Finite Group Databasebeta

← #16857222

Subgroups — #16857222

Order 3968.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758519C211yesmaximal, characteristic, complemented
2C22×#92481C211yesmaximal, characteristic, complemented
3#14521695C211yesmaximal, characteristic
4#14521441C211yesmaximal, characteristic, complemented
5#14521297C211yesmaximal, characteristic, complemented
6#14521288C211yesmaximal, characteristic, complemented
7#14521271C211yesmaximal, characteristic, complemented
8#14521251C211yesmaximal, characteristic, complemented
9#14521209C211yesmaximal, characteristic, complemented
10#14521205C211yesmaximal, characteristic, complemented
11#14521199C211yesmaximal, characteristic, complemented
12#14521188C211yesmaximal, characteristic, complemented
13#14521147C211yesmaximal, characteristic, complemented
14#14521121C211yesmaximal, characteristic, complemented
15D4×Q8×C31C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic, complemented
18C2×#92495C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C31×C4⋊Q8—21no—
21C31×C4⋊Q8C2211yescharacteristic, complemented
22C31×C22⋊Q8—21no—
23C31×C22⋊Q8—21no—
24C31×C22⋊Q8C2211yescharacteristic
25C31×C22⋊Q8C2211yescharacteristic, complemented
26C4×Q8×C31C2211yescharacteristic, complemented
27C4×D4×C31—21no—
28C4×D4×C31C2211yescharacteristic
29C2×#92482C2211yescharacteristic, complemented
30C2×#92482C2211yescharacteristic, complemented
31C2×#92482C2211yescharacteristic, complemented
32#11758519—24no—
33C2×#92481—24no—
34C2×#92481C2211yescharacteristic
35C2×#92481C2211yescharacteristic
36C2×#92481C2211yescharacteristic
37C2×#92481C2211yescharacteristic
38C2×#92481C2211yescharacteristic
39C2×#92481C2211yescharacteristic
40#11758514—21no—
41#11758507C2211yescharacteristic
42#11758501C2211yescharacteristic
43C2×#92474C2211yescharacteristic, complemented
44C22×C31⋊C8C2211yescharacteristic
45#11758471—21no—
46#11758461—21no—
47C22⋊C4×D31—21no—
48#11758452C2211yescharacteristic
49C4×D124C2211yescharacteristic
50#11758423C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758423C2211yescharacteristic, complemented
53C22⋊(C31⋊C8)C2211yescharacteristic
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic
57#11758397C2211yescharacteristic
58#11758395C2211yescharacteristic, complemented
59C4⋊(C31⋊C8)C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—22no—
65C2×Q8×C31—22no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31C2311yescharacteristic
69C2×Q8×C31C2311yescharacteristic
70C2×Q8×C31C2311yescharacteristic
71C2×D4×C31D411yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31C2311yescharacteristic
75C22×C4×C31C2311yescharacteristic
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78#92495—22no—
79#92495—41no—
80C2×D124—22no—
81C2×D124—21no—
82C2×D124—41no—
83C2×D124C2311yescharacteristic
84C2×D124C2311yescharacteristic
85C2×C4×D31—21no—
86C2×C4×D31—21no—
87C2×Dic62C2311yescharacteristic
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31D411yescharacteristic
94C4⋊C4×C31C2311yescharacteristic
95C22⋊C4×C31—21no—
96C22⋊C4×C31—21no—
97C22⋊C4×C31D411yescharacteristic
98C22⋊C4×C31D411yescharacteristic
99C42×C31—21no—
100C42×C31C2311yescharacteristic
101C22⋊Dic31—21no—
102#92482—22no—
103#92482—22no—
104#92482—41no—
105#92481—22no—
106#92481—22no—
107#92481—22no—
108#92481—22no—
109#92481—41no—
110#92481D414yescomplemented
111(C2×D31)⋊C4—21no—
112(C2×D31)⋊C4—21no—
113(C2×D31)⋊C4—21no—
114(C2×D31)⋊C4—21no—
115C4⋊Dic31C2311yescharacteristic
116Dic31⋊C4—21no—
117#92474—22no—
118C2×C31⋊C8—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121C2×C31⋊C8C2311yescharacteristic
122C2×C31⋊C8C2311yescharacteristic
123C23×C31C2×D411yescharacteristic
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—42no—
128C22×D31—42no—
129C22×D31—81no—
130Q8×C31—21no—
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135Q8×C31—42no—
136Q8×C31—42no—
137Q8×C31C2×D412yes—
138D4×C31—22no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31C2×D411yescharacteristic
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2×D411yescharacteristic
153C2×C4×C31C2411yescharacteristic
154C22⋊D31—42no—
155C22⋊D31—81no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158D124—21no—
159D124—21no—
160D124—21no—
161D124—42no—
162D124C2×D412yes—
163C4×D31—42no—
164C4×D31—81no—
165Dic62—21no—
166Dic62—21no—
167C31⋊C8—21no—
168C31⋊C8—21no—
169C31⋊C8—21no—
170C31⋊C8C2×D412yes—
171#3266—311nomaximal, Sylow
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31#13712yescomplemented
178C22×C31C22×D411yescharacteristic
179C2×D31—22no—
180C2×D31—41no—
181C2×D31—41no—
182C2×D31—41no—
183C2×D31—41no—
184C2×D31—41no—
185C2×D31—84no—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—22no—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—41no—
192C4×C31—41no—
193C4×C31—41no—
194C4×C31C22×D411yescharacteristic
195C4×C31H(5,𝔽2)11yescharacteristic
196Dic31—41no—
197Dic31—41no—
198C2×#137—311no—
199C22×SD16—311no—
200D4×Q8D3111yescharacteristic, complemented
201#546—311no—
202#497—311no—
203#468—311no—
204#467—311no—
205#464—311no—
206#461—311no—
207#452—311no—
208#451—311no—
209#450—311no—
210#448—311no—
211#440—311no—
212#435—311no—
213C2×C31—22no—
214C2×C31—41no—
215C2×C31D4211yescharacteristic
216C2×C31C2×#13711yescharacteristic
217C2×C31#57711yescharacteristic
218D31—42no—
219D31—81no—
220C2×#41—311no—
221C22×Q8C2×D3111yescharacteristic
222C22×Q8C2×D3111yescharacteristic, complemented
223C22×D4—311no—
224#137—624no—
225C2×Q16—311no—
226C2×Q16—311no—
227C2×Q16—311no—
228C2×SD16—311no—
229C2×SD16—311no—
230C2×SD16—311no—
231C2×SD16—311no—
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—624no—
235C2×M16—311no—
236C22×C8—311no—
237C4⋊Q8—21no—
238C4⋊Q8C2×D3111yescharacteristic, complemented
239#124—311no—
240#123—621no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—311no—
244C22⋊Q8C2×D3111yescharacteristic
245C22⋊Q8C2×D3111yescharacteristic, complemented
246#121—311no—
247#121—621no—
248#120—621no—
249C4×Q8C2×D3111yescharacteristic, complemented
250C4×D4—21no—
251C4×D4—311no—
252C4×D4C2×D3111yescharacteristic
253C2×C22⋊C4—621no—
254#107—311no—
255C4⋊C8—311no—
256#103—311no—
257#103—311no—
258#103—311no—
259#103—311no—
260#102—311no—
261#102—311no—
262C22⋊C8—311no—
263C22⋊C8—311no—
264#97—311no—
265C31#326611yescharacteristic, complemented, Sylow
266C24—621no—
267#41—622no—
268#41—1241no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—311no—
276C2×Q8C22×D3111yescharacteristic
277C2×Q8C22×D3111yescharacteristic
278C2×Q8C22×D3111yescharacteristic
279C2×D4—311no—
280C2×D4—311no—
281C2×D4—622no—
282C2×D4—621no—
283C2×D4—621no—
284C2×D4—621no—
285C2×D4—1241no—
286C2×D4C22⋊D3111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—621no—
290C22×C4—621no—
291C22×C4C22×D3111yescharacteristic
292C22×C4C22×D3111yescharacteristic
293Q16—622no—
294Q16—622no—
295Q16—1241no—
296SD16—314no—
297SD16—622no—
298SD16—622no—
299SD16—622no—
300SD16—622no—
301SD16—1241no—
302M16—622no—
303C2×C8—311no—
304C2×C8—311no—
305C2×C8—311no—
306C2×C8—311no—
307C2×C8—622no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—311no—
314C4⋊C4—621no—
315C4⋊C4C22⋊D3111yescharacteristic
316C4⋊C4C22×D3111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—621no—
320C22⋊C4—621no—
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4C22⋊D3111yescharacteristic
325C22⋊C4C22⋊D3111yescharacteristic
326C42—21no—
327C42C22×D3111yescharacteristic
328C23—621no—
329C23—621no—
330C23—1242no—
331C23—1242no—
332C23—2481no—
333C23C2×C22⋊D3111yescharacteristic
334C23C2×C22⋊D3111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—621no—
343Q8—621no—
344Q8D4×D3112yes—
345D4—22no—
346D4—312no—
347D4—621no—
348D4—621no—
349D4—621no—
350D4—1242no—
351D4—1242no—
352D4—2481no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—1242no—
366C2×C4—2481no—
367C2×C4C2×C22⋊D3111yescharacteristic
368C2×C4C2×C22⋊D3111yescharacteristic
369C2×C4C2×C22⋊D3111yescharacteristic
370C2×C4C2×C22⋊D3111yescharacteristic
371C2×C4C23×D3111yescharacteristic
372C8—312no—
373C8—621no—
374C8—621no—
375C8—621no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—622no—
382C22—1241no—
383C22—1241no—
384C22—1241no—
385C22—1241no—
386C22—1241no—
387C22—2484no—
388C22#1175851912yescomplemented
389C22C22×C22⋊D3111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—1241no—
399C4—1241no—
400C4C2×D4×D3111yescharacteristic
401C4#1175856111yescharacteristic
402C2—22no—
403C2—41no—
404C2—1242no—
405C2—2481no—
406C2D4×C22⋊D3111yescharacteristic
407C2C2×#1175851911yescharacteristic
408C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingD4×Q8×C31C2
socleC22×C31C22×D4
prime radical#14521695C2
hypercenterD4×Q8D31
hypocenterC31#3266
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11758519C2semidirect
2. C22×#92481C2semidirect
4. #14521441C2semidirect
5. #14521297C2semidirect
6. #14521288C2semidirect
7. #14521271C2semidirect
8. #14521251C2semidirect
9. #14521209C2semidirect
10. #14521205C2semidirect
11. #14521199C2semidirect
12. #14521188C2semidirect
13. #14521147C2semidirect
14. #14521121C2semidirect
15. D4×Q8×C31C2semidirect
17. C22×Q8×C31C22semidirect
21. C31×C4⋊Q8C22semidirect
25. C31×C22⋊Q8C22semidirect
26. C4×Q8×C31C22semidirect
29. C2×#92482C22semidirect
30. C2×#92482C22semidirect
31. C2×#92482C22semidirect
43. C2×#92474C22semidirect
50. #11758423C22semidirect
51. #11758423C22semidirect
52. #11758423C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
55. #11758398C22semidirect
58. #11758395C22semidirect
59. C4⋊(C31⋊C8)C22semidirect
60. #11758391C22semidirect
110. #92481D4semidirect
177. C22×C31#137semidirect
200. D4×Q8D31semidirect
222. C22×Q8C2×D31semidirect
238. C4⋊Q8C2×D31semidirect
245. C22⋊Q8C2×D31semidirect
249. C4×Q8C2×D31semidirect
265. C31#3266semidirect
388. C22#11758519semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
C22×#924812
#145216952
#145214412
#145212972
#145212882
#145212712
#145212512
#145212092
#145212052
#145211992
#145211882
#145211472
#145211212
D4×Q8×C312
#326631