Finite Group Databasebeta

← #46845862

Subgroups — #46845862

Order 34688.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576184C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774460C211yesmaximal, characteristic, complemented
4#31774369C211yesmaximal, characteristic, complemented
5#31774113C211yesmaximal, characteristic, complemented
6#31774112C211yesmaximal, characteristic, complemented
7#31774111C211yesmaximal, characteristic, complemented
8#31774110C211yesmaximal, characteristic, complemented
9#31774091C211yesmaximal, characteristic, complemented
10#31774090C211yesmaximal, characteristic, complemented
11#31774089C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#459×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576184—24no—
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×D2168C2211yescharacteristic
23C2×#16864639C2211yescharacteristic
24C2×#16864639C2211yescharacteristic
25C2×#16864639C2211yescharacteristic, complemented
26D8×D271—24no—
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576156C2211yescharacteristic, complemented
30#22576148C2211yescharacteristic, complemented
31#22576147C2211yescharacteristic, complemented
32#22576132—21no—
33#22576132—21no—
34#22576128—21no—
35#22576128—21no—
36C4×D1084C2211yescharacteristic, complemented
37D4×Dic271C2211yescharacteristic, complemented
38#22576122C2211yescharacteristic, complemented
39C4×C22⋊D271—21no—
40(C22⋊D271)⋊C4—21no—
41#22576118C2211yescharacteristic
42#22576117C2211yescharacteristic
43C2×(C2×D271)⋊C4—21no—
44C22⋊C4×D271—21no—
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic, complemented
56#127×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864653—22no—
64#16864653—22no—
65C2×Dic542C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73D4×D271—22no—
74D4×D271—22no—
75D4×D271—22no—
76D4×D271D414yescomplemented
77C22×Dic271—21no—
78C22×Dic271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271C2311yescharacteristic
82#16864643—22no—
83#16864642—22no—
84D2168—22no—
85#16864639—22no—
86#16864639—22no—
87#16864639—22no—
88#16864637—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C8×D271—22no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4D411yescharacteristic
94Dic271⋊C4D411yescharacteristic
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4D411yescharacteristic
104(C2×D271)⋊C4D411yescharacteristic
105C4×Dic271—21no—
106C4×Dic271C2311yescharacteristic
107C2×D4×C271—21no—
108C2×D4×C271—21no—
109C2×D4×C271C2311yescharacteristic
110C2×D4×C271C2311yescharacteristic
111C2×D4×C271C2311yescharacteristic
112C22×C4×C271—21no—
113D8×C271—22no—
114D8×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271—24no—
122C22×D271—21no—
123C22×D271—42no—
124C22×D271—81no—
125C22×D271—81no—
126C22×D271—81no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542—21no—
131D1084—21no—
132D1084—21no—
133D1084—21no—
134D1084C2×D412yes—
135C22⋊D271—24no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—81no—
139C22⋊D271—81no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—42no—
143C2×Dic271—81no—
144C2×Dic271C2×D411yescharacteristic
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C4×D271—81no—
148C4×D271—81no—
149C4×D271C2×D412yes—
150C4×D271#4112yes—
151C271⋊C8—21no—
152C271⋊C8—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—81no—
162D4×C271—81no—
163D4×C271C2×D412yes—
164C2×C4×C271—21no—
165C2×C4×C271—81no—
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×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271C22×D411yescharacteristic
181C2×D271—21no—
182C2×D271—21no—
183C2×D271—22no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—84no—
190C2×D271—84no—
191C2×D271#12112yes—
192C4×C271—21no—
193C4×C271—21no—
194C4×C271—41no—
195C4×C271C22×D411yescharacteristic
196C4×C271C2×#4111yescharacteristic
197Dic271—21no—
198Dic271—21no—
199Dic271—41no—
200Dic271—41no—
201Dic271#12112yes—
202C2×C271—42no—
203C2×C271—81no—
204C2×C271—81no—
205C2×C271C2×#12111yescharacteristic
206C2×C271#57611yescharacteristic
207C2×C271#57611yescharacteristic
208D271—22no—
209D271—42no—
210D271—81no—
211C271#304111yescharacteristic, complemented, Sylow
212#3041—2711nomaximal, Sylow
213C2×AGL(1,ℤ/8)—2711no—
214C22×D8—2711no—
215#535—2711no—
216#518—2711no—
217#463—2711no—
218#463—2711no—
219#463—2711no—
220#463—2711no—
221#459—2711no—
222#459—2711no—
223#459—2711no—
224#459D27111yescharacteristic, complemented
225#424—2711no—
226#418—2711no—
227#416—2711no—
228C2×#41—2711no—
229C22×D4—2711no—
230C22×D4—2711no—
231AGL(1,ℤ/8)—5424no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×D8—2711no—
235C2×D8—2711no—
236C2×D8—2711no—
237C2×D8—2711no—
238C2×D8—5424no—
239C2×D8C2×D27111yescharacteristic
240C2×D8C2×D27111yescharacteristic, complemented
241C2×M16—2711no—
242C22×C8—2711no—
243#127—2711no—
244#127C2×D27111yescharacteristic, complemented
245#124—2711no—
246#124—2711no—
247#121—5421no—
248#121—5421no—
249#120—5421no—
250#120—5421no—
251C4×D4—2711no—
252C4×D4—2711no—
253C4×D4—2711no—
254C4×D4—5421no—
255C4×D4—5421no—
256C4×D4C2×D27111yescharacteristic, complemented
257#117—2711no—
258#117—2711no—
259C2×C22⋊C4—5421no—
260C2×C22⋊C4—5421no—
261C4⋊C8—2711no—
262C4⋊C8—2711no—
263C4⋊C8—2711no—
264C4⋊C8C2×D27111yescharacteristic, complemented
265#103—2711no—
266#103—2711no—
267#102—2711no—
268#102—2711no—
269#102—2711no—
270#102—2711no—
271#102C2×D27111yescharacteristic, complemented
272#102C2×D27111yescharacteristic, complemented
273C24—5421no—
274C24—5421no—
275#41—5422no—
276#41—5422no—
277C2×Q8—2711no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—2714no—
281C2×D4—2711no—
282C2×D4—2711no—
283C2×D4—5422no—
284C2×D4—5422no—
285C2×D4—5422no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4C22×D27111yescharacteristic
292C2×D4C22×D27111yescharacteristic
293C2×D4C22×D27111yescharacteristic
294C22×C4—21no—
295C22×C4—2711no—
296C22×C4—5421no—
297C22×C4—5421no—
298C22×C4—5421no—
299C22×C4—5421no—
300SD16—5422no—
301SD16—5422no—
302D8—22no—
303D8—22no—
304D8—5422no—
305D8—5422no—
306D8—5422no—
307D8—5422no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8—5422no—
312C2×C8C22×D27111yescharacteristic
313C2×C8C22×D27111yescharacteristic
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—2711no—
317C4⋊C4C22×D27111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—2711no—
320C22⋊C4—2711no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C42—2711no—
330C42—5421no—
331C42C22×D27111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—2711no—
336C23—5424no—
337C23—5421no—
338C23—5421no—
339C23—10842no—
340C23—21681no—
341C23—21681no—
342C23—21681no—
343C23—21681no—
344Q8—5421no—
345Q8—5421no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—2712no—
354D4—5424no—
355D4—5421no—
356D4—5421no—
357D4—5421no—
358D4—10842no—
359D4—10842no—
360D4—21681no—
361D4—21681no—
362D4D4×D27112yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—2711no—
366C2×C4—2712no—
367C2×C4—2712no—
368C2×C4—2711no—
369C2×C4—2711no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—10842no—
373C2×C4—21681no—
374C2×C4—21681no—
375C2×C4—21681no—
376C2×C4D4×D27111yescharacteristic
377C2×C4D4×D27111yescharacteristic
378C2×C4C23×D27111yescharacteristic
379C8—21no—
380C8—21no—
381C8—5421no—
382C8—5421no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—2712no—
393C22—5421no—
394C22—5421no—
395C22—5422no—
396C22—10841no—
397C22—10841no—
398C22—10841no—
399C22—10841no—
400C22—10841no—
401C22—21684no—
402C22—21684no—
403C22C2×D4×D27111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—2712no—
408C4—5421no—
409C4—5421no—
410C4—10841no—
411C4—10841no—
412C4C2×D4×D27111yescharacteristic
413C4#41×D27111yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—5422no—
418C2—10842no—
419C2—21681no—
420C2#121×D27111yescharacteristic
421C2#3177469811yescharacteristic
422C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3041
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×D8×D271C2semidirect
3. #31774460C2semidirect
4. #31774369C2semidirect
5. #31774113C2semidirect
6. #31774112C2semidirect
7. #31774111C2semidirect
8. #31774110C2semidirect
9. #31774091C2semidirect
10. #31774090C2semidirect
11. #31774089C2semidirect
12. #31773922C2semidirect
13. #31773900C2semidirect
14. #31773890C2semidirect
15. #459×C271C2semidirect
20. C2×#16864643C22semidirect
21. C2×#16864642C22semidirect
25. C2×#16864639C22semidirect
29. #22576156C22semidirect
30. #22576148C22semidirect
31. #22576147C22semidirect
36. C4×D1084C22semidirect
37. D4×Dic271C22semidirect
38. #22576122C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. #22576081C22semidirect
53. #22576080C22semidirect
55. C2×D8×C271C22semidirect
56. #127×C271C22semidirect
57. C4×D4×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
76. D4×D271D4semidirect
211. C271#3041semidirect
224. #459D271semidirect
240. C2×D8C2×D271semidirect
244. #127C2×D271semidirect
256. C4×D4C2×D271semidirect
264. C4⋊C8C2×D271semidirect
271. #102C2×D271semidirect
272. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×D8×D2712
#317744602
#317743692
#317741132
#317741122
#317741112
#317741102
#317740912
#317740902
#317740892
#317739222
#317739002
#317738902
#459×C2712
#3041271