Finite Group Databasebeta

← #16857237

Subgroups — #16857237

Order 3968.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758526C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3#14521695C211yesmaximal, characteristic, complemented
4#14521441C211yesmaximal, characteristic, complemented
5#14521305C211yesmaximal, characteristic, complemented
6#14521280C211yesmaximal, characteristic, complemented
7#14521264C211yesmaximal, characteristic, complemented
8#14521251C211yesmaximal, characteristic, complemented
9#14521211C211yesmaximal, characteristic, complemented
10#14521209C211yesmaximal, characteristic, complemented
11#14521199C211yesmaximal, characteristic, complemented
12#14521194C211yesmaximal, characteristic, complemented
13#14521159C211yesmaximal, characteristic, complemented
14#14521121C211yesmaximal, characteristic, complemented
15C31×#547C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×#41×C31C2211yescharacteristic, complemented
18C2×#92495C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C31×C4⋊Q8C2211yescharacteristic, complemented
21C31×#123—21no—
22C31×#123—21no—
23C31×C22⋊Q8C2211yescharacteristic
24C31×C22⋊Q8C2211yescharacteristic, complemented
25C31×#121—21no—
26C4×D4×C31C2211yescharacteristic
27C4×D4×C31C2211yescharacteristic, complemented
28C2×C4⋊C4×C31—21no—
29#11758527—24no—
30#11758526—24no—
31C2×#92482C2211yescharacteristic, complemented
32C2×#92482C2211yescharacteristic, complemented
33C2×#92481C2211yescharacteristic
34C2×#92481C2211yescharacteristic
35C2×#92481C2211yescharacteristic
36C2×#92481C2211yescharacteristic
37#11758514—21no—
38C2×#92480C2211yescharacteristic, complemented
39C2×#92479C2211yescharacteristic
40C2×#92479C2211yescharacteristic
41#11758507C2211yescharacteristic
42#11758501C2211yescharacteristic
43C2×#92474C2211yescharacteristic
44C22×C31⋊C8C2211yescharacteristic, complemented
45#11758471—21no—
46#11758461—21no—
47C22⋊C4×D31—21no—
48#11758452C2211yescharacteristic
49C4×D124C2211yescharacteristic
50#11758423C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic, complemented
53C22⋊(C31⋊C8)C2211yescharacteristic
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic
57#11758397C2211yescharacteristic
58#11758396C2211yescharacteristic, complemented
59C4⋊(C31⋊C8)C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62#41×C31—22no—
63#41×C31—22no—
64#41×C31—41no—
65#41×C31—41no—
66C2×Q8×C31C2311yescharacteristic
67C2×Q8×C31C2311yescharacteristic
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31D411yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31C2311yescharacteristic
76C22×C4×C31C2311yescharacteristic
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79#92495—22no—
80#92495—41no—
81C2×D124—22no—
82C2×D124—21no—
83C2×D124—41no—
84C2×D124C2311yescharacteristic
85C2×D124C2311yescharacteristic
86C2×C4×D31—21no—
87C2×C4×D31—21no—
88C2×Dic62C2311yescharacteristic
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×C31—21no—
98C22⋊C4×C31D411yescharacteristic
99C22⋊C4×C31D411yescharacteristic
100C42×C31C2311yescharacteristic
101C22⋊Dic31—21no—
102#92482—22no—
103#92482—41no—
104#92481—22no—
105#92481—22no—
106#92481—22no—
107#92481—41no—
108#92480—22no—
109#92479—22no—
110#92479D414yescomplemented
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×C31—21no—
124C23×C31C2×D411yescharacteristic
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—81no—
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135D4×C31—21no—
136D4×C31—41no—
137D4×C31—42no—
138D4×C31—42no—
139D4×C31C2×D412yes—
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×C31—42no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2×D411yescharacteristic
153C2×C4×C31C2×D411yescharacteristic
154C2×C4×C31C2411yescharacteristic
155C22⋊D31—42no—
156C22⋊D31—81no—
157C2×Dic31—21no—
158C2×Dic31—21no—
159D124—21no—
160D124—21no—
161D124—21no—
162D124—42no—
163D124C2×D412yes—
164C4×D31—42no—
165C4×D31—81no—
166Dic62—21no—
167Dic62—21no—
168C31⋊C8—21no—
169C31⋊C8—21no—
170C31⋊C8—21no—
171C31⋊C8C2×D412yes—
172#3267—311nomaximal, Sylow
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—22no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31C22×D411yescharacteristic
183C2×D31—22no—
184C2×D31—41no—
185C2×D31—41no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C2×D31—84no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—21no—
194C4×C31—41no—
195C4×C31—41no—
196C4×C31—41no—
197C4×C31C22×D411yescharacteristic
198C4×C31H(5,𝔽2)11yescharacteristic
199Dic31—41no—
200Dic31—41no—
201C2×AGL(1,ℤ/8)—311no—
202C2×#135—311no—
203#547D3111yescharacteristic, complemented
204#546—311no—
205#497—311no—
206#470—311no—
207#465—311no—
208#463—311no—
209#461—311no—
210#452—311no—
211#452—311no—
212#450—311no—
213#449—311no—
214#442—311no—
215#435—311no—
216C2×C31—41no—
217C2×C31—41no—
218C2×C31—42no—
219C2×C31D4211yescharacteristic
220C2×C31#57511yescharacteristic
221C2×C31#57711yescharacteristic
222D31—42no—
223D31—81no—
224C2×#41—311no—
225C2×#41C2×D3111yescharacteristic
226C2×#41C2×D3111yescharacteristic, complemented
227C22×D4—311no—
228AGL(1,ℤ/8)—624no—
229#135—624no—
230C2×Q16—311no—
231C2×Q16—311no—
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16—311no—
236C2×SD16—311no—
237C2×D8—311no—
238C2×D8—311no—
239C2×M16—311no—
240C22×C8—311no—
241C4⋊Q8C2×D3111yescharacteristic, complemented
242#124—311no—
243#123—21no—
244#123—21no—
245#123—621no—
246C22⋊Q8—311no—
247C22⋊Q8C2×D3111yescharacteristic
248C22⋊Q8C2×D3111yescharacteristic, complemented
249#121—21no—
250#121—311no—
251#121—621no—
252#120—621no—
253C4×D4—311no—
254C4×D4C2×D3111yescharacteristic
255C4×D4C2×D3111yescharacteristic, complemented
256C2×C4⋊C4—21no—
257C2×C22⋊C4—621no—
258#106—311no—
259C4⋊C8—311no—
260#103—311no—
261#103—311no—
262#103—311no—
263#102—311no—
264#102—311no—
265#102—311no—
266C22⋊C8—311no—
267C22⋊C8—311no—
268#97—311no—
269C31#326711yescharacteristic, complemented, Sylow
270C24—621no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—622no—
276#41—1241no—
277C2×Q8—311no—
278C2×Q8C22×D3111yescharacteristic
279C2×Q8C22×D3111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—311no—
283C2×D4—311no—
284C2×D4—622no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—1241no—
289C2×D4C22⋊D3111yescharacteristic
290C2×D4C22×D3111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4C22×D3111yescharacteristic
297C22×C4C22×D3111yescharacteristic
298Q16—622no—
299Q16—1241no—
300SD16—622no—
301SD16—622no—
302SD16—622no—
303SD16—622no—
304SD16—1241no—
305D8—314no—
306D8—622no—
307M16—622no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—311no—
311C2×C8—311no—
312C2×C8—622no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—311no—
318C4⋊C4—621no—
319C4⋊C4C22⋊D3111yescharacteristic
320C4⋊C4C22×D3111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C22⋊C4C22⋊D3111yescharacteristic
330C22⋊C4C22⋊D3111yescharacteristic
331C42C22×D3111yescharacteristic
332C23—21no—
333C23—621no—
334C23—621no—
335C23—1242no—
336C23—1242no—
337C23—2481no—
338C23C2×C22⋊D3111yescharacteristic
339C23C2×C22⋊D3111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—621no—
345Q8—621no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—312no—
351D4—621no—
352D4—621no—
353D4—621no—
354D4—1242no—
355D4—1242no—
356D4—2481no—
357D4D4×D3112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—621no—
369C2×C4—621no—
370C2×C4—1242no—
371C2×C4—2481no—
372C2×C4C2×C22⋊D3111yescharacteristic
373C2×C4C2×C22⋊D3111yescharacteristic
374C2×C4C2×C22⋊D3111yescharacteristic
375C2×C4C2×C22⋊D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—312no—
378C8—621no—
379C8—621no—
380C8—621no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—622no—
391C22—1241no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C22×C22⋊D3111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—1241no—
406C4—1241no—
407C4C2×D4×D3111yescharacteristic
408C4#1175856111yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—1242no—
413C2—2481no—
414C2D4×C22⋊D3111yescharacteristic
415C2#1452184711yescharacteristic
416C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3267
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. C2×#11758526C2semidirect
2. C2×#11758527C2semidirect
3. #14521695C2semidirect
4. #14521441C2semidirect
5. #14521305C2semidirect
6. #14521280C2semidirect
7. #14521264C2semidirect
8. #14521251C2semidirect
9. #14521211C2semidirect
10. #14521209C2semidirect
11. #14521199C2semidirect
12. #14521194C2semidirect
13. #14521159C2semidirect
14. #14521121C2semidirect
15. C31×#547C2semidirect
17. C2×#41×C31C22semidirect
20. C31×C4⋊Q8C22semidirect
24. C31×C22⋊Q8C22semidirect
27. C4×D4×C31C22semidirect
31. C2×#92482C22semidirect
32. C2×#92482C22semidirect
38. C2×#92480C22semidirect
44. C22×C31⋊C8C22semidirect
50. #11758423C22semidirect
51. #11758423C22semidirect
52. #11758420C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
55. #11758398C22semidirect
58. #11758396C22semidirect
59. C4⋊(C31⋊C8)C22semidirect
60. #11758391C22semidirect
110. #92479D4semidirect
203. #547D31semidirect
226. C2×#41C2×D31semidirect
241. C4⋊Q8C2×D31semidirect
248. C22⋊Q8C2×D31semidirect
255. C4×D4C2×D31semidirect
269. C31#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
C2×#117585272
#145216952
#145214412
#145213052
#145212802
#145212642
#145212512
#145212112
#145212092
#145211992
#145211942
#145211592
#145211212
C31×#5472
#326731