Finite Group Databasebeta

← #46846825

Subgroups — #46846825

Order 34688.

572 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2C2×#22576189C211yesmaximal, characteristic, complemented
3C2×#22576187C211yesmaximal, characteristic, complemented
4C2×#22576184C211yesmaximal, characteristic, complemented
5#31774446C211yesmaximal, characteristic, complemented
6#124×D271C211yesmaximal, characteristic, complemented
7#31774290C211yesmaximal, characteristic, complemented
8#31774288C211yesmaximal, characteristic, complemented
9#31774285C211yesmaximal, characteristic, complemented
10#31774283C211yesmaximal, characteristic, complemented
11#31774281C211yesmaximal, characteristic, complemented
12#31774279C211yesmaximal, characteristic, complemented
13#31774279C211yesmaximal, characteristic, complemented
14C4×#16864637C211yesmaximal, characteristic, complemented
15#495×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic
20#22576192—24no—
21#22576189—24no—
22#22576187—24no—
23#22576184—24no—
24C2×Dic1084C2211yescharacteristic, complemented
25C2×#16864644C2211yescharacteristic, complemented
26C2×#16864644C2211yescharacteristic, complemented
27C2×#16864643C2211yescharacteristic
28C2×#16864643C2211yescharacteristic, complemented
29C2×#16864642C2211yescharacteristic
30C2×#16864642C2211yescharacteristic, complemented
31C2×#16864641C2211yescharacteristic
32C2×#16864641C2211yescharacteristic, complemented
33C2×D2168C2211yescharacteristic
34C2×#16864639C2211yescharacteristic
35C2×#16864639C2211yescharacteristic
36C2×#16864637C2211yescharacteristic
37C2×#16864637C2211yescharacteristic
38#22576158C2211yescharacteristic, complemented
39#22576156C2211yescharacteristic, complemented
40#22576149C2211yescharacteristic
41#22576148C2211yescharacteristic
42#22576148C2211yescharacteristic, complemented
43#22576147C2211yescharacteristic
44#22576146—22no—
45#22576133—22no—
46(C22⋊D271)⋊C4—22no—
47C22⋊C4×D271—22no—
48C42×D271C2211yescharacteristic
49#22576070C2211yescharacteristic, complemented
50#22576070C2211yescharacteristic, complemented
51C4×C271⋊C8C2211yescharacteristic, complemented
52C2×Q16×C271C2211yescharacteristic, complemented
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57#124×C271C2211yescharacteristic, complemented
58C4×C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—22no—
62#16864653—22no—
63#16864653—22no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69C2×D1084—21no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—21no—
75D4×D271—22no—
76D4×D271—22no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271D412yes—
81C2×C4×D271C2311yescharacteristic
82Dic1084—22no—
83#16864644—22no—
84#16864644—22no—
85#16864643—22no—
86#16864643—22no—
87#16864642—22no—
88#16864642—22no—
89#16864641—22no—
90#16864641—22no—
91D2168—22no—
92#16864639—22no—
93#16864639—22no—
94#16864637D414yescomplemented
95#16864637D414yescomplemented
96C2×C271⋊C8C2311yescharacteristic
97C2×C271⋊C8C2311yescharacteristic
98Dic271⋊C4—22no—
99C22⋊Dic271—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103C4×Dic271D412yes—
104C4×Dic271C2311yescharacteristic
105C2×Q8×C271C2311yescharacteristic
106C2×Q8×C271C2311yescharacteristic
107C2×D4×C271C2311yescharacteristic
108C2×D4×C271C2311yescharacteristic
109Q16×C271—22no—
110SD16×C271—22no—
111SD16×C271—22no—
112D8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C22⋊C4×C271—22no—
116C22⋊C4×C271—22no—
117C42×C271C2311yescharacteristic
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—44no—
121C22×D271—42no—
122C22×D271C2×D411yescharacteristic
123Dic542—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128D1084—21no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—42no—
133C22⋊D271—44no—
134C22⋊D271—42no—
135C22⋊D271—42no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—42no—
139C2×Dic271C2×D412yes—
140C2×Dic271C2×D411yescharacteristic
141C4×D271—42no—
142C4×D271—42no—
143C4×D271—42no—
144C4×D271C2×D412yes—
145C4×D271C2×D412yes—
146C271⋊C8C2×D412yes—
147C271⋊C8C2×D412yes—
148C23×C271—21no—
149C23×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271C2×D412yes—
161C2×C4×C271C2411yescharacteristic
162C8×C271C2×D412yes—
163C8×C271C2×D412yes—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—42no—
167C22×C271—42no—
168C22×C271C22×D411yescharacteristic
169C2×D271—22no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—42no—
173C2×D271—42no—
174C2×D271—84no—
175C2×D271#12712yes—
176C4×C271—22no—
177C4×C271—41no—
178C4×C271—41no—
179C4×C271C22×D411yescharacteristic
180C4×C271C22×D411yescharacteristic
181Dic271—22no—
182Dic271—41no—
183Dic271—41no—
184Dic271#12712yes—
185C2×C271—81no—
186C2×C271—81no—
187C2×C271C2×#12711yescharacteristic
188C2×C271#57512yes—
189D271—22no—
190D271—81no—
191D271—81no—
192C271#315311yescharacteristic, complemented, Sylow
193#3153—2711nomaximal, Sylow
194C2×#137—2711no—
195C2×#137—2711no—
196C2×AGL(1,ℤ/8)—2711no—
197C2×AGL(1,ℤ/8)—2711no—
198#532—2711no—
199C2×#124—2711no—
200#497—2711no—
201#497—2711no—
202#496—2711no—
203#496—2711no—
204#495—2711no—
205#495—2711no—
206#495—2711no—
207#495D27111yescharacteristic, complemented
208C4×M16—2711no—
209C2×#41—2711no—
210C2×#41—2711no—
211C22×Q8—2711no—
212C22×D4—2711no—
213#137—5424no—
214#137—5424no—
215AGL(1,ℤ/8)—5424no—
216AGL(1,ℤ/8)—5424no—
217C2×Q16—2711no—
218C2×Q16—2711no—
219C2×Q16—2711no—
220C2×Q16C2×D27111yescharacteristic, complemented
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16—2711no—
226C2×SD16—2711no—
227C2×SD16C2×D27111yescharacteristic
228C2×SD16C2×D27111yescharacteristic, complemented
229C2×D8—2711no—
230C2×D8—2711no—
231C2×D8—2711no—
232C2×D8C2×D27111yescharacteristic
233C2×M16—2711no—
234C2×M16—2711no—
235C4⋊Q8—2711no—
236#127—2711no—
237#124—2711no—
238#124—2711no—
239#124—2711no—
240#124—2711no—
241#124—5422no—
242#124C2×D27111yescharacteristic
243#124C2×D27111yescharacteristic, complemented
244#121—5422no—
245C4×D4—5422no—
246C2×C22⋊C4—5422no—
247C2×C42—2711no—
248#97—2711no—
249#97—2711no—
250C4×C8—2711no—
251C4×C8C2×D27111yescharacteristic, complemented
252C24—5421no—
253#41—5422no—
254#41—5422no—
255#41—5422no—
256#41—5422no—
257C2×Q8—2711no—
258C2×Q8—2711no—
259C2×Q8—5421no—
260C2×Q8—5422no—
261C2×Q8—5422no—
262C2×Q8C22×D27111yescharacteristic
263C2×Q8C22×D27111yescharacteristic
264C2×D4—2711no—
265C2×D4—2711no—
266C2×D4—5421no—
267C2×D4—5422no—
268C2×D4—5422no—
269C2×D4—5422no—
270C2×D4—5421no—
271C2×D4—5421no—
272C2×D4C22×D27111yescharacteristic
273C2×D4C22×D27111yescharacteristic
274C22×C4—2711no—
275C22×C4—2712no—
276C22×C4—5421no—
277C22×C4—5421no—
278C22×C4—5421no—
279Q16—22no—
280Q16—5422no—
281Q16—5422no—
282Q16—5422no—
283SD16—22no—
284SD16—22no—
285SD16—5422no—
286SD16—5422no—
287SD16—5422no—
288SD16—5422no—
289SD16—5422no—
290SD16—5422no—
291D8—22no—
292D8—5422no—
293D8—5422no—
294D8—5422no—
295M16—2714no—
296M16—2714no—
297C2×C8—2711no—
298C2×C8—2711no—
299C2×C8C22×D27111yescharacteristic
300C2×C8C22×D27111yescharacteristic
301C4⋊C4—5422no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—5422no—
305C22⋊C4—5422no—
306C22⋊C4—5422no—
307C22⋊C4—5422no—
308C42—2712no—
309C42—2711no—
310C42C22×D27111yescharacteristic
311C23—21no—
312C23—21no—
313C23—2711no—
314C23—5421no—
315C23—5421no—
316C23—10842no—
317C23—10844no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—5421no—
323Q8—5421no—
324Q8—5421no—
325Q8—5421no—
326Q8—10842no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—5421no—
332D4—5421no—
333D4—5421no—
334D4—5421no—
335D4—10842no—
336D4—10842no—
337D4—10844no—
338D4—10842no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—2711no—
342C2×C4—2712no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—10842no—
348C2×C4—10842no—
349C2×C4—10842no—
350C2×C4—10842no—
351C2×C4D4×D27112yes—
352C2×C4C23×D27111yescharacteristic
353C8—2712no—
354C8—2712no—
355C8D4×D27112yes—
356C8D4×D27112yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—2712no—
362C22—5422no—
363C22—10841no—
364C22—10841no—
365C22—10842no—
366C22—10842no—
367C22—21684no—
368C22C2×D4×D27111yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—2712no—
373C4—5422no—
374C4—10841no—
375C4—10841no—
376C4C2×D4×D27111yescharacteristic
377C4C2×D4×D27111yescharacteristic
378C2—81no—
379C2—81no—
380C2—5422no—
381C2—21681no—
382C2—21681no—
383C2#127×D27111yescharacteristic
384C2#3177469412yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#495×C271C2
socleC22×C271C22×D4
hypercenter#495D271
hypocenterC271#3153
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. C2×#22576189C2semidirect
3. C2×#22576187C2semidirect
4. C2×#22576184C2semidirect
5. #31774446C2semidirect
6. #124×D271C2semidirect
7. #31774290C2semidirect
8. #31774288C2semidirect
9. #31774285C2semidirect
10. #31774283C2semidirect
11. #31774281C2semidirect
12. #31774279C2semidirect
13. #31774279C2semidirect
14. C4×#16864637C2semidirect
15. #495×C271C2semidirect
24. C2×Dic1084C22semidirect
25. C2×#16864644C22semidirect
26. C2×#16864644C22semidirect
28. C2×#16864643C22semidirect
30. C2×#16864642C22semidirect
32. C2×#16864641C22semidirect
38. #22576158C22semidirect
39. #22576156C22semidirect
42. #22576148C22semidirect
49. #22576070C22semidirect
50. #22576070C22semidirect
51. C4×C271⋊C8C22semidirect
52. C2×Q16×C271C22semidirect
54. C2×SD16×C271C22semidirect
57. #124×C271C22semidirect
58. C4×C8×C271C22semidirect
94. #16864637D4semidirect
95. #16864637D4semidirect
192. C271#3153semidirect
207. #495D271semidirect
220. C2×Q16C2×D271semidirect
228. C2×SD16C2×D271semidirect
243. #124C2×D271semidirect
251. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×#225761892
C2×#225761872
C2×#225761842
#317744462
#124×D2712
#317742902
#317742882
#317742852
#317742832
#317742812
#317742792
C4×#168646372
#495×C2712
#3153271