Finite Group Databasebeta

← #46845980

Subgroups — #46845980

Order 34688.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576184C211yesmaximal, characteristic, complemented
2C22×#16864637C211yesmaximal, characteristic, complemented
3#31774407C211yesmaximal, characteristic, complemented
4#121×D271C211yesmaximal, characteristic, complemented
5#31774149C211yesmaximal, characteristic, complemented
6#31774147C211yesmaximal, characteristic, complemented
7#31774144C211yesmaximal, characteristic, complemented
8#31774142C211yesmaximal, characteristic, complemented
9#31774131C211yesmaximal, characteristic, complemented
10#31774126C211yesmaximal, characteristic, complemented
11#31774124C211yesmaximal, characteristic, complemented
12#31773938C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#31773895C211yesmaximal, characteristic, complemented
15#466×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271—21no—
18C2×D4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576184—24no—
21C2×#16864643C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864639C2211yescharacteristic, complemented
24C2×#16864637—24no—
25C2×#16864637C2211yescharacteristic
26C2×#16864637C2211yescharacteristic
27C2×#16864637C2211yescharacteristic, complemented
28C2×#16864637C2211yescharacteristic, complemented
29C22×C271⋊C8C2211yescharacteristic, complemented
30#22576142—21no—
31#22576140C2211yescharacteristic, complemented
32#22576139C2211yescharacteristic, complemented
33#22576135C2211yescharacteristic
34#22576134C2211yescharacteristic
35#22576133—21no—
36#22576132—21no—
37#22576131C2211yescharacteristic
38#22576131C2211yescharacteristic, complemented
39#22576128—21no—
40D4×Dic271—21no—
41(C22⋊D271)⋊C4—21no—
42#22576118C2211yescharacteristic
43C4⋊C4×D271C2211yescharacteristic
44C22⋊C4×D271—21no—
45#22576093C2211yescharacteristic, complemented
46#22576092C2211yescharacteristic, complemented
47#22576091C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic, complemented
56#121×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic, complemented
58#107×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271D411yescharacteristic
63#16864653—22no—
64#16864653—22no—
65C2×Dic542C2311yescharacteristic
66C2×D1084—21no—
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—44no—
76C22×Dic271—21no—
77C22×Dic271D411yescharacteristic
78C2×C4×D271—22no—
79C2×C4×D271—22no—
80C2×C4×D271—21no—
81C2×C4×D271D411yescharacteristic
82C2×C4×D271D411yescharacteristic
83C2×C4×D271C2311yescharacteristic
84C2×C4×D271C2311yescharacteristic
85#16864643—22no—
86#16864642—22no—
87#16864639—22no—
88#16864637—22no—
89#16864637—22no—
90#16864637—22no—
91#16864637D414yescomplemented
92C2×C271⋊C8—22no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C4⋊Dic271—21no—
96C4⋊Dic271C2311yescharacteristic
97C4⋊Dic271C2311yescharacteristic
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107C4×Dic271—21no—
108C2×D4×C271—21no—
109C2×D4×C271—21no—
110C2×D4×C271C2311yescharacteristic
111C2×D4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113D8×C271—22no—
114C2×C8×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C4⋊C4×C271C2311yescharacteristic
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C22×D271—22no—
122C22×D271—22no—
123C22×D271—21no—
124C22×D271—41no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271C2×D411yescharacteristic
129C22×D271C2×D411yescharacteristic
130Dic542—21no—
131Dic542—21no—
132D1084—21no—
133D1084—21no—
134D1084—42no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C22⋊D271—42no—
140C2×Dic271—22no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—42no—
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C4×D271—22no—
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—22no—
151C4×D271—42no—
152C4×D271C2×D412yes—
153C4×D271#4112yes—
154C271⋊C8—21no—
155C271⋊C8C2×D412yescomplemented
156C23×C271—21no—
157C23×C271—21no—
158C23×C271C2×D411yescharacteristic
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—42no—
164D4×C271—42no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2411yescharacteristic
171C8×C271—21no—
172C8×C271C2×D412yescomplemented
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271AGL(1,ℤ/8)12yescomplemented
182C22×C271C22×D411yescharacteristic
183C2×D271—21no—
184C2×D271—21no—
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—44no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—84no—
193C2×D271#12112yes—
194C4×C271—21no—
195C4×C271—41no—
196C4×C271—41no—
197C4×C271C22×D411yescharacteristic
198C4×C271C2×#4111yescharacteristic
199Dic271—21no—
200Dic271—41no—
201Dic271—41no—
202Dic271—41no—
203Dic271#12112yes—
204C2×C271—22no—
205C2×C271—81no—
206C2×C271—81no—
207C2×C271C2×#12111yescharacteristic
208C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C271#57511yescharacteristic
210D271—22no—
211D271—41no—
212D271—81no—
213C271#305511yescharacteristic, complemented, Sylow
214#3055—2711nomaximal, Sylow
215C2×AGL(1,ℤ/8)—2711no—
216C22×M16—2711no—
217#525—2711no—
218C2×#121—2711no—
219#469—2711no—
220#469—2711no—
221#468—2711no—
222#468—2711no—
223#466—2711no—
224#466D27111yescharacteristic, complemented
225#465—2711no—
226#465—2711no—
227#428—2711no—
228#418—2711no—
229#417—2711no—
230C2×#41—2711no—
231C22×D4—2711no—
232C22×D4—5421no—
233C23×C4—2711no—
234AGL(1,ℤ/8)—5424no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×D8—2711no—
238C2×D8C2×D27111yescharacteristic, complemented
239C2×M16—2711no—
240C2×M16—2711no—
241C2×M16—2711no—
242C2×M16—2711no—
243C2×M16—5424no—
244C22×C8—2711no—
245C22×C8C2×D27111yescharacteristic, complemented
246#123—5421no—
247C22⋊Q8—2711no—
248C22⋊Q8—2711no—
249#121—2711no—
250#121—2711no—
251#121—2711no—
252#121—2711no—
253#121—5421no—
254#121—5421no—
255#121C2×D27111yescharacteristic
256#121C2×D27111yescharacteristic, complemented
257#120—5421no—
258C4×D4—5421no—
259C4×D4—5421no—
260#117—2711no—
261C2×C4⋊C4—2711no—
262C2×C22⋊C4—5421no—
263#107—2711no—
264#107C2×D27111yescharacteristic, complemented
265#106—2711no—
266#106—2711no—
267#103—2711no—
268#103—2711no—
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273#102C2×D27111yescharacteristic, complemented
274#102C2×D27111yescharacteristic, complemented
275C24—2711no—
276C24—5421no—
277#41—5422no—
278#41—5422no—
279C2×Q8—2711no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2711no—
283C2×D4—5421no—
284C2×D4—5422no—
285C2×D4—5422no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4—10844no—
292C2×D4C22×D27111yescharacteristic
293C2×D4C22×D27111yescharacteristic
294C22×C4—2711no—
295C22×C4—2711no—
296C22×C4—2711no—
297C22×C4—2711no—
298C22×C4—2711no—
299C22×C4—5422no—
300C22×C4—5422no—
301C22×C4—5421no—
302C22×C4—5421no—
303C22×C4C22×D27111yescharacteristic
304SD16—5422no—
305SD16—5422no—
306D8—22no—
307D8—5422no—
308M16—2714no—
309M16—5422no—
310M16—5422no—
311M16—5422no—
312C2×C8—22no—
313C2×C8—2711no—
314C2×C8—2711no—
315C2×C8—5422no—
316C2×C8C22×D27111yescharacteristic
317C2×C8C22×D27111yescharacteristic
318C4⋊C4—2711no—
319C4⋊C4—2711no—
320C4⋊C4—5421no—
321C4⋊C4—5421no—
322C4⋊C4—5421no—
323C4⋊C4C22×D27111yescharacteristic
324C4⋊C4C22×D27111yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C42—5421no—
335C23—21no—
336C23—21no—
337C23—2711no—
338C23—2711no—
339C23—5422no—
340C23—5422no—
341C23—5421no—
342C23—10841no—
343C23—10842no—
344C23—10842no—
345C23—10842no—
346C23D4×D27111yescharacteristic
347Q8—5421no—
348Q8—5421no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—42no—
355D4—5421no—
356D4—5421no—
357D4—10842no—
358D4—10842no—
359D4—10842no—
360D4—10842no—
361D4—10842no—
362D4—10842no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—2711no—
368C2×C4—2712no—
369C2×C4—2712no—
370C2×C4—2711no—
371C2×C4—5422no—
372C2×C4—5422no—
373C2×C4—5422no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—10842no—
380C2×C4—10842no—
381C2×C4D4×D27111yescharacteristic
382C2×C4C23×D27111yescharacteristic
383C8—21no—
384C8—2712no—
385C8—5421no—
386C8D4×D27112yescomplemented
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—2712no—
396C22—5421no—
397C22—5421no—
398C22—5421no—
399C22—5421no—
400C22—5421no—
401C22—10844no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—21684no—
406C22#2257618412yescomplemented
407C22C2×D4×D27111yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—2712no—
412C4—5421no—
413C4—10841no—
414C4—10841no—
415C4—10841no—
416C4C2×D4×D27111yescharacteristic
417C4#41×D27111yescharacteristic
418C2—22no—
419C2—81no—
420C2—81no—
421C2—5422no—
422C2—10841no—
423C2—21681no—
424C2#121×D27111yescharacteristic
425C2C2×#2257618411yescharacteristic
426C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#466×C271C2
socleC22×C271C22×D4
hypercenter#466D271
hypocenterC271#3055
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C22×#16864637C2semidirect
3. #31774407C2semidirect
4. #121×D271C2semidirect
5. #31774149C2semidirect
6. #31774147C2semidirect
7. #31774144C2semidirect
8. #31774142C2semidirect
9. #31774131C2semidirect
10. #31774126C2semidirect
11. #31774124C2semidirect
12. #31773938C2semidirect
13. #31773900C2semidirect
14. #31773895C2semidirect
15. #466×C271C2semidirect
21. C2×#16864643C22semidirect
22. C2×#16864642C22semidirect
23. C2×#16864639C22semidirect
27. C2×#16864637C22semidirect
28. C2×#16864637C22semidirect
29. C22×C271⋊C8C22semidirect
31. #22576140C22semidirect
32. #22576139C22semidirect
38. #22576131C22semidirect
45. #22576093C22semidirect
46. #22576092C22semidirect
47. #22576091C22semidirect
48. #22576085C22semidirect
49. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. #22576081C22semidirect
53. #22576080C22semidirect
54. C2×D8×C271C22semidirect
55. C22×C8×C271C22semidirect
57. #121×C271C22semidirect
58. #107×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
91. #16864637D4semidirect
155. C271⋊C8C2×D4semidirect
172. C8×C271C2×D4semidirect
181. C22×C271AGL(1,ℤ/8)semidirect
213. C271#3055semidirect
224. #466D271semidirect
238. C2×D8C2×D271semidirect
245. C22×C8C2×D271semidirect
256. #121C2×D271semidirect
264. #107C2×D271semidirect
273. #102C2×D271semidirect
274. #102C2×D271semidirect
386. C8D4×D271semidirect
406. C22#22576184semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C22×#168646372
#317744072
#121×D2712
#317741492
#317741472
#317741442
#317741422
#317741312
#317741262
#317741242
#317739382
#317739002
#317738952
#466×C2712
#3055271