Finite Group Databasebeta

← #46845855

Subgroups — #46845855

Order 34688.

500 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×#22576187C211yesmaximal, characteristic
3#124×D271C211yesmaximal, characteristic, complemented
4C4×#16864655C211yesmaximal, characteristic, complemented
5#31774121C211yesmaximal, characteristic, complemented
6#31774119C211yesmaximal, characteristic, complemented
7#31774117C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774113C211yesmaximal, characteristic, complemented
10#31774111C211yesmaximal, characteristic, complemented
11#31774109C211yesmaximal, characteristic, complemented
12#31773919C211yesmaximal, characteristic, complemented
13#31773904C211yesmaximal, characteristic, complemented
14#31773900C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576192—24no—
20#22576187—24no—
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864641C2211yescharacteristic
24C2×#16864641C2211yescharacteristic, complemented
25C2×D2168C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×#16864637C2211yescharacteristic
29#22576149C2211yescharacteristic
30#22576148C2211yescharacteristic
31#22576147C2211yescharacteristic
32#22576146—21no—
33#22576146—21no—
34Q8×Dic271C2211yescharacteristic, complemented
35C4×D1084—21no—
36C4×D1084C2211yescharacteristic, complemented
37#22576122—21no—
38#22576122C2211yescharacteristic, complemented
39#22576118—21no—
40#22576118C2211yescharacteristic
41C22⋊C4×D271—21no—
42C22⋊C4×D271—21no—
43C42×D271—21no—
44C42×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic
52#22576081C2211yescharacteristic
53#22576080C2211yescharacteristic
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#16864655D414yes—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66Q8×D271—22no—
67Q8×D271—22no—
68C2×D1084—21no—
69C2×D1084C2311yescharacteristic
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72D4×D271—22no—
73D4×D271—22no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271D411yescharacteristic
79C2×C4×D271D411yescharacteristic
80C2×C4×D271C2311yescharacteristic
81#16864644—22no—
82#16864643—22no—
83#16864641—22no—
84#16864641—22no—
85D2168—22no—
86#16864639—22no—
87#16864637—22no—
88#16864637—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102C4×Dic271—21no—
103C4×Dic271—21no—
104C4×Dic271D411yescharacteristic
105C4×Dic271D411yescharacteristic
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—42no—
124C22×D271—42no—
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129D1084—24no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084C2×D412yes—
134C22⋊D271—42no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271C2×D411yescharacteristic
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C4×D271—24no—
141C4×D271—41no—
142C4×D271—41no—
143C4×D271—42no—
144C4×D271—42no—
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—22no—
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—21no—
187Dic271—21no—
188Dic271—41no—
189Dic271—41no—
190Dic271#12112yes—
191C2×C271—81no—
192C2×C271C2×#12111yescharacteristic
193C2×C271#57511yescharacteristic
194C2×C271#57511yescharacteristic
195D271—22no—
196D271—42no—
197D271—81no—
198C271#304011yescharacteristic, complemented, Sylow
199#3040—2711nomaximal, Sylow
200C2×#137—2711no—
201C2×AGL(1,ℤ/8)—2711no—
202C2×#124—2711no—
203C4×#41—2711no—
204#464—2711no—
205#464—2711no—
206#464—2711no—
207#464D27111yescharacteristic, complemented
208#463—2711no—
209#463—2711no—
210#463—2711no—
211#463—2711no—
212#423—2711no—
213#419—2711no—
214#418—2711no—
215C2×#41—2711no—
216C22×Q8—2711no—
217C22×D4—2711no—
218#137—5424no—
219AGL(1,ℤ/8)—5424no—
220C2×Q16—2711no—
221C2×Q16C2×D27111yescharacteristic, complemented
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16C2×D27111yescharacteristic
226C2×D8—2711no—
227C2×D8—2711no—
228C2×M16—2711no—
229C2×M16—2711no—
230#124—2711no—
231#124—2711no—
232#124—2711no—
233#124—5421no—
234#124—5421no—
235#124C2×D27111yescharacteristic
236C4×Q8—2711no—
237C4×Q8C2×D27111yescharacteristic, complemented
238C4×D4—2711no—
239C4×D4—2711no—
240C4×D4—5421no—
241C4×D4—5421no—
242#117—2711no—
243#117—5421no—
244C2×C22⋊C4—5421no—
245C2×C22⋊C4—5421no—
246C2×C42—2711no—
247C2×C42—5421no—
248C4⋊C8—2711no—
249C4⋊C8—2711no—
250C4⋊C8—2711no—
251C4⋊C8C2×D27111yescharacteristic, complemented
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#103C2×D27111yescharacteristic, complemented
256#102—2711no—
257#102—2711no—
258#102—2711no—
259#102C2×D27111yescharacteristic
260C24—5421no—
261#41—2714no—
262#41—5422no—
263C2×Q8—2711no—
264C2×Q8—5421no—
265C2×Q8—5422no—
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×D4C22×D27111yescharacteristic
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—5421no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283Q16—22no—
284Q16—5422no—
285SD16—22no—
286SD16—5422no—
287SD16—5422no—
288SD16—5422no—
289D8—5422no—
290D8—5422no—
291M16—5422no—
292M16—5422no—
293C2×C8—2711no—
294C2×C8—2711no—
295C2×C8C22×D27111yescharacteristic
296C2×C8C22×D27111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2711no—
299C4⋊C4—5421no—
300C4⋊C4C22×D27111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C22⋊C4—5421no—
309C22⋊C4—5421no—
310C22⋊C4—5421no—
311C22⋊C4—5421no—
312C42—21no—
313C42—2711no—
314C42—2711no—
315C42—2711no—
316C42—5421no—
317C42—5421no—
318C42C22×D27111yescharacteristic
319C23—21no—
320C23—2711no—
321C23—5421no—
322C23—5421no—
323C23—10842no—
324C23—10842no—
325C23—10842no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—5421no—
330Q8—5421no—
331Q8—10842no—
332Q8D4×D27112yes—
333D4—21no—
334D4—21no—
335D4—2712no—
336D4—5424no—
337D4—5421no—
338D4—5421no—
339D4—5421no—
340D4—10842no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2711no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—2711no—
348C2×C4—2711no—
349C2×C4—5424no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—10841no—
353C2×C4—10841no—
354C2×C4—10842no—
355C2×C4—10842no—
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—5422no—
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—5421no—
385C4—5421no—
386C4—10841no—
387C4—10841no—
388C4C2×D4×D27111yescharacteristic
389C4#41×D27111yescharacteristic
390C2—81no—
391C2—5422no—
392C2—10842no—
393C2—21681no—
394C2#121×D27111yescharacteristic
395C2#3177469411yescharacteristic
396C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576187C2
hypercenter#464D271
hypocenterC271#3040
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
3. #124×D271C2semidirect
4. C4×#16864655C2semidirect
5. #31774121C2semidirect
6. #31774119C2semidirect
7. #31774117C2semidirect
8. #31774115C2semidirect
9. #31774113C2semidirect
10. #31774111C2semidirect
11. #31774109C2semidirect
12. #31773919C2semidirect
13. #31773904C2semidirect
14. #31773900C2semidirect
15. #464×C271C2semidirect
21. C2×#16864644C22semidirect
22. C2×#16864643C22semidirect
24. C2×#16864641C22semidirect
34. Q8×Dic271C22semidirect
36. C4×D1084C22semidirect
38. #22576122C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
54. C2×Q16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
198. C271#3040semidirect
207. #464D271semidirect
221. C2×Q16C2×D271semidirect
237. C4×Q8C2×D271semidirect
251. C4⋊C8C2×D271semidirect
255. #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×#225761872
#124×D2712
C4×#168646552
#317741212
#317741192
#317741172
#317741152
#317741132
#317741112
#317741092
#317739192
#317739042
#317739002
#464×C2712
#3040271