no—
379C4#41×D3111yescharacteristic
380C4#1175856611yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—1242no—
385C2—2481no—
386C2#1452168711yescharacteristic
387C2C2×#1175850911yescharacteristic
388C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3306
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #14521695C2semidirect
2. #14521411C2semidirect
3. #14521379C2semidirect
4. #14521367C2semidirect
5. #14521336C2semidirect
6. #14521300C2semidirect
7. #14521276C2semidirect
8. #14521264C2semidirect
9. #14521192C2semidirect
10. #14521182C2semidirect
11. #14521159C2semidirect
12. #14521121C2semidirect
13. #14521048C2semidirect
14. C2×#11758397C2semidirect
15. C31×#547C2semidirect
19. C31×C4⋊Q8C22semidirect
24. C31×#121C22semidirect
26. C4×D4×C31C22semidirect
28. C2×C4⋊C4×C31C22semidirect
30. C2×#92480C22semidirect
35. C2×#92474C22semidirect
42. #11758420C22semidirect
44. C22⋊(C31⋊C8)C22semidirect
45. #11758398C22semidirect
46. #11758398C22semidirect
47. #11758398C22semidirect
54. #11758396C22semidirect
55. #11758396C22semidirect
56. #11758396C22semidirect
57. C4⋊(C31⋊C8)C22semidirect
58. #11758391C22semidirect
149. D124#41semidirect
187. #547D31semidirect
218. C4⋊Q8C2×D31semidirect
228. #121C2×D31semidirect
232. C4×D4C2×D31semidirect
234. C2×C4⋊C4C2×D31semidirect
253. C31#3306semidirect

Maximal subgroups

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

subgroupindex
#145216952
#145214112
#145213792
#145213672
#145213362
#145213002
#145212762
#145212642
#145211922
#145211822
#145211592
#145211212
#145210482
C2×#117583972
C31×#5472
#330631