Finite Group Databasebeta

← #44409196

Subgroups — #44409196

Order 29056.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300315C211yesmaximal, characteristic, complemented
2C2×#20300312C211yesmaximal, characteristic, complemented
3#28409581C211yesmaximal, characteristic, complemented
4C4×D4×D227C211yesmaximal, characteristic, complemented
5#28409248C211yesmaximal, characteristic, complemented
6#28409247C211yesmaximal, characteristic, complemented
7#28409244C211yesmaximal, characteristic, complemented
8#28409236C211yesmaximal, characteristic, complemented
9#28409235C211yesmaximal, characteristic, complemented
10#28409226C211yesmaximal, characteristic, complemented
11#28409225C211yesmaximal, characteristic, complemented
12#28409054C211yesmaximal, characteristic, complemented
13#28409032C211yesmaximal, characteristic, complemented
14#28409030C211yesmaximal, characteristic, complemented
15#463×C227C211yesmaximal, characteristic, complemented
16C2×#16813885C2211yescharacteristic
17C2×#16813883C2211yescharacteristic
18C2×D4×D227C2211yescharacteristic
19C22×C4×D227—21no—
20#20300315—24no—
21#20300312—24no—
22C2×#16813873C2211yescharacteristic
23C2×#16813872C2211yescharacteristic
24C2×#16813871C2211yescharacteristic
25C2×D1816C2211yescharacteristic
26C2×#16813869C2211yescharacteristic
27C2×#16813869C2211yescharacteristic
28C2×#16813867C2211yescharacteristic
29C2×#16813867C2211yescharacteristic
30#20300286C2211yescharacteristic, complemented
31#20300284C2211yescharacteristic, complemented
32#20300276C2211yescharacteristic, complemented
33#20300261—21no—
34#20300261—21no—
35C4×D908C2211yescharacteristic
36D4×Dic227C2211yescharacteristic
37#20300250C2211yescharacteristic
38C4×C22⋊D227—21no—
39(C22⋊D227)⋊C4—21no—
40(C22⋊D227)⋊C4—21no—
41(C22⋊D227)⋊C4—21no—
42C4⋊C4×D227C2211yescharacteristic
43C22⋊C4×D227—21no—
44C42×D227C2211yescharacteristic
45C4⋊(C227⋊C8)C2211yescharacteristic, complemented
46Dic227⋊C8C2211yescharacteristic, complemented
47Dic227⋊C8C2211yescharacteristic, complemented
48#20300213C2211yescharacteristic, complemented
49#20300212C2211yescharacteristic, complemented
50#20300211C2211yescharacteristic, complemented
51#20300210C2211yescharacteristic, complemented
52#20300209C2211yescharacteristic, complemented
53#20300208C2211yescharacteristic, complemented
54C2×SD16×C227C2211yescharacteristic
55C2×D8×C227C2211yescharacteristic
56#124×C227C2211yescharacteristic, complemented
57C4×D4×C227C2211yescharacteristic
58C4⋊C8×C227C2211yescharacteristic, complemented
59#103×C227C2211yescharacteristic, complemented
60#102×C227C2211yescharacteristic, complemented
61C23×D227—21no—
62#16813885—22no—
63#16813885—22no—
64#16813883—22no—
65#16813883—22no—
66C2×Dic454C2311yescharacteristic
67C2×D908—21no—
68C2×D908C2311yescharacteristic
69C2×D908C2311yescharacteristic
70C2×C22⋊D227—21no—
71C2×C22⋊D227—21no—
72C2×C22⋊D227—21no—
73C2×C22⋊D227—21no—
74D4×D227—22no—
75D4×D227D414yescomplemented
76C22×Dic227—21no—
77C22×Dic227—21no—
78C2×C4×D227—21no—
79C2×C4×D227—21no—
80C2×C4×D227—21no—
81C2×C4×D227—21no—
82C2×C4×D227—44no—
83C2×C4×D227D411yescharacteristic
84C2×C4×D227D411yescharacteristic
85C2×C4×D227C2311yescharacteristic
86#16813873—22no—
87#16813872—22no—
88#16813871—22no—
89D1816—22no—
90#16813869—22no—
91#16813869—22no—
92#16813867—22no—
93#16813867—22no—
94C2×C227⋊C8C2311yescharacteristic
95C2×C227⋊C8C2311yescharacteristic
96C4⋊Dic227C2311yescharacteristic
97Dic227⋊C4—21no—
98Dic227⋊C4—21no—
99Dic227⋊C4—21no—
100C22⋊Dic227—21no—
101(C2×D227)⋊C4—21no—
102(C2×D227)⋊C4—21no—
103(C2×D227)⋊C4—21no—
104(C2×D227)⋊C4—21no—
105C4×Dic227D411yescharacteristic
106C4×Dic227D411yescharacteristic
107C4×Dic227C2311yescharacteristic
108C2×Q8×C227C2311yescharacteristic
109C2×D4×C227C2311yescharacteristic
110C2×D4×C227C2311yescharacteristic
111C22×C4×C227—21no—
112SD16×C227—22no—
113D8×C227—22no—
114C2×C8×C227C2311yescharacteristic
115C2×C8×C227C2311yescharacteristic
116C4⋊C4×C227C2311yescharacteristic
117C22⋊C4×C227—21no—
118C22⋊C4×C227—21no—
119C22⋊C4×C227—21no—
120C42×C227C2311yescharacteristic
121C22×D227—21no—
122C22×D227—24no—
123C22×D227—21no—
124C22×D227—42no—
125C22×D227—42no—
126C22×D227C2×D411yescharacteristic
127Dic454—21no—
128Dic454—21no—
129D908—21no—
130D908—21no—
131D908—21no—
132D908—42no—
133D908C2×D412yes—
134C22⋊D227—24no—
135C22⋊D227—42no—
136C22⋊D227—42no—
137C22⋊D227—42no—
138C2×Dic227—21no—
139C2×Dic227—21no—
140C2×Dic227—42no—
141C2×Dic227—42no—
142C2×Dic227C2×D411yescharacteristic
143C2×Dic227C2×D411yescharacteristic
144C2×Dic227C2×D411yescharacteristic
145C4×D227—22no—
146C4×D227—41no—
147C4×D227—42no—
148C4×D227—42no—
149C4×D227—42no—
150C4×D227—42no—
151C4×D227C2×D412yes—
152C4×D227#4112yes—
153C227⋊C8—21no—
154C227⋊C8—21no—
155C23×C227—21no—
156C23×C227—21no—
157Q8×C227—21no—
158Q8×C227—21no—
159D4×C227—21no—
160D4×C227—21no—
161D4×C227—21no—
162D4×C227C2×D412yes—
163C2×C4×C227—21no—
164C2×C4×C227—21no—
165C2×C4×C227—42no—
166C2×C4×C227C2×D411yescharacteristic
167C2×C4×C227C2×D411yescharacteristic
168C2×C4×C227C2411yescharacteristic
169C8×C227—21no—
170C8×C227—21no—
171C22×C227—22no—
172C22×C227—41no—
173C22×C227—41no—
174C22×C227—41no—
175C22×C227—41no—
176C22×C227—41no—
177C22×C227C22×D411yescharacteristic
178C2×D227—21no—
179C2×D227—21no—
180C2×D227—22no—
181C2×D227—48no—
182C2×D227—41no—
183C2×D227—41no—
184C2×D227—41no—
185C2×D227—41no—
186C2×D227—41no—
187C2×D227#12112yes—
188C4×C227—21no—
189C4×C227—21no—
190C4×C227—41no—
191C4×C227—41no—
192C4×C227C22×D411yescharacteristic
193C4×C227C2×#4111yescharacteristic
194Dic227—21no—
195Dic227—41no—
196Dic227—41no—
197Dic227#12112yes—
198Dic227AGL(1,ℤ/8)12yescomplemented
199C2×C227—42no—
200C2×C227—81no—
201C2×C227C2×#12111yescharacteristic
202C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C227#57511yescharacteristic
204D227—22no—
205D227—42no—
206D227—81no—
207C227#303611yescharacteristic, complemented, Sylow
208#3036—2271nomaximal, Sylow
209C2×AGL(1,ℤ/8)—2271no—
210C2×AGL(1,ℤ/8)—2271no—
211#532—2271no—
212C2×C4×D4—2271no—
213#463—2271no—
214#463—2271no—
215#463—2271no—
216#463D22711yescharacteristic, complemented
217#461—2271no—
218#461—2271no—
219#459—2271no—
220#459—2271no—
221#423—2271no—
222#417—2271no—
223#417—2271no—
224C2×#41—2271no—
225C2×#41—2271no—
226C22×D4—2271no—
227C23×C4—4541no—
228AGL(1,ℤ/8)—4544no—
229AGL(1,ℤ/8)—4544no—
230C2×SD16—2271no—
231C2×SD16—2271no—
232C2×SD16—2271no—
233C2×SD16C2×D22711yescharacteristic
234C2×D8—2271no—
235C2×D8—2271no—
236C2×D8—2271no—
237C2×D8C2×D22711yescharacteristic
238C2×M16—2271no—
239C2×M16—2271no—
240C4⋊Q8—2271no—
241#127—2271no—
242#124—2271no—
243#124C2×D22711yescharacteristic, complemented
244#121—4541no—
245#121—4541no—
246C4×D4—4541no—
247C4×D4—4541no—
248C4×D4—4541no—
249C4×D4—4541no—
250C4×D4—2271no—
251C4×D4—2271no—
252C4×D4—2271no—
253C4×D4C2×D22711yescharacteristic
254C2×C4⋊C4—2271no—
255C2×C22⋊C4—4541no—
256C2×C42—2271no—
257C4⋊C8—2271no—
258C4⋊C8—2271no—
259C4⋊C8—2271no—
260C4⋊C8C2×D22711yescharacteristic, complemented
261#103—2271no—
262#103—2271no—
263#103—2271no—
264#103C2×D22711yescharacteristic, complemented
265#102—2271no—
266#102—2271no—
267#102—2271no—
268#102C2×D22711yescharacteristic, complemented
269C24—4541no—
270#41—4542no—
271#41—4542no—
272#41—4542no—
273#41—4542no—
274C2×Q8—2271no—
275C2×Q8C22×D22711yescharacteristic
276C2×D4—4541no—
277C2×D4—4541no—
278C2×D4—4542no—
279C2×D4—4541no—
280C2×D4—4541no—
281C2×D4—4541no—
282C2×D4—2274no—
283C2×D4—2271no—
284C2×D4—2271no—
285C2×D4C22×D22711yescharacteristic
286C2×D4C22×D22711yescharacteristic
287C22×C4—21no—
288C22×C4—9084no—
289C22×C4—4541no—
290C22×C4—4541no—
291C22×C4—4541no—
292C22×C4—4541no—
293C22×C4—4541no—
294C22×C4—4541no—
295C22×C4—2271no—
296C22×C4—2271no—
297C22×C4—2271no—
298SD16—22no—
299SD16—4542no—
300SD16—4542no—
301SD16—4542no—
302D8—22no—
303D8—4542no—
304D8—4542no—
305D8—4542no—
306M16—4542no—
307M16—4542no—
308C2×C8—2271no—
309C2×C8—2271no—
310C2×C8C22×D22711yescharacteristic
311C2×C8C22×D22711yescharacteristic
312C4⋊C4—4541no—
313C4⋊C4—4541no—
314C4⋊C4—4541no—
315C4⋊C4—2271no—
316C4⋊C4C22×D22711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—4541no—
321C22⋊C4—4541no—
322C22⋊C4—4541no—
323C22⋊C4—4541no—
324C22⋊C4—4541no—
325C42—2271no—
326C42—2271no—
327C42—2271no—
328C42C22×D22711yescharacteristic
329C23—21no—
330C23—21no—
331C23—9082no—
332C23—9082no—
333C23—4544no—
334C23—4541no—
335C23—4541no—
336C23—2271no—
337Q8—21no—
338Q8—21no—
339Q8—4541no—
340Q8—4541no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—9082no—
345D4—9082no—
346D4—9082no—
347D4—9082no—
348D4—4544no—
349D4—4541no—
350D4—4541no—
351D4—4541no—
352D4—2272no—
353D4D4×D22712yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—9082no—
358C2×C4—9081no—
359C2×C4—9082no—
360C2×C4—9082no—
361C2×C4—9082no—
362C2×C4—9082no—
363C2×C4—9082no—
364C2×C4—4542no—
365C2×C4—4541no—
366C2×C4—4541no—
367C2×C4—2271no—
368C2×C4—2271no—
369C2×C4—2271no—
370C2×C4—2272no—
371C2×C4—2272no—
372C2×C4D4×D22711yescharacteristic
373C2×C4D4×D22711yescharacteristic
374C2×C4C23×D22711yescharacteristic
375C8—21no—
376C8—21no—
377C8—4541no—
378C8—4541no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—9088no—
386C22—9081no—
387C22—9081no—
388C22—9081no—
389C22—9081no—
390C22—9081no—
391C22—4542no—
392C22—4541no—
393C22—4541no—
394C22—2272no—
395C22C2×D4×D22711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—9081no—
401C4—9081no—
402C4—4541no—
403C4—2272no—
404C4—2272no—
405C4C2×D4×D22711yescharacteristic
406C4#41×D22711yescharacteristic
407C2—42no—
408C2—81no—
409C2—18161no—
410C2—9082no—
411C2—4542no—
412C2#121×D22711yescharacteristic
413C2AGL(1,ℤ/8)×D22711yescharacteristic
414C2#2840982911yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#463×C227C2
socleC22×C227C22×D4
hypercenter#463D227
hypocenterC227#3036
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20300315C2semidirect
2. C2×#20300312C2semidirect
3. #28409581C2semidirect
4. C4×D4×D227C2semidirect
5. #28409248C2semidirect
6. #28409247C2semidirect
7. #28409244C2semidirect
8. #28409236C2semidirect
9. #28409235C2semidirect
10. #28409226C2semidirect
11. #28409225C2semidirect
12. #28409054C2semidirect
13. #28409032C2semidirect
14. #28409030C2semidirect
15. #463×C227C2semidirect
30. #20300286C22semidirect
31. #20300284C22semidirect
32. #20300276C22semidirect
45. C4⋊(C227⋊C8)C22semidirect
46. Dic227⋊C8C22semidirect
47. Dic227⋊C8C22semidirect
48. #20300213C22semidirect
49. #20300212C22semidirect
50. #20300211C22semidirect
51. #20300210C22semidirect
52. #20300209C22semidirect
53. #20300208C22semidirect
56. #124×C227C22semidirect
58. C4⋊C8×C227C22semidirect
59. #103×C227C22semidirect
60. #102×C227C22semidirect
75. D4×D227D4semidirect
198. Dic227AGL(1,ℤ/8)semidirect
207. C227#3036semidirect
216. #463D227semidirect
243. #124C2×D227semidirect
260. C4⋊C8C2×D227semidirect
264. #103C2×D227semidirect
268. #102C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003152
C2×#203003122
#284095812
C4×D4×D2272
#284092482
#284092472
#284092442
#284092362
#284092352
#284092262
#284092252
#284090542
#284090322
#284090302
#463×C2272
#3036227