Finite Group Databasebeta

← #46846833

Subgroups — #46846833

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×#22576189C211yesmaximal, characteristic, complemented
2C2×#22576187C211yesmaximal, characteristic, complemented
3C2×#22576181C211yesmaximal, characteristic, complemented
4C2×SD16×D271C211yesmaximal, characteristic, complemented
5#31774472C211yesmaximal, characteristic, complemented
6#31774460C211yesmaximal, characteristic, complemented
7#31774289C211yesmaximal, characteristic, complemented
8#31774288C211yesmaximal, characteristic, complemented
9#31774284C211yesmaximal, characteristic, complemented
10#31774283C211yesmaximal, characteristic, complemented
11#31774281C211yesmaximal, characteristic, complemented
12#31774278C211yesmaximal, characteristic, complemented
13#31774268C211yesmaximal, characteristic, complemented
14#31773851C211yesmaximal, characteristic, complemented
15#492×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic
20#22576189—24no—
21#22576187—24no—
22#22576181—24no—
23C2×Dic1084C2211yescharacteristic, complemented
24C2×#16864644C2211yescharacteristic, complemented
25C2×#16864644C2211yescharacteristic, complemented
26C2×#16864643C2211yescharacteristic
27C2×#16864643C2211yescharacteristic, complemented
28C2×#16864642C2211yescharacteristic
29C2×#16864642C2211yescharacteristic, complemented
30C2×#16864641C2211yescharacteristic
31C2×#16864641C2211yescharacteristic
32SD16×D271—24no—
33C2×D2168C2211yescharacteristic
34C2×#16864639C2211yescharacteristic
35C2×#16864639C2211yescharacteristic, complemented
36C2×#16864637C2211yescharacteristic
37C2×C8×D271C2211yescharacteristic
38#22576158C2211yescharacteristic, complemented
39#22576156C2211yescharacteristic
40#22576149C2211yescharacteristic, complemented
41#22576148C2211yescharacteristic
42#22576148C2211yescharacteristic, complemented
43#22576147C2211yescharacteristic
44#22576145—22no—
45#22576138—22no—
46#22576132—22no—
47#22576128—22no—
48#22576117C2211yescharacteristic
49#22576071C2211yescharacteristic, complemented
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C4⋊Q8×C271C2211yescharacteristic, complemented
57#127×C271C2211yescharacteristic
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—21no—
74C2×C22⋊D271—22no—
75D4×D271—22no—
76D4×D271—22no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271C2311yescharacteristic
81Dic1084—22no—
82#16864644—22no—
83#16864644—22no—
84#16864643—22no—
85#16864643—22no—
86#16864642—22no—
87#16864642—22no—
88#16864641—22no—
89#16864641—22no—
90D2168—22no—
91#16864639—22no—
92#16864639—22no—
93#16864637D414yescomplemented
94C2×C271⋊C8C2311yescharacteristic
95C2×C271⋊C8C2311yescharacteristic
96C8×D271D414yescomplemented
97Dic271⋊C4—22no—
98Dic271⋊C4D412yes—
99C22⋊Dic271—22no—
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×D4×C271—22no—
108C2×D4×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110SD16×C271—22no—
111SD16×C271—22no—
112SD16×C271—22no—
113SD16×C271—22no—
114C2×C8×C271C2311yescharacteristic
115C2×C8×C271C2311yescharacteristic
116C4⋊C4×C271—22no—
117C42×C271C2311yescharacteristic
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—42no—
121C22×D271—82no—
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—42no—
134C22⋊D271—42no—
135C22⋊D271—82no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—42no—
139C2×Dic271C2×D412yes—
140C2×Dic271C2×D411yescharacteristic
141C4×D271—42no—
142C4×D271—42no—
143C4×D271C2×D412yes—
144C4×D271C2×D412yes—
145C271⋊C8C2×D412yes—
146C271⋊C8C2×D412yes—
147C23×C271—21no—
148C23×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153D4×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—82no—
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#57712yes—
189D271—22no—
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#497—2711no—
202#496—2711no—
203#496—2711no—
204#495—2711no—
205#495—2711no—
206#492—2711no—
207#492D27111yescharacteristic, complemented
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×Q16—2711no—
218C2×Q16—2711no—
219C2×SD16—2711no—
220C2×SD16—2711no—
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16—5424no—
226C2×SD16C2×D27111yescharacteristic
227C2×SD16C2×D27111yescharacteristic
228C2×SD16C2×D27111yescharacteristic, complemented
229C2×SD16C2×D27111yescharacteristic, complemented
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#127C2×D27111yescharacteristic
239#124—2711no—
240#124—2711no—
241#124—2711no—
242#124—2711no—
243#123—5422no—
244C22⋊Q8—5422no—
245#121—5422no—
246#120—5422no—
247#117—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—22no—
265C2×D4—2711no—
266C2×D4—2711no—
267C2×D4—5421no—
268C2×D4—5422no—
269C2×D4—5422no—
270C2×D4—5421no—
271C2×D4—5422no—
272C2×D4—5421no—
273C2×D4C22×D27111yescharacteristic
274C2×D4C22×D27111yescharacteristic
275C22×C4—2711no—
276C22×C4—5421no—
277C22×C4—5421no—
278C22×C4—5421no—
279Q16—5422no—
280Q16—5422no—
281Q16—5422no—
282SD16—22no—
283SD16—22no—
284SD16—22no—
285SD16—22no—
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—2712no—
305C22⋊C4—5422no—
306C22⋊C4—5422no—
307C22⋊C4—5422no—
308C22⋊C4—5422no—
309C42—2711no—
310C42C22×D27111yescharacteristic
311C23—21no—
312C23—21no—
313C23—2711no—
314C23—5421no—
315C23—5421no—
316C23—10842no—
317C23—21682no—
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—82no—
332D4—5421no—
333D4—5421no—
334D4—5421no—
335D4—5421no—
336D4—10842no—
337D4—10842no—
338D4—10842no—
339D4—21682no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2711no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—5421no—
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—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#3177470312yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#492×C271C2
socleC22×C271C22×D4
hypercenter#492D271
hypocenterC271#3154
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. C2×#22576187C2semidirect
3. C2×#22576181C2semidirect
4. C2×SD16×D271C2semidirect
5. #31774472C2semidirect
6. #31774460C2semidirect
7. #31774289C2semidirect
8. #31774288C2semidirect
9. #31774284C2semidirect
10. #31774283C2semidirect
11. #31774281C2semidirect
12. #31774278C2semidirect
13. #31774268C2semidirect
14. #31773851C2semidirect
15. #492×C271C2semidirect
23. C2×Dic1084C22semidirect
24. C2×#16864644C22semidirect
25. C2×#16864644C22semidirect
27. C2×#16864643C22semidirect
29. C2×#16864642C22semidirect
35. C2×#16864639C22semidirect
38. #22576158C22semidirect
40. #22576149C22semidirect
42. #22576148C22semidirect
49. #22576071C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×SD16×C271C22semidirect
56. C4⋊Q8×C271C22semidirect
58. C4×C8×C271C22semidirect
93. #16864637D4semidirect
96. C8×D271D4semidirect
192. C271#3154semidirect
207. #492D271semidirect
228. C2×SD16C2×D271semidirect
229. C2×SD16C2×D271semidirect
236. C4⋊Q8C2×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×#225761892
C2×#225761872
C2×#225761812
C2×SD16×D2712
#317744722
#317744602
#317742892
#317742882
#317742842
#317742832
#317742812
#317742782
#317742682
#317738512
#492×C2712
#3154271