Finite Group Databasebeta

← #46846001

Subgroups — #46846001

Order 34688.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D271C211yesmaximal, characteristic, complemented
2C2×#22576168C211yesmaximal, characteristic, complemented
3#31774470C211yesmaximal, characteristic, complemented
4#31774463C211yesmaximal, characteristic
5#31774144C211yesmaximal, characteristic, complemented
6#31774143C211yesmaximal, characteristic, complemented
7#31774140C211yesmaximal, characteristic, complemented
8#31774129C211yesmaximal, characteristic, complemented
9#31774128C211yesmaximal, characteristic, complemented
10#31774125C211yesmaximal, characteristic, complemented
11#31774124C211yesmaximal, characteristic, complemented
12#31773934C211yesmaximal, characteristic, complemented
13#31773904C211yesmaximal, characteristic, complemented
14#31773900C211yesmaximal, characteristic, complemented
15#468×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19C2×#16864643C2211yescharacteristic
20C2×#16864642C2211yescharacteristic
21C2×#16864641C2211yescharacteristic
22SD16×D271—24no—
23#22576168—24no—
24C2×#16864637C2211yescharacteristic
25C2×#16864637C2211yescharacteristic, complemented
26C22×C271⋊C8C2211yescharacteristic, complemented
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic, complemented
29#22576159—21no—
30#22576156—21no—
31#22576146—21no—
32#22576146—21no—
33#22576145—21no—
34#22576142—21no—
35#22576140C2211yescharacteristic, complemented
36#22576139C2211yescharacteristic, complemented
37#22576136C2211yescharacteristic, complemented
38#22576135C2211yescharacteristic
39#22576134C2211yescharacteristic
40#22576131C2211yescharacteristic
41#22576130—21no—
42#22576128—21no—
43#22576118C2211yescharacteristic
44#22576118C2211yescharacteristic
45#22576094C2211yescharacteristic, complemented
46#22576091C2211yescharacteristic, complemented
47#22576091C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic
52#22576081C2211yescharacteristic
53#22576080C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C2×M16×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#107×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864654—22no—
63#16864654—22no—
64C2×Dic542—21no—
65C2×Dic542D411yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69C2×D1084—21no—
70C2×D1084D411yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271D411yescharacteristic
76C2×C22⋊D271D411yescharacteristic
77D4×D271—22no—
78D4×D271—22no—
79C2×C4×D271—21no—
80C2×C4×D271C2311yescharacteristic
81C2×C4×D271C2311yescharacteristic
82#16864643—22no—
83#16864642—22no—
84#16864641—22no—
85#16864637—22no—
86#16864637—22no—
87C2×C271⋊C8—22no—
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C8×D271—22no—
91C8×D271D414yescomplemented
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94C4⋊Dic271C2311yescharacteristic
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105C4×Dic271—21no—
106C4×Dic271—21no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—21no—
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111SD16×C271—22no—
112M16×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271—21no—
116C4⋊C4×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22×D271—21no—
121C22×D271—42no—
122C22×D271—81no—
123C22×D271—81no—
124C22×D271C2×D411yescharacteristic
125C22×D271C2×D411yescharacteristic
126Dic542—22no—
127Dic542—21no—
128Dic542—21no—
129Dic542—42no—
130D1084—22no—
131D1084—21no—
132D1084—21no—
133D1084—81no—
134C22⋊D271—22no—
135C22⋊D271—22no—
136C22⋊D271—42no—
137C22⋊D271—81no—
138C22⋊D271—81no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C4×D271—21no—
145C4×D271—21no—
146C4×D271—42no—
147C4×D271C2×D412yes—
148C4×D271#4112yes—
149C271⋊C8—21no—
150C271⋊C8C2×D412yes—
151C23×C271—21no—
152C23×C271C2×D411yescharacteristic
153Q8×C271—21no—
154Q8×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—81no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2411yescharacteristic
165C8×C271—21no—
166C8×C271C2×D412yes—
167C22×C271—21no—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271C22×D411yescharacteristic
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—21no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—84no—
183C2×D271#12112yes—
184C4×C271—21no—
185C4×C271—41no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271C22×D411yescharacteristic
189C4×C271C2×#4111yescharacteristic
190Dic271—21no—
191Dic271—41no—
192Dic271—41no—
193Dic271—41no—
194Dic271#12112yes—
195C2×C271—41no—
196C2×C271—81no—
197C2×C271C2×#12111yescharacteristic
198C2×C271#57711yescharacteristic
199C2×C271#57711yescharacteristic
200D271—22no—
201D271—41no—
202D271—81no—
203C271#305711yescharacteristic, complemented, Sylow
204#3057—2711nomaximal, Sylow
205C22×SD16—2711no—
206C2×#131—2711no—
207#536—2711no—
208#535—2711no—
209#468—2711no—
210#468—2711no—
211#468—2711no—
212#468D27111yescharacteristic, complemented
213#465—2711no—
214#465—2711no—
215#465—2711no—
216#465—2711no—
217#427—2711no—
218#419—2711no—
219#418—2711no—
220C2×#41—2711no—
221C22×Q8—2711no—
222C22×D4—2711no—
223C2×SD16—2711no—
224C2×SD16—2711no—
225C2×SD16—2711no—
226C2×SD16—5424no—
227C2×SD16C2×D27111yescharacteristic
228#131—5424no—
229C2×M16—2711no—
230C2×M16—2711no—
231C2×M16C2×D27111yescharacteristic, complemented
232C22×C8—2711no—
233C22×C8—2711no—
234C22×C8—2711no—
235C4⋊Q8—5421no—
236#127—5421no—
237#124—5421no—
238#124—5421no—
239#123—5421no—
240#123—5421no—
241C22⋊Q8—2711no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8C2×D27111yescharacteristic, complemented
245#121—2711no—
246#121—2711no—
247#121—2711no—
248#121C2×D27111yescharacteristic
249#120—5421no—
250#120—5421no—
251#117—2711no—
252#117—2711no—
253#107—2711no—
254#107C2×D27111yescharacteristic, complemented
255#106—2711no—
256#106—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103C2×D27111yescharacteristic, complemented
261#102—2711no—
262#102—2711no—
263#102—2711no—
264#102C2×D27111yescharacteristic
265C24—5421no—
266#41—5422no—
267#41—5422no—
268C2×Q8—2711no—
269C2×Q8—2711no—
270C2×Q8—5421no—
271C2×Q8—5422no—
272C2×Q8—5422no—
273C2×Q8C22×D27111yescharacteristic
274C2×D4—21no—
275C2×D4—2711no—
276C2×D4—2711no—
277C2×D4—2711no—
278C2×D4—2711no—
279C2×D4—5421no—
280C2×D4—5422no—
281C2×D4—5422no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4C22×D27111yescharacteristic
286C22×C4—2711no—
287C22×C4—2711no—
288C22×C4—5421no—
289C22×C4C22×D27111yescharacteristic
290SD16—22no—
291SD16—5422no—
292SD16—5422no—
293SD16—5422no—
294M16—22no—
295M16—5422no—
296M16—5422no—
297C2×C8—2711no—
298C2×C8—2714no—
299C2×C8—2711no—
300C2×C8—5422no—
301C2×C8—5422no—
302C2×C8C22×D27111yescharacteristic
303C2×C8C22×D27111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—2711no—
306C4⋊C4—2711no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4C22×D27111yescharacteristic
311C4⋊C4C22×D27111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C42—5421no—
323C42—5421no—
324C23—21no—
325C23—2711no—
326C23—2711no—
327C23—5421no—
328C23—10842no—
329C23—21681no—
330C23—21681no—
331C23D4×D27111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—5422no—
335Q8—5421no—
336Q8—5421no—
337Q8—10842no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—5422no—
342D4—5422no—
343D4—5422no—
344D4—5421no—
345D4—5421no—
346D4—10842no—
347D4—21681no—
348D4—21681no—
349D4—21681no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2711no—
356C2×C4—2712no—
357C2×C4—2712no—
358C2×C4—2711no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—10842no—
365C2×C4D4×D27111yescharacteristic
366C2×C4C23×D27111yescharacteristic
367C8—21no—
368C8—2712no—
369C8—5421no—
370C8D4×D27112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2712no—
378C22—5421no—
379C22—5421no—
380C22—5421no—
381C22—5421no—
382C22—5421no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—21684no—
387C22C2×D4×D27111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2712no—
393C4—5421no—
394C4—10841no—
395C4—10841no—
396C4—10841no—
397C4C2×D4×D27111yescharacteristic
398C4#41×D27111yescharacteristic
399C2—41no—
400C2—81no—
401C2—5422no—
402C2—10841no—
403C2—21681no—
404C2#121×D27111yescharacteristic
405C2#3177470611yescharacteristic
406C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#468×C271C2
socleC22×C271C22×D4
prime radical#31774463C2
hypercenter#468D271
hypocenterC271#3057
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D271C2semidirect
2. C2×#22576168C2semidirect
3. #31774470C2semidirect
5. #31774144C2semidirect
6. #31774143C2semidirect
7. #31774140C2semidirect
8. #31774129C2semidirect
9. #31774128C2semidirect
10. #31774125C2semidirect
11. #31774124C2semidirect
12. #31773934C2semidirect
13. #31773904C2semidirect
14. #31773900C2semidirect
15. #468×C271C2semidirect
25. C2×#16864637C22semidirect
26. C22×C271⋊C8C22semidirect
28. C2×C8×D271C22semidirect
35. #22576140C22semidirect
36. #22576139C22semidirect
37. #22576136C22semidirect
45. #22576094C22semidirect
46. #22576091C22semidirect
47. #22576091C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
55. C2×M16×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #107×C271C22semidirect
59. #103×C271C22semidirect
91. C8×D271D4semidirect
203. C271#3057semidirect
212. #468D271semidirect
231. C2×M16C2×D271semidirect
244. C22⋊Q8C2×D271semidirect
254. #107C2×D271semidirect
260. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2712
C2×#225761682
#317744702
#317744632
#317741442
#317741432
#317741402
#317741292
#317741282
#317741252
#317741242
#317739342
#317739042
#317739002
#468×C2712
#3057271