Finite Group Databasebeta

← #46847977

Subgroups — #46847977

Order 34688.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576185C211yesmaximal, characteristic, complemented
3#31774566C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic, complemented
5#31774272C211yesmaximal, characteristic, complemented
6#31774149C211yesmaximal, characteristic, complemented
7#31774148C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774091C211yesmaximal, characteristic, complemented
10#31774045C211yesmaximal, characteristic, complemented
11#31774033C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14#31773972C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20#22576187—24no—
21#22576185—24no—
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic, complemented
24C2×D2168C2211yescharacteristic
25C2×D2168C2211yescharacteristic, complemented
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic, complemented
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864638C2211yescharacteristic, complemented
30C2×#16864637C2211yescharacteristic, complemented
31#22576156—21no—
32#22576156C2211yescharacteristic
33#22576156C2211yescharacteristic, complemented
34#22576135C2211yescharacteristic
35#22576134—21no—
36#22576134C2211yescharacteristic, complemented
37#22576133—21no—
38#22576131—21no—
39#22576131C2211yescharacteristic, complemented
40#22576130—21no—
41#22576128—21no—
42Q8×Dic271C2211yescharacteristic, complemented
43#22576122C2211yescharacteristic
44C4×C22⋊D271—21no—
45C4×C22⋊D271C2211yescharacteristic
46#22576091C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C8×Dic271C2211yescharacteristic, complemented
54C2×#41×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×D8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864655—22no—
64#16864655—41no—
65C2×D1084—22no—
66C2×D1084—21no—
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084—41no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C2×C22⋊D271D411yescharacteristic
81D4×D271—22no—
82D4×D271—41no—
83D4×D271—44no—
84C2×C4×D271—21no—
85C2×C4×D271—21no—
86C2×C4×D271C2311yescharacteristic
87#16864641—22no—
88#16864641D414yescomplemented
89D2168—22no—
90D2168—41no—
91#16864639—22no—
92#16864639—22no—
93#16864639—22no—
94#16864638—22no—
95#16864637—22no—
96C2×C271⋊C8C2311yescharacteristic
97C2×C271⋊C8C2311yescharacteristic
98C4⋊Dic271—21no—
99C4⋊Dic271C2311yescharacteristic
100Dic271⋊C4D411yescharacteristic
101C22⋊Dic271—21no—
102C22⋊Dic271D411yescharacteristic
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4D411yescharacteristic
107C4×Dic271—21no—
108C4×Dic271C2311yescharacteristic
109#41×C271—22no—
110#41×C271—41no—
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271—21no—
113C2×D4×C271—21no—
114C2×D4×C271C2311yescharacteristic
115C2×D4×C271C2311yescharacteristic
116C22×C4×C271—21no—
117C22×C4×C271C2311yescharacteristic
118SD16×C271—22no—
119D8×C271—41no—
120C2×C8×C271C2311yescharacteristic
121C2×C8×C271C2311yescharacteristic
122C4⋊C4×C271C2311yescharacteristic
123C22⋊C4×C271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—42no—
131C22×D271—81no—
132C22×D271—81no—
133C22×D271C2×D411yescharacteristic
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—42no—
140D1084—42no—
141D1084—42no—
142D1084C2×D412yes—
143C22⋊D271—22no—
144C22⋊D271—42no—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—81no—
148C22⋊D271—81no—
149C22⋊D271—81no—
150C22⋊D271—81no—
151C2×Dic271—21no—
152C2×Dic271C2×D411yescharacteristic
153C2×Dic271C2×D411yescharacteristic
154C2×Dic271C2×D411yescharacteristic
155C4×D271—21no—
156C4×D271—21no—
157C4×D271—42no—
158C4×D271—42no—
159C271⋊C8—21no—
160C271⋊C8C2×D412yescomplemented
161C23×C271—21no—
162C23×C271—21no—
163C23×C271C2×D411yescharacteristic
164Q8×C271—21no—
165Q8×C271C2×D412yescomplemented
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—21no—
170D4×C271—42no—
171D4×C271—42no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—42no—
177C2×C4×C271C2×D411yescharacteristic
178C2×C4×C271C2411yescharacteristic
179C8×C271—21no—
180C8×C271—21no—
181C22×C271—21no—
182C22×C271—21no—
183C22×C271—21no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271C22×D411yescharacteristic
191C2×D271—21no—
192C2×D271—22no—
193C2×D271—21no—
194C2×D271—21no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—41no—
203C2×D271—84no—
204C2×D271—84no—
205C4×C271—21no—
206C4×C271—22no—
207C4×C271—41no—
208C4×C271C22×D411yescharacteristic
209C4×C271H(5,𝔽2)11yescharacteristic
210Dic271—21no—
211Dic271—21no—
212Dic271—41no—
213Dic271AGL(1,ℤ/8)12yescomplemented
214C2×C271—41no—
215C2×C271—81no—
216C2×C271—81no—
217C2×C271D4211yescharacteristic
218C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C271#57611yescharacteristic
220D271—41no—
221D271—42no—
222D271—81no—
223D271—81no—
224C271#327411yescharacteristic, complemented, Sylow
225#3274—2711nomaximal, Sylow
226C2×AGL(1,ℤ/8)—2711no—
227C2×AGL(1,ℤ/8)—2711no—
228#550—2711no—
229D42—2711no—
230#493—2711no—
231#469—2711no—
232#469—2711no—
233#460—2711no—
234#459—2711no—
235#449—2711no—
236#449D27111yescharacteristic, complemented
237#447—2711no—
238#447—2711no—
239C4×SD16—2711no—
240#436—2711no—
241C2×#41—2711no—
242C2×#41C2×D27111yescharacteristic, complemented
243C22×D4—2711no—
244C22×D4—2711no—
245C22×D4—5421no—
246AGL(1,ℤ/8)—5424no—
247AGL(1,ℤ/8)—5424no—
248C2×SD16—2711no—
249C2×SD16—2711no—
250C2×SD16C2×D27111yescharacteristic, complemented
251C2×D8—2711no—
252C2×D8—2711no—
253C2×D8—2711no—
254C2×D8—2711no—
255C2×D8—2711no—
256C2×D8C2×D27111yescharacteristic, complemented
257C2×M16—2711no—
258C2×M16—2711no—
259#127—2711no—
260#127—2711no—
261#127—5421no—
262#121—2711no—
263#121—2711no—
264#121—2711no—
265#121—5421no—
266#121—5421no—
267#121—5421no—
268#121C2×D27111yescharacteristic
269#120—5421no—
270#120—5421no—
271C4×Q8—2711no—
272C4×D4—2711no—
273C4×D4—2711no—
274C4×D4—5421no—
275#106—2711no—
276C4⋊C8—2711no—
277#103C2×D27111yescharacteristic, complemented
278#102—2711no—
279#102—2711no—
280#102—2711no—
281#102—2711no—
282#102C2×D27111yescharacteristic, complemented
283C22⋊C8—2711no—
284C22⋊C8C2×D27111yescharacteristic, complemented
285C4×C8—2711no—
286C24—5421no—
287C24—5421no—
288#41—22no—
289#41—41no—
290#41—5422no—
291#41—10841no—
292C2×Q8C22×D27111yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2711no—
296C2×D4—2711no—
297C2×D4—2711no—
298C2×D4—2711no—
299C2×D4—5422no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—5421no—
304C2×D4—5421no—
305C2×D4—5421no—
306C2×D4—5421no—
307C2×D4—5422no—
308C2×D4—5421no—
309C2×D4—5421no—
310C2×D4—5421no—
311C2×D4—10841no—
312C2×D4—10841no—
313C2×D4—10844no—
314C2×D4C22×D27111yescharacteristic
315C2×D4C22×D27111yescharacteristic
316C22×C4—21no—
317C22×C4—2711no—
318C22×C4—5421no—
319C22×C4—5421no—
320C22×C4C22×D27111yescharacteristic
321SD16—22no—
322SD16—2714no—
323SD16—5422no—
324D8—41no—
325D8—5422no—
326D8—5422no—
327D8—5422no—
328D8—5422no—
329D8—10841no—
330M16—5422no—
331M16—5422no—
332C2×C8—2711no—
333C2×C8—2711no—
334C2×C8C22×D27111yescharacteristic
335C2×C8C22×D27111yescharacteristic
336C4⋊C4—2711no—
337C4⋊C4—2711no—
338C4⋊C4—5421no—
339C4⋊C4C22×D27111yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—2711no—
342C22⋊C4—2711no—
343C22⋊C4—5421no—
344C22⋊C4—5421no—
345C22⋊C4—5421no—
346C22⋊C4—5421no—
347C42—2711no—
348C42—5421no—
349C23—21no—
350C23—21no—
351C23—2711no—
352C23—5421no—
353C23—5421no—
354C23—5421no—
355C23—10842no—
356C23—10842no—
357C23—10842no—
358C23—10842no—
359C23—21681no—
360C23—21681no—
361C23D4×D27111yescharacteristic
362Q8—21no—
363Q8D4×D27112yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—2712no—
371D4—5422no—
372D4—5421no—
373D4—5421no—
374D4—5421no—
375D4—5421no—
376D4—5421no—
377D4—10842no—
378D4—10842no—
379D4—10842no—
380D4—10842no—
381D4—10842no—
382D4—10842no—
383D4—21681no—
384D4—21681no—
385D4—21681no—
386D4—21681no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—2711no—
393C2×C4—2711no—
394C2×C4—2711no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—10842no—
399C2×C4—10842no—
400C2×C4D4×D27111yescharacteristic
401C2×C4C23×D27111yescharacteristic
402C8—21no—
403C8—21no—
404C8—2712no—
405C8—5421no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—5421no—
416C22—5422no—
417C22—5421no—
418C22—5421no—
419C22—10841no—
420C22—10841no—
421C22—10841no—
422C22—10841no—
423C22—10841no—
424C22—10841no—
425C22—10841no—
426C22—10841no—
427C22—21684no—
428C22—21684no—
429C22C2×D4×D27111yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—2712no—
434C4—5421no—
435C4—5421no—
436C4—10841no—
437C4C2×D4×D27111yescharacteristic
438C4#2257620511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—10841no—
443C2—10842no—
444C2—21681no—
445C2—21681no—
446C2#3177452911yescharacteristic
447C2AGL(1,ℤ/8)×D27111yescharacteristic
448C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576185C2semidirect
3. #31774566C2semidirect
4. #31774531C2semidirect
5. #31774272C2semidirect
6. #31774149C2semidirect
7. #31774148C2semidirect
8. #31774095C2semidirect
9. #31774091C2semidirect
10. #31774045C2semidirect
11. #31774033C2semidirect
12. #31774029C2semidirect
13. #31773982C2semidirect
14. #31773972C2semidirect
15. #449×C271C2semidirect
16. C2×#16864655C22semidirect
23. C2×#16864641C22semidirect
25. C2×D2168C22semidirect
27. C2×#16864639C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864638C22semidirect
30. C2×#16864637C22semidirect
33. #22576156C22semidirect
36. #22576134C22semidirect
39. #22576131C22semidirect
42. Q8×Dic271C22semidirect
46. #22576091C22semidirect
47. Dic271⋊C8C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576080C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C8×Dic271C22semidirect
54. C2×#41×C271C22semidirect
55. C2×SD16×C271C22semidirect
56. C2×D8×C271C22semidirect
58. #103×C271C22semidirect
59. #102×C271C22semidirect
60. C22⋊C8×C271C22semidirect
88. #16864641D4semidirect
160. C271⋊C8C2×D4semidirect
165. Q8×C271C2×D4semidirect
213. Dic271AGL(1,ℤ/8)semidirect
224. C271#3274semidirect
236. #449D271semidirect
242. C2×#41C2×D271semidirect
250. C2×SD16C2×D271semidirect
256. C2×D8C2×D271semidirect
277. #103C2×D271semidirect
282. #102C2×D271semidirect
284. C22⋊C8C2×D271semidirect
363. Q8D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761852
#317745662
#317745312
#317742722
#317741492
#317741482
#317740952
#317740912
#317740452
#317740332
#317740292
#317739822
#317739722
#449×C2712
#3274271