Finite Group Databasebeta

← #46846839

Subgroups — #46846839

Order 34688.

564 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×#22576187C211yesmaximal, characteristic, complemented
3C2×#22576183C211yesmaximal, characteristic, complemented
4C2×SD16×D271C211yesmaximal, characteristic, complemented
5#31774472C211yesmaximal, characteristic, complemented
6#31774465C211yesmaximal, characteristic, complemented
7#31774290C211yesmaximal, characteristic, complemented
8#31774286C211yesmaximal, characteristic, complemented
9#31774283C211yesmaximal, characteristic, complemented
10#31774282C211yesmaximal, characteristic, complemented
11#31774279C211yesmaximal, characteristic, complemented
12#31774271C211yesmaximal, characteristic, complemented
13#31774268C211yesmaximal, characteristic, complemented
14#31773854C211yesmaximal, characteristic, complemented
15#497×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic
20#22576192—24no—
21#22576187—24no—
22#22576183—24no—
23C2×#16864644C2211yescharacteristic, complemented
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864642C2211yescharacteristic
27C2×#16864641C2211yescharacteristic
28C2×#16864641C2211yescharacteristic
29C2×#16864641C2211yescharacteristic, complemented
30C2×#16864641C2211yescharacteristic, complemented
31C2×#16864641C2211yescharacteristic, complemented
32SD16×D271—24no—
33C2×D2168C2211yescharacteristic
34C2×D2168C2211yescharacteristic, complemented
35C2×#16864639C2211yescharacteristic
36C2×#16864637C2211yescharacteristic
37C2×C8×D271C2211yescharacteristic
38#22576158C2211yescharacteristic, complemented
39#22576157C2211yescharacteristic
40#22576156C2211yescharacteristic
41#22576149C2211yescharacteristic
42#22576149C2211yescharacteristic, complemented
43#22576148C2211yescharacteristic, complemented
44#22576145—22no—
45#22576138—22no—
46#22576133—22no—
47#22576130—22no—
48#22576117C2211yescharacteristic
49#22576070C2211yescharacteristic, complemented
50C4×C271⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×Q16×C271C2211yescharacteristic, complemented
53C2×Q16×C271C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C2×SD16×C271C2211yescharacteristic
56C4⋊Q8×C271C2211yescharacteristic, complemented
57#124×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—22no—
62#16864655—22no—
63#16864655—22no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66Q8×D271—22no—
67Q8×D271—22no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084—22no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—22no—
76D4×D271—22no—
77D4×D271—22no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271C2311yescharacteristic
82#16864644—22no—
83#16864643—22no—
84#16864643—22no—
85#16864642—22no—
86#16864641—22no—
87#16864641—22no—
88#16864641—22no—
89#16864641—22no—
90#16864641—22no—
91D2168—22no—
92D2168—22no—
93#16864639—22no—
94#16864637D414yescomplemented
95C2×C271⋊C8C2311yescharacteristic
96C2×C271⋊C8C2311yescharacteristic
97C8×D271D414yescomplemented
98Dic271⋊C4—22no—
99Dic271⋊C4D412yes—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4D412yes—
104C4×Dic271C2311yescharacteristic
105C2×Q8×C271C2311yescharacteristic
106C2×Q8×C271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271C2311yescharacteristic
109Q16×C271—22no—
110Q16×C271—22no—
111SD16×C271—22no—
112SD16×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271—22no—
116C22⋊C4×C271—22no—
117C42×C271C2311yescharacteristic
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—42no—
122C22×D271—82no—
123C22×D271C2×D411yescharacteristic
124Dic542—21no—
125Dic542—21no—
126Dic542—42no—
127D1084—21no—
128D1084—21no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—42no—
134D1084—42no—
135D1084—82no—
136C22⋊D271—42no—
137C22⋊D271—82no—
138C2×Dic271—21no—
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—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157C2×C4×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—42no—
166C22×C271C22×D411yescharacteristic
167C2×D271—22no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—42no—
172C2×D271—42no—
173C2×D271—42no—
174C2×D271—84no—
175C2×D271#12712yes—
176C4×C271—22no—
177C4×C271—41no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271C22×D411yescharacteristic
181C4×C271C22×D411yescharacteristic
182Dic271—22no—
183Dic271—41no—
184Dic271#12712yes—
185C2×C271—81no—
186C2×C271C2×#12711yescharacteristic
187C2×C271#57712yes—
188D271—22no—
189D271—81no—
190D271—81no—
191D271—81no—
192C271#315411yescharacteristic, complemented, Sylow
193#3154—2711nomaximal, Sylow
194C2×#137—2711no—
195C2×AGL(1,ℤ/8)—2711no—
196C2×#135—2711no—
197C22×SD16—2711no—
198#536—2711no—
199#535—2711no—
200#497—2711no—
201#497D27111yescharacteristic, complemented
202#496—2711no—
203#496—2711no—
204#495—2711no—
205#495—2711no—
206#492—2711no—
207#492—2711no—
208#405—2711no—
209C2×#41—2711no—
210C2×#41—2711no—
211C22×Q8—2711no—
212C22×D4—2711no—
213#137—5424no—
214AGL(1,ℤ/8)—5424no—
215#135—5424no—
216C2×Q16—2711no—
217C2×Q16C2×D27111yescharacteristic, complemented
218C2×Q16C2×D27111yescharacteristic, complemented
219C2×SD16—2711no—
220C2×SD16—2711no—
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16—2711no—
226C2×SD16—2711no—
227C2×SD16—5424no—
228C2×SD16C2×D27111yescharacteristic
229C2×SD16C2×D27111yescharacteristic
230C2×D8—2711no—
231C2×D8—2711no—
232C2×D8—2711no—
233C2×M16—2711no—
234C22×C8—2711no—
235C4⋊Q8—2711no—
236C4⋊Q8C2×D27111yescharacteristic, complemented
237#127—2711no—
238#127—2711no—
239#124—2711no—
240#124—2711no—
241#124—2711no—
242#124C2×D27111yescharacteristic
243#123—5422no—
244C22⋊Q8—5422no—
245#121—5422no—
246#120—5422no—
247#117—2711no—
248#97—2711no—
249#97C2×D27111yescharacteristic, complemented
250C4×C8—2711no—
251C4×C8—2711no—
252C24—5421no—
253#41—5422no—
254#41—5422no—
255#41—5422no—
256#41—5422no—
257C2×Q8—2711no—
258C2×Q8—5421no—
259C2×Q8—5422no—
260C2×Q8—5422no—
261C2×Q8C22×D27111yescharacteristic
262C2×Q8C22×D27111yescharacteristic
263C2×Q8C22×D27111yescharacteristic
264C2×D4—2711no—
265C2×D4—2711no—
266C2×D4—2711no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—5422no—
270C2×D4—5422no—
271C2×D4—5421no—
272C2×D4—5422no—
273C2×D4—5422no—
274C2×D4C22×D27111yescharacteristic
275C22×C4—2711no—
276C22×C4—5421no—
277C22×C4—5421no—
278C22×C4—5421no—
279Q16—22no—
280Q16—22no—
281Q16—5422no—
282SD16—22no—
283SD16—22no—
284SD16—5422no—
285SD16—5422no—
286SD16—5422no—
287SD16—5422no—
288SD16—5422no—
289SD16—5422no—
290SD16—5422no—
291SD16—5422no—
292D8—5422no—
293D8—5422no—
294D8—5422no—
295M16—2714no—
296C2×C8—2711no—
297C2×C8—2711no—
298C2×C8—2714no—
299C2×C8C22×D27111yescharacteristic
300C2×C8C22×D27111yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—2712no—
303C4⋊C4—5422no—
304C22⋊C4—22no—
305C22⋊C4—2712no—
306C22⋊C4—5422no—
307C22⋊C4—5422no—
308C22⋊C4—5422no—
309C42—2711no—
310C42C22×D27111yescharacteristic
311C23—21no—
312C23—2711no—
313C23—5421no—
314C23—5421no—
315C23—5421no—
316C23—10842no—
317C23—21682no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—5421no—
325Q8—5421no—
326Q8—10842no—
327D4—21no—
328D4—21no—
329D4—5421no—
330D4—5421no—
331D4—5421no—
332D4—5421no—
333D4—5421no—
334D4—5421no—
335D4—10842no—
336D4—10842no—
337D4—10842no—
338D4—21682no—
339D4—21682no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—2711no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—5421no—
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—42no—
359C22—2712no—
360C22—5422no—
361C22—10841no—
362C22—10841no—
363C22—10841no—
364C22—10842no—
365C22—10842no—
366C22—10842no—
367C22—21684no—
368C22C2×D4×D27111yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2712no—
374C4—5422no—
375C4—10841no—
376C4C2×D4×D27111yescharacteristic
377C4C2×D4×D27111yescharacteristic
378C2—81no—
379C2—5422no—
380C2—21681no—
381C2—21681no—
382C2—21681no—
383C2#127×D27111yescharacteristic
384C2#3177470612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#497×C271C2
socleC22×C271C22×D4
hypercenter#497D271
hypocenterC271#3154
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. C2×#22576187C2semidirect
3. C2×#22576183C2semidirect
4. C2×SD16×D271C2semidirect
5. #31774472C2semidirect
6. #31774465C2semidirect
7. #31774290C2semidirect
8. #31774286C2semidirect
9. #31774283C2semidirect
10. #31774282C2semidirect
11. #31774279C2semidirect
12. #31774271C2semidirect
13. #31774268C2semidirect
14. #31773854C2semidirect
15. #497×C271C2semidirect
23. C2×#16864644C22semidirect
25. C2×#16864643C22semidirect
29. C2×#16864641C22semidirect
30. C2×#16864641C22semidirect
31. C2×#16864641C22semidirect
34. C2×D2168C22semidirect
38. #22576158C22semidirect
42. #22576149C22semidirect
43. #22576148C22semidirect
49. #22576070C22semidirect
50. C4×C271⋊C8C22semidirect
51. C8×Dic271C22semidirect
52. C2×Q16×C271C22semidirect
53. C2×Q16×C271C22semidirect
56. C4⋊Q8×C271C22semidirect
58. #97×C271C22semidirect
94. #16864637D4semidirect
97. C8×D271D4semidirect
192. C271#3154semidirect
201. #497D271semidirect
217. C2×Q16C2×D271semidirect
218. C2×Q16C2×D271semidirect
236. C4⋊Q8C2×D271semidirect
249. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×#225761872
C2×#225761832
C2×SD16×D2712
#317744722
#317744652
#317742902
#317742862
#317742832
#317742822
#317742792
#317742712
#317742682
#317738542
#497×C2712
#3154271