Finite Group Databasebeta

← #46847894

Subgroups — #46847894

Order 34688.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576188C211yesmaximal, characteristic, complemented
2C2×#22576182C211yesmaximal, characteristic, complemented
3#31774552C211yesmaximal, characteristic, complemented
4#31774543C211yesmaximal, characteristic, complemented
5#31774291C211yesmaximal, characteristic, complemented
6#31774155C211yesmaximal, characteristic, complemented
7#31774126C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774103C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774061C211yesmaximal, characteristic, complemented
12#31774050C211yesmaximal, characteristic, complemented
13#31774044C211yesmaximal, characteristic, complemented
14#31774010C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×#16864654C2211yescharacteristic, complemented
19C22×D1084C2211yescharacteristic
20#22576188—24no—
21#22576182—24no—
22C2×Dic1084C2211yescharacteristic, complemented
23C2×Dic1084C2211yescharacteristic, complemented
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic
26C2×#16864643C2211yescharacteristic
27C2×#16864643C2211yescharacteristic
28C2×#16864643C2211yescharacteristic, complemented
29C2×D2168C2211yescharacteristic
30C2×D2168C2211yescharacteristic
31#22576160C2211yescharacteristic, complemented
32#22576148C2211yescharacteristic
33#22576145—21no—
34#22576142—21no—
35#22576142—21no—
36#22576139C2211yescharacteristic
37#22576139C2211yescharacteristic
38#22576139C2211yescharacteristic, complemented
39#22576134C2211yescharacteristic
40#22576133—21no—
41#22576131—21no—
42#22576128—21no—
43C4×D1084C2211yescharacteristic
44C4×D1084C2211yescharacteristic, complemented
45C4⋊C4×D271—21no—
46C22⋊C4×D271—21no—
47#22576092C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576085C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic
52#22576082C2211yescharacteristic
53#22576082C2211yescharacteristic, complemented
54C2×M16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic, complemented
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C22⋊C8×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864654—22no—
63#16864654—22no—
64#16864654—22no—
65#16864654—41no—
66#16864654—41no—
67#16864654—41no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C2×Dic542C2311yescharacteristic
71C2×D1084—22no—
72C2×D1084—21no—
73C2×D1084—41no—
74C2×D1084C2311yescharacteristic
75C2×D1084C2311yescharacteristic
76C2×D1084C2311yescharacteristic
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271—21no—
84C2×C4×D271—21no—
85C2×C4×D271—21no—
86Dic1084—22no—
87Dic1084—41no—
88#16864643—22no—
89#16864643—22no—
90#16864643—22no—
91#16864643—22no—
92#16864643—41no—
93D2168—22no—
94D2168D414yescomplemented
95C4⋊Dic271—21no—
96C4⋊Dic271—21no—
97C4⋊Dic271C2311yescharacteristic
98C4⋊Dic271C2311yescharacteristic
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104C22⋊Dic271—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109(C2×D271)⋊C4—21no—
110C2×D4×C271D411yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113M16×C271—22no—
114C2×C8×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—81no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—21no—
133Dic542—21no—
134Dic542—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—42no—
140D1084C2×D412yes—
141D1084C2×D412yes—
142C22⋊D271—42no—
143C22⋊D271—42no—
144C22⋊D271—42no—
145C22⋊D271—81no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C4×D271—42no—
152C4×D271—42no—
153C4×D271—42no—
154C4×D271—42no—
155C4×D271—81no—
156C23×C271C2×D411yescharacteristic
157C23×C271C2×D411yescharacteristic
158D4×C271—41no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2411yescharacteristic
168C8×C271—21no—
169C8×C271—21no—
170C8×C271—21no—
171C8×C271C2×D412yes—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C271C22×D411yescharacteristic
179C2×D271—22no—
180C2×D271—22no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—84no—
189C4×C271—21no—
190C4×C271—21no—
191C4×C271—21no—
192C4×C271—21no—
193C4×C271C22×D411yescharacteristic
194C4×C271H(5,𝔽2)11yescharacteristic
195Dic271—41no—
196Dic271—41no—
197Dic271—41no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—41no—
201C2×C271—41no—
202C2×C271D4211yescharacteristic
203C2×C271#57511yescharacteristic
204C2×C271#57711yescharacteristic
205D271—42no—
206D271—42no—
207D271—81no—
208C271#326711yescharacteristic, complemented, Sylow
209#3267—2711nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2711no—
211C2×#135—2711no—
212#547—2711no—
213#546—2711no—
214#497—2711no—
215#470—2711no—
216#465—2711no—
217#463—2711no—
218#461—2711no—
219#452—2711no—
220#452—2711no—
221#450—2711no—
222#449—2711no—
223#442—2711no—
224#435D27111yescharacteristic, complemented
225C2×#41—2711no—
226C2×#41—2711no—
227C2×#41—2711no—
228C22×D4—2711no—
229AGL(1,ℤ/8)—5424no—
230#135—5424no—
231C2×Q16—2711no—
232C2×Q16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×SD16—2711no—
238C2×D8—2711no—
239C2×D8—2711no—
240C2×M16C2×D27111yescharacteristic
241C22×C8C2×D27111yescharacteristic, complemented
242C4⋊Q8—2711no—
243#124—2711no—
244#123—5421no—
245#123—5421no—
246#123—5421no—
247C22⋊Q8—2711no—
248C22⋊Q8—2711no—
249C22⋊Q8—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#120—5421no—
254C4×D4—2711no—
255C4×D4—2711no—
256C4×D4C2×D27111yescharacteristic
257C2×C4⋊C4—5421no—
258C2×C22⋊C4—5421no—
259#106—2711no—
260C4⋊C8C2×D27111yescharacteristic, complemented
261#103—2711no—
262#103—2711no—
263#103—2711no—
264#102—2711no—
265#102—2711no—
266#102—2711no—
267C22⋊C8C2×D27111yescharacteristic
268C22⋊C8C2×D27111yescharacteristic, complemented
269#97C2×D27111yescharacteristic, complemented
270C24—5421no—
271#41—5422no—
272#41—5422no—
273#41—5422no—
274#41—10841no—
275#41—10841no—
276#41—10841no—
277C2×Q8—2711no—
278C2×Q8—2711no—
279C2×Q8—2711no—
280C2×D4—2711no—
281C2×D4—2711no—
282C2×D4—2711no—
283C2×D4—5422no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4—10841no—
290C2×D4D108411yescharacteristic
291C22×C4—5421no—
292C22×C4—5421no—
293C22×C4—5421no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4C22×D27111yescharacteristic
297C22×C4C22×D27111yescharacteristic
298Q16—5422no—
299Q16—10841no—
300SD16—5422no—
301SD16—5422no—
302SD16—5422no—
303SD16—5422no—
304SD16—10841no—
305D8—2714no—
306D8—5422no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D27111yescharacteristic
310C2×C8C22×D27111yescharacteristic
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—2711no—
314C4⋊C4—2711no—
315C4⋊C4—5421no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C4⋊C4—5421no—
320C4⋊C4D108411yescharacteristic
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4D108411yescharacteristic
330C22⋊C4D108411yescharacteristic
331C42C22×D27111yescharacteristic
332C23—5421no—
333C23—5421no—
334C23—5421no—
335C23—10842no—
336C23—10842no—
337C23—21681no—
338C23C2×D108411yescharacteristic
339C23C2×D108411yescharacteristic
340Q8—5421no—
341Q8—5421no—
342Q8—5421no—
343Q8—5421no—
344Q8—5421no—
345Q8—5421no—
346D4—41no—
347D4—2712no—
348D4—2712no—
349D4—5421no—
350D4—5421no—
351D4—5421no—
352D4—5421no—
353D4—10842no—
354D4—10842no—
355D4—10842no—
356D4—10842no—
357D4—21681no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—10842no—
368C2×C4—10842no—
369C2×C4—10842no—
370C2×C4—10842no—
371C2×C4—21681no—
372C2×C4C2×D108411yescharacteristic
373C2×C4C2×D108411yescharacteristic
374C2×C4C2×D108411yescharacteristic
375C2×C4C2×D108411yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D27112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—5422no—
388C22—5422no—
389C22—10841no—
390C22—10841no—
391C22—10841no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22—10841no—
396C22—21684no—
397C22C22×D108411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—10841no—
403C4—10841no—
404C4—10841no—
405C4—10841no—
406C4—10841no—
407C4C2×D4×D27111yescharacteristic
408C4#2257620611yescharacteristic
409C2—41no—
410C2—41no—
411C2—10842no—
412C2—10842no—
413C2—21681no—
414C2D4×D108411yescharacteristic
415C2#3177469711yescharacteristic
416C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
hypercenter#435D271
hypocenterC271#3267
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×#22576182C2semidirect
3. #31774552C2semidirect
4. #31774543C2semidirect
5. #31774291C2semidirect
6. #31774155C2semidirect
7. #31774126C2semidirect
8. #31774113C2semidirect
9. #31774103C2semidirect
10. #31774061C2semidirect
11. #31774061C2semidirect
12. #31774050C2semidirect
13. #31774044C2semidirect
14. #31774010C2semidirect
15. #435×C271C2semidirect
18. C2×#16864654C22semidirect
22. C2×Dic1084C22semidirect
23. C2×Dic1084C22semidirect
28. C2×#16864643C22semidirect
31. #22576160C22semidirect
38. #22576139C22semidirect
44. C4×D1084C22semidirect
47. #22576092C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
50. #22576085C22semidirect
53. #22576082C22semidirect
55. C22×C8×C271C22semidirect
57. C4⋊C8×C271C22semidirect
59. C22⋊C8×C271C22semidirect
60. #97×C271C22semidirect
94. D2168D4semidirect
208. C271#3267semidirect
224. #435D271semidirect
241. C22×C8C2×D271semidirect
260. C4⋊C8C2×D271semidirect
268. C22⋊C8C2×D271semidirect
269. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C2×#225761822
#317745522
#317745432
#317742912
#317741552
#317741262
#317741132
#317741032
#317740612
#317740502
#317740442
#317740102
#435×C2712
#3267271