Finite Group Databasebeta

← #46845826

Subgroups — #46845826

Order 34688.

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×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3#31774446C211yesmaximal, characteristic, complemented
4C4×D4×D271C211yesmaximal, characteristic, complemented
5#31774113C211yesmaximal, characteristic, complemented
6#31774112C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774101C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774091C211yesmaximal, characteristic, complemented
11#31774090C211yesmaximal, characteristic, complemented
12#31773919C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#31773895C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19C22×C4×D271—21no—
20#22576187—24no—
21#22576184—24no—
22C2×#16864643C2211yescharacteristic
23C2×#16864642C2211yescharacteristic
24C2×#16864641C2211yescharacteristic
25C2×D2168C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864637C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30#22576158C2211yescharacteristic, complemented
31#22576156C2211yescharacteristic, complemented
32#22576148C2211yescharacteristic, complemented
33#22576133—21no—
34#22576133—21no—
35C4×D1084C2211yescharacteristic
36D4×Dic271C2211yescharacteristic
37#22576122C2211yescharacteristic
38C4×C22⋊D271—21no—
39(C22⋊D271)⋊C4—21no—
40(C22⋊D271)⋊C4—21no—
41(C22⋊D271)⋊C4—21no—
42C4⋊C4×D271C2211yescharacteristic
43C22⋊C4×D271—21no—
44C42×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64#16864653—22no—
65#16864653—22no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74D4×D271—22no—
75D4×D271D414yescomplemented
76C22×Dic271—21no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—44no—
83C2×C4×D271D411yescharacteristic
84C2×C4×D271D411yescharacteristic
85C2×C4×D271C2311yescharacteristic
86#16864643—22no—
87#16864642—22no—
88#16864641—22no—
89D2168—22no—
90#16864639—22no—
91#16864639—22no—
92#16864637—22no—
93#16864637—22no—
94C2×C271⋊C8C2311yescharacteristic
95C2×C271⋊C8C2311yescharacteristic
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100C22⋊Dic271—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105C4×Dic271D411yescharacteristic
106C4×Dic271D411yescharacteristic
107C4×Dic271C2311yescharacteristic
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110C2×D4×C271C2311yescharacteristic
111C22×C4×C271—21no—
112SD16×C271—22no—
113D8×C271—22no—
114C2×C8×C271C2311yescharacteristic
115C2×C8×C271C2311yescharacteristic
116C4⋊C4×C271C2311yescharacteristic
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C42×C271C2311yescharacteristic
121C22×D271—21no—
122C22×D271—24no—
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271C2×D411yescharacteristic
127Dic542—21no—
128Dic542—21no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—42no—
133D1084C2×D412yes—
134C22⋊D271—24no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—42no—
141C2×Dic271—42no—
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C4×D271—22no—
146C4×D271—41no—
147C4×D271—42no—
148C4×D271—42no—
149C4×D271—42no—
150C4×D271—42no—
151C4×D271C2×D412yes—
152C4×D271#4112yes—
153C271⋊C8—21no—
154C271⋊C8—21no—
155C23×C271—21no—
156C23×C271—21no—
157Q8×C271—21no—
158Q8×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271C2×D412yes—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—42no—
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2411yescharacteristic
169C8×C271—21no—
170C8×C271—21no—
171C22×C271—22no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271C22×D411yescharacteristic
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—22no—
181C2×D271—48no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271#12112yes—
188C4×C271—21no—
189C4×C271—21no—
190C4×C271—41no—
191C4×C271—41no—
192C4×C271C22×D411yescharacteristic
193C4×C271C2×#4111yescharacteristic
194Dic271—21no—
195Dic271—41no—
196Dic271—41no—
197Dic271#12112yes—
198Dic271AGL(1,ℤ/8)12yescomplemented
199C2×C271—42no—
200C2×C271—81no—
201C2×C271C2×#12111yescharacteristic
202C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C271#57511yescharacteristic
204D271—22no—
205D271—42no—
206D271—81no—
207C271#303611yescharacteristic, complemented, Sylow
208#3036—2711nomaximal, Sylow
209C2×AGL(1,ℤ/8)—2711no—
210C2×AGL(1,ℤ/8)—2711no—
211#532—2711no—
212C2×C4×D4—2711no—
213#463—2711no—
214#463—2711no—
215#463—2711no—
216#463D27111yescharacteristic, complemented
217#461—2711no—
218#461—2711no—
219#459—2711no—
220#459—2711no—
221#423—2711no—
222#417—2711no—
223#417—2711no—
224C2×#41—2711no—
225C2×#41—2711no—
226C22×D4—2711no—
227C23×C4—5421no—
228AGL(1,ℤ/8)—5424no—
229AGL(1,ℤ/8)—5424no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16C2×D27111yescharacteristic
234C2×D8—2711no—
235C2×D8—2711no—
236C2×D8—2711no—
237C2×D8C2×D27111yescharacteristic
238C2×M16—2711no—
239C2×M16—2711no—
240C4⋊Q8—2711no—
241#127—2711no—
242#124—2711no—
243#124C2×D27111yescharacteristic, complemented
244#121—5421no—
245#121—5421no—
246C4×D4—2711no—
247C4×D4—2711no—
248C4×D4—2711no—
249C4×D4—5421no—
250C4×D4—5421no—
251C4×D4—5421no—
252C4×D4—5421no—
253C4×D4C2×D27111yescharacteristic
254C2×C4⋊C4—2711no—
255C2×C22⋊C4—5421no—
256C2×C42—2711no—
257C4⋊C8—2711no—
258C4⋊C8—2711no—
259C4⋊C8—2711no—
260C4⋊C8C2×D27111yescharacteristic, complemented
261#103—2711no—
262#103—2711no—
263#103—2711no—
264#103C2×D27111yescharacteristic, complemented
265#102—2711no—
266#102—2711no—
267#102—2711no—
268#102C2×D27111yescharacteristic, complemented
269C24—5421no—
270#41—5422no—
271#41—5422no—
272#41—5422no—
273#41—5422no—
274C2×Q8—2711no—
275C2×Q8C22×D27111yescharacteristic
276C2×D4—2714no—
277C2×D4—2711no—
278C2×D4—2711no—
279C2×D4—5421no—
280C2×D4—5422no—
281C2×D4—5421no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4C22×D27111yescharacteristic
286C2×D4C22×D27111yescharacteristic
287C22×C4—21no—
288C22×C4—2711no—
289C22×C4—2711no—
290C22×C4—2711no—
291C22×C4—5421no—
292C22×C4—5421no—
293C22×C4—5421no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4—10844no—
298SD16—22no—
299SD16—5422no—
300SD16—5422no—
301SD16—5422no—
302D8—22no—
303D8—5422no—
304D8—5422no—
305D8—5422no—
306M16—5422no—
307M16—5422no—
308C2×C8—2711no—
309C2×C8—2711no—
310C2×C8C22×D27111yescharacteristic
311C2×C8C22×D27111yescharacteristic
312C4⋊C4—2711no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4—5421no—
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C42—2711no—
326C42—2711no—
327C42—2711no—
328C42C22×D27111yescharacteristic
329C23—21no—
330C23—21no—
331C23—2711no—
332C23—5424no—
333C23—5421no—
334C23—5421no—
335C23—10842no—
336C23—10842no—
337Q8—21no—
338Q8—21no—
339Q8—5421no—
340Q8—5421no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—2712no—
345D4—5424no—
346D4—5421no—
347D4—5421no—
348D4—5421no—
349D4—10842no—
350D4—10842no—
351D4—10842no—
352D4—10842no—
353D4D4×D27112yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—2711no—
358C2×C4—2711no—
359C2×C4—2712no—
360C2×C4—2712no—
361C2×C4—2711no—
362C2×C4—5422no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—10841no—
366C2×C4—10842no—
367C2×C4—10842no—
368C2×C4—10842no—
369C2×C4—10842no—
370C2×C4—10842no—
371C2×C4—10842no—
372C2×C4D4×D27111yescharacteristic
373C2×C4D4×D27111yescharacteristic
374C2×C4C23×D27111yescharacteristic
375C8—21no—
376C8—21no—
377C8—5421no—
378C8—5421no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—2712no—
386C22—5421no—
387C22—5421no—
388C22—5422no—
389C22—10848no—
390C22—10841no—
391C22—10841no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22C2×D4×D27111yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—2712no—
401C4—2712no—
402C4—5421no—
403C4—10841no—
404C4—10841no—
405C4C2×D4×D27111yescharacteristic
406C4#41×D27111yescharacteristic
407C2—42no—
408C2—81no—
409C2—5422no—
410C2—10842no—
411C2—21681no—
412C2#121×D27111yescharacteristic
413C2AGL(1,ℤ/8)×D27111yescharacteristic
414C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3036
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576184C2semidirect
3. #31774446C2semidirect
4. C4×D4×D271C2semidirect
5. #31774113C2semidirect
6. #31774112C2semidirect
7. #31774109C2semidirect
8. #31774101C2semidirect
9. #31774100C2semidirect
10. #31774091C2semidirect
11. #31774090C2semidirect
12. #31773919C2semidirect
13. #31773897C2semidirect
14. #31773895C2semidirect
15. #463×C271C2semidirect
30. #22576158C22semidirect
31. #22576156C22semidirect
32. #22576148C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
51. #22576082C22semidirect
52. #22576081C22semidirect
53. #22576080C22semidirect
56. #124×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
75. D4×D271D4semidirect
198. Dic271AGL(1,ℤ/8)semidirect
207. C271#3036semidirect
216. #463D271semidirect
243. #124C2×D271semidirect
260. C4⋊C8C2×D271semidirect
264. #103C2×D271semidirect
268. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761842
#317744462
C4×D4×D2712
#317741132
#317741122
#317741092
#317741012
#317741002
#317740912
#317740902
#317739192
#317738972
#317738952
#463×C2712
#3036271