Finite Group Databasebeta

← #46845883

Subgroups — #46845883

Order 34688.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774465C211yesmaximal, characteristic
4#31774373C211yesmaximal, characteristic, complemented
5#31774122C211yesmaximal, characteristic, complemented
6#31774121C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774098C211yesmaximal, characteristic, complemented
9#31774097C211yesmaximal, characteristic, complemented
10#31774096C211yesmaximal, characteristic, complemented
11#31774095C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#31773892C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576192—24no—
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864643C2211yescharacteristic
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864642C2211yescharacteristic
24C2×#16864641C2211yescharacteristic
25C2×#16864641C2211yescharacteristic, complemented
26SD16×D271—24no—
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576157C2211yescharacteristic
30#22576156C2211yescharacteristic
31#22576149C2211yescharacteristic
32#22576133—21no—
33#22576133—21no—
34#22576130—21no—
35#22576130—21no—
36Q8×Dic271C2211yescharacteristic, complemented
37C4×Dic542—21no—
38C4×Dic542C2211yescharacteristic, complemented
39#22576125—21no—
40#22576125C2211yescharacteristic, complemented
41#22576118—21no—
42#22576118C2211yescharacteristic
43#22576117—21no—
44#22576117C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic
52#22576080C2211yescharacteristic
53#22576080C2211yescharacteristic, complemented
54C2×Q16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66Q8×D271—22no—
67Q8×D271D414yescomplemented
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76D4×D271—22no—
77D4×D271—22no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271C2311yescharacteristic
81#16864644—22no—
82#16864643—22no—
83#16864643—22no—
84#16864642—22no—
85#16864641—22no—
86#16864641—22no—
87#16864637—22no—
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C8×D271—22no—
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4D411yescharacteristic
96Dic271⋊C4D411yescharacteristic
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4D411yescharacteristic
101(C2×D271)⋊C4D411yescharacteristic
102C4×Dic271—21no—
103C4×Dic271—21no—
104C4×Dic271—21no—
105C4×Dic271—21no—
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110Q16×C271—22no—
111SD16×C271—22no—
112C2×C8×C271C2311yescharacteristic
113C2×C8×C271C2311yescharacteristic
114C4⋊C4×C271—21no—
115C4⋊C4×C271C2311yescharacteristic
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C42×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—42no—
123C22×D271—81no—
124C22×D271—81no—
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542—24no—
128Dic542C2×D412yes—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—42no—
134D1084—81no—
135D1084—81no—
136C22⋊D271—42no—
137C22⋊D271—81no—
138C22⋊D271—81no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271C2×D411yescharacteristic
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C4×D271—24no—
145C4×D271—42no—
146C4×D271C2×D412yes—
147C4×D271#4112yes—
148C271⋊C8—21no—
149C271⋊C8—21no—
150C23×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271C2×D412yes—
155D4×C271—21no—
156D4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2×D411yescharacteristic
162C2×C4×C271C2411yescharacteristic
163C8×C271—21no—
164C8×C271—21no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271C22×D411yescharacteristic
169C2×D271—21no—
170C2×D271—21no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—84no—
178C2×D271#12112yes—
179C4×C271—22no—
180C4×C271—21no—
181C4×C271—21no—
182C4×C271—41no—
183C4×C271—41no—
184C4×C271C22×D411yescharacteristic
185C4×C271C2×#4111yescharacteristic
186Dic271—22no—
187Dic271—21no—
188Dic271—21no—
189Dic271—41no—
190Dic271#12112yes—
191C2×C271—81no—
192C2×C271C2×#12111yescharacteristic
193C2×C271#57711yescharacteristic
194C2×C271#57711yescharacteristic
195D271—22no—
196D271—81no—
197D271—81no—
198C271#304311yescharacteristic, complemented, Sylow
199#3043—2711nomaximal, Sylow
200C2×#137—2711no—
201C22×SD16—2711no—
202#535—2711no—
203#519—2711no—
204#464—2711no—
205#464—2711no—
206#464—2711no—
207#464D27111yescharacteristic, complemented
208#460—2711no—
209#460—2711no—
210#460—2711no—
211#460—2711no—
212#424—2711no—
213#418—2711no—
214#416—2711no—
215C2×#41—2711no—
216C22×Q8—2711no—
217C22×D4—2711no—
218#137—5424no—
219C2×Q16—2711no—
220C2×Q16C2×D27111yescharacteristic, complemented
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16—2711no—
226C2×SD16—5424no—
227C2×SD16C2×D27111yescharacteristic
228C2×M16—2711no—
229C22×C8—2711no—
230#127—2711no—
231#127—2711no—
232#124—2711no—
233#124C2×D27111yescharacteristic
234#121—5421no—
235#121—5421no—
236#120—5421no—
237#120—5421no—
238C4×Q8—2711no—
239C4×Q8—2711no—
240C4×Q8—2711no—
241C4×Q8—5421no—
242C4×Q8—5421no—
243C4×Q8C2×D27111yescharacteristic, complemented
244#117—2711no—
245#117—2711no—
246#117—5421no—
247#117—5421no—
248C4⋊C8—2711no—
249C4⋊C8—2711no—
250C4⋊C8—2711no—
251C4⋊C8C2×D27111yescharacteristic, complemented
252#103—2711no—
253#103C2×D27111yescharacteristic, complemented
254#102—2711no—
255#102—2711no—
256#102—2711no—
257#102—2711no—
258#102—2711no—
259#102C2×D27111yescharacteristic
260C24—5421no—
261#41—5422no—
262#41—5422no—
263C2×Q8—2714no—
264C2×Q8—2711no—
265C2×Q8—5421no—
266C2×Q8—5422no—
267C2×Q8C22×D27111yescharacteristic
268C2×Q8C22×D27111yescharacteristic
269C2×D4—2711no—
270C2×D4—2711no—
271C2×D4—5421no—
272C2×D4—5422no—
273C2×D4—5422no—
274C2×D4—5421no—
275C2×D4—5421no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4—5421no—
279C2×D4C22×D27111yescharacteristic
280C22×C4—2711no—
281C22×C4—5421no—
282C22×C4—5421no—
283Q16—22no—
284Q16—5422no—
285SD16—22no—
286SD16—5422no—
287SD16—5422no—
288SD16—5422no—
289SD16—5422no—
290SD16—5422no—
291M16—5422no—
292C2×C8—2711no—
293C2×C8—2711no—
294C2×C8—5422no—
295C2×C8C22×D27111yescharacteristic
296C2×C8C22×D27111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2711no—
299C4⋊C4—2711no—
300C4⋊C4—2711no—
301C4⋊C4—5421no—
302C4⋊C4—5421no—
303C4⋊C4—5421no—
304C4⋊C4C22×D27111yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—2711no—
308C22⋊C4—2711no—
309C22⋊C4—5421no—
310C22⋊C4—5421no—
311C22⋊C4—5421no—
312C42—21no—
313C42—2711no—
314C42—5421no—
315C42—5421no—
316C42—5421no—
317C42—5421no—
318C42C22×D27111yescharacteristic
319C23—21no—
320C23—2711no—
321C23—5421no—
322C23—5421no—
323C23—10842no—
324C23—21681no—
325C23—21681no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—2712no—
330Q8—5424no—
331Q8—5421no—
332Q8D4×D27112yes—
333D4—21no—
334D4—21no—
335D4—5421no—
336D4—5421no—
337D4—5421no—
338D4—5421no—
339D4—10842no—
340D4—10842no—
341D4—21681no—
342D4—21681no—
343D4—21681no—
344D4—21681no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—2711no—
349C2×C4—2712no—
350C2×C4—2712no—
351C2×C4—2711no—
352C2×C4—2711no—
353C2×C4—5424no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—10842no—
357C2×C4D4×D27111yescharacteristic
358C2×C4D4×D27111yescharacteristic
359C2×C4C23×D27111yescharacteristic
360C8—21no—
361C8—21no—
362C8—5421no—
363C8—5421no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2712no—
368C22—5421no—
369C22—5421no—
370C22—10841no—
371C22—10841no—
372C22—10841no—
373C22—10841no—
374C22—10841no—
375C22—10841no—
376C22—21684no—
377C22C2×D4×D27111yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—2712no—
384C4—5422no—
385C4—5421no—
386C4—5421no—
387C4—10841no—
388C4C2×D4×D27111yescharacteristic
389C4#41×D27111yescharacteristic
390C2—81no—
391C2—5422no—
392C2—21681no—
393C2—21681no—
394C2#121×D27111yescharacteristic
395C2#3177470411yescharacteristic
396C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
prime radical#31774465C2
hypercenter#464D271
hypocenterC271#3043
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. C2×SD16×D271C2semidirect
4. #31774373C2semidirect
5. #31774122C2semidirect
6. #31774121C2semidirect
7. #31774116C2semidirect
8. #31774098C2semidirect
9. #31774097C2semidirect
10. #31774096C2semidirect
11. #31774095C2semidirect
12. #31773922C2semidirect
13. #31773900C2semidirect
14. #31773892C2semidirect
15. #464×C271C2semidirect
20. C2×#16864644C22semidirect
22. C2×#16864643C22semidirect
25. C2×#16864641C22semidirect
36. Q8×Dic271C22semidirect
38. C4×Dic542C22semidirect
40. #22576125C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576084C22semidirect
50. #22576082C22semidirect
53. #22576080C22semidirect
54. C2×Q16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
67. Q8×D271D4semidirect
198. C271#3043semidirect
207. #464D271semidirect
220. C2×Q16C2×D271semidirect
243. C4×Q8C2×D271semidirect
251. C4⋊C8C2×D271semidirect
253. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×SD16×D2712
#317744652
#317743732
#317741222
#317741212
#317741162
#317740982
#317740972
#317740962
#317740952
#317739222
#317739002
#317738922
#464×C2712
#3043271