Finite Group Databasebeta

← #46847897

Subgroups — #46847897

Order 34688.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3#31774566C211yesmaximal, characteristic, complemented
4#31774544C211yesmaximal, characteristic, complemented
5#31774285C211yesmaximal, characteristic, complemented
6#31774142C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774117C211yesmaximal, characteristic, complemented
9#31774091C211yesmaximal, characteristic, complemented
10#31774062C211yesmaximal, characteristic, complemented
11#31774044C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31774000C211yesmaximal, characteristic, complemented
14#31773965C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576187—24no—
20#22576180—24no—
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic
25C2×D2168C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864637C2211yescharacteristic
29C22×C271⋊C8C2211yescharacteristic, complemented
30#22576156—21no—
31#22576156C2211yescharacteristic, complemented
32#22576147C2211yescharacteristic
33#22576142—21no—
34#22576140C2211yescharacteristic
35#22576134—21no—
36#22576134C2211yescharacteristic, complemented
37#22576131—21no—
38#22576131—21no—
39#22576131C2211yescharacteristic
40#22576128—21no—
41Q8×Dic271C2211yescharacteristic, complemented
42#22576122C2211yescharacteristic
43C4×C22⋊D271—21no—
44C4×C22⋊D271C2211yescharacteristic
45C22⋊C4×D271—21no—
46#22576093C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576081C2211yescharacteristic
51#22576080C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53#22576070C2211yescharacteristic, complemented
54C2×#41×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic
56C2×D8×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864655—22no—
63#16864655—41no—
64#16864654—22no—
65#16864654—41no—
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—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271D411yescharacteristic
77D4×D271—22no—
78D4×D271—41no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83#16864644—22no—
84#16864643—41no—
85#16864642—22no—
86#16864641D414yescomplemented
87D2168—22no—
88#16864639—22no—
89#16864639—22no—
90#16864637—22no—
91C2×C271⋊C8—22no—
92C2×C271⋊C8C2311yescharacteristic
93C2×C271⋊C8C2311yescharacteristic
94C4⋊Dic271—21no—
95C4⋊Dic271—21no—
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4D411yescharacteristic
98C22⋊Dic271—21no—
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271D411yescharacteristic
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271C2311yescharacteristic
108#41×C271—22no—
109#41×C271—41no—
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271—21no—
116C22×C4×C271C2311yescharacteristic
117SD16×C271—22no—
118D8×C271—41no—
119C2×C8×C271C2311yescharacteristic
120C2×C8×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542—21no—
131D1084—21no—
132D1084—21no—
133D1084—21no—
134D1084—42no—
135D1084C2×D412yes—
136C22⋊D271—41no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C22⋊D271—42no—
140C22⋊D271—81no—
141C22⋊D271—81no—
142C22⋊D271—81no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—42no—
151C4×D271—42no—
152C4×D271—81no—
153C271⋊C8—21no—
154C271⋊C8C2×D412yes—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271C2×D411yescharacteristic
158Q8×C271—21no—
159Q8×C271C2×D412yes—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—21no—
164D4×C271—42no—
165D4×C271—81no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—42no—
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271—21no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—22no—
187C2×D271—21no—
188C2×D271—21no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C4×C271—21no—
196C4×C271—22no—
197C4×C271—41no—
198C4×C271C22×D411yescharacteristic
199C4×C271H(5,𝔽2)11yescharacteristic
200Dic271—21no—
201Dic271—21no—
202Dic271—21no—
203Dic271—41no—
204Dic271—41no—
205C2×C271—41no—
206C2×C271—81no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271#57511yescharacteristic
210C2×C271#57611yescharacteristic
211D271—41no—
212D271—42no—
213D271—81no—
214C271#326811yescharacteristic, complemented, Sylow
215#3268—2711nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2711no—
217C2×#135—2711no—
218#550—2711no—
219#546—2711no—
220#496—2711no—
221#468—2711no—
222#466—2711no—
223#464—2711no—
224#459—2711no—
225#452—2711no—
226#449—2711no—
227#449D27111yescharacteristic, complemented
228#447—2711no—
229#440—2711no—
230#435—2711no—
231C2×#41—2711no—
232C2×#41—2711no—
233C2×#41C2×D27111yescharacteristic, complemented
234C22×D4—2711no—
235AGL(1,ℤ/8)—5424no—
236#135—5424no—
237C2×Q16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16C2×D27111yescharacteristic
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8C2×D27111yescharacteristic
246C2×M16—2711no—
247C22×C8—2711no—
248#127—2711no—
249#127—5421no—
250#124—2711no—
251#123—5421no—
252C22⋊Q8—2711no—
253#121—2711no—
254#121—2711no—
255#121—5421no—
256#121—5421no—
257#121—5421no—
258#121C2×D27111yescharacteristic
259#120—5421no—
260C4×Q8—2711no—
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4—5421no—
264C2×C22⋊C4—5421no—
265#107—2711no—
266C4⋊C8—2711no—
267#103—2711no—
268#103C2×D27111yescharacteristic, complemented
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102C2×D27111yescharacteristic, complemented
273C22⋊C8—2711no—
274C22⋊C8C2×D27111yescharacteristic, complemented
275#97—2711no—
276C24—5421no—
277#41—22no—
278#41—41no—
279#41—5422no—
280#41—5422no—
281#41—10841no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—2711no—
290C2×D4—5421no—
291C2×D4—5421no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5422no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—10841no—
299C2×D4C22×D27111yescharacteristic
300C2×D4C22×D27111yescharacteristic
301C22×C4—21no—
302C22×C4—2711no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4C22×D27111yescharacteristic
307Q16—5422no—
308SD16—22no—
309SD16—2714no—
310SD16—5422no—
311SD16—10841no—
312D8—41no—
313D8—5422no—
314D8—5422no—
315D8—5422no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C4⋊C4C22×D27111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2711no—
329C22⋊C4—2711no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C42—2711no—
337C42—5421no—
338C23—21no—
339C23—21no—
340C23—2711no—
341C23—5421no—
342C23—5421no—
343C23—10842no—
344C23—10842no—
345C23—21681no—
346C23D4×D27111yescharacteristic
347Q8—21no—
348Q8—5421no—
349Q8—5421no—
350Q8D4×D27112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—2712no—
358D4—5421no—
359D4—5421no—
360D4—5421no—
361D4—10842no—
362D4—10841no—
363D4—10842no—
364D4—10842no—
365D4—10842no—
366D4—21681no—
367D4—21681no—
368D4—21681no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2711no—
375C2×C4—2711no—
376C2×C4—2711no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—10842no—
382C2×C4—10842no—
383C2×C4—21681no—
384C2×C4D4×D27111yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—21no—
387C8—21no—
388C8—2712no—
389C8—5421no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5421no—
400C22—5422no—
401C22—5421no—
402C22—5421no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—21684no—
409C22C2×D4×D27111yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—5421no—
414C4—5421no—
415C4—5421no—
416C4—10841no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620511yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—10841no—
424C2—10842no—
425C2—21681no—
426C2#3177452911yescharacteristic
427C2#3177469411yescharacteristic
428C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3268
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×#22576180C2semidirect
3. #31774566C2semidirect
4. #31774544C2semidirect
5. #31774285C2semidirect
6. #31774142C2semidirect
7. #31774131C2semidirect
8. #31774117C2semidirect
9. #31774091C2semidirect
10. #31774062C2semidirect
11. #31774044C2semidirect
12. #31774029C2semidirect
13. #31774000C2semidirect
14. #31773965C2semidirect
15. #449×C271C2semidirect
21. C2×#16864644C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864642C22semidirect
29. C22×C271⋊C8C22semidirect
31. #22576156C22semidirect
36. #22576134C22semidirect
41. Q8×Dic271C22semidirect
46. #22576093C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
51. #22576080C22semidirect
53. #22576070C22semidirect
54. C2×#41×C271C22semidirect
58. #103×C271C22semidirect
59. #102×C271C22semidirect
60. C22⋊C8×C271C22semidirect
86. #16864641D4semidirect
214. C271#3268semidirect
227. #449D271semidirect
233. C2×#41C2×D271semidirect
268. #103C2×D271semidirect
272. #102C2×D271semidirect
274. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761802
#317745662
#317745442
#317742852
#317741422
#317741312
#317741172
#317740912
#317740622
#317740442
#317740292
#317740002
#317739652
#449×C2712
#3268271