Finite Group Databasebeta

← #46847879

Subgroups — #46847879

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576193C211yesmaximal, characteristic, complemented
2C22×#16864643C211yesmaximal, characteristic, complemented
3D4×Dic542C211yesmaximal, characteristic, complemented
4#31774543C211yesmaximal, characteristic
5#31774291C211yesmaximal, characteristic, complemented
6#31774143C211yesmaximal, characteristic, complemented
7#31774138C211yesmaximal, characteristic, complemented
8#31774120C211yesmaximal, characteristic, complemented
9#31774103C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774055C211yesmaximal, characteristic, complemented
12#31774050C211yesmaximal, characteristic, complemented
13#31774038C211yesmaximal, characteristic, complemented
14#31773999C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C22×Dic542C2211yescharacteristic, complemented
19C22×D1084C2211yescharacteristic
20#22576193—24no—
21C2×Dic1084C2211yescharacteristic, complemented
22C2×Dic1084C2211yescharacteristic, complemented
23C2×Dic1084C2211yescharacteristic, complemented
24C2×#16864643—24no—
25C2×#16864643C2211yescharacteristic
26C2×#16864643C2211yescharacteristic
27C2×#16864643C2211yescharacteristic
28C2×#16864643C2211yescharacteristic
29C2×#16864643C2211yescharacteristic
30C2×#16864643C2211yescharacteristic
31#22576160C2211yescharacteristic, complemented
32#22576159—21no—
33#22576148C2211yescharacteristic
34#22576145—21no—
35#22576139C2211yescharacteristic
36#22576139C2211yescharacteristic
37#22576139C2211yescharacteristic, complemented
38#22576137—21no—
39#22576137—21no—
40#22576134C2211yescharacteristic
41#22576133—21no—
42#22576128—21no—
43C4×Dic542C2211yescharacteristic, complemented
44C4×D1084C2211yescharacteristic
45D4×Dic271—21no—
46C22⋊C4×D271—21no—
47#22576094C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576085C2211yescharacteristic, complemented
51#22576085C2211yescharacteristic, complemented
52#22576082C2211yescharacteristic
53#22576082C2211yescharacteristic
54C2×M16×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C22⋊C8×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864654—22no—
63#16864654—41no—
64C2×Dic542—22no—
65C2×Dic542—22no—
66C2×Dic542—21no—
67C2×Dic542—21no—
68C2×Dic542—41no—
69C2×Dic542—41no—
70C2×Dic542C2311yescharacteristic
71C2×Dic542C2311yescharacteristic
72C2×Dic542C2311yescharacteristic
73C2×Dic542C2311yescharacteristic
74C2×D1084—22no—
75C2×D1084—21no—
76C2×D1084—41no—
77C2×D1084C2311yescharacteristic
78C2×D1084C2311yescharacteristic
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81C22×Dic271—21no—
82C22×Dic271—21no—
83C2×C4×D271—21no—
84C2×C4×D271—21no—
85Dic1084—22no—
86Dic1084—22no—
87Dic1084—41no—
88#16864643—22no—
89#16864643—22no—
90#16864643—22no—
91#16864643—22no—
92#16864643—41no—
93#16864643D414yescomplemented
94C4⋊Dic271—21no—
95C4⋊Dic271—21no—
96C4⋊Dic271C2311yescharacteristic
97C4⋊Dic271C2311yescharacteristic
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104C22⋊Dic271—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109C4×Dic271—21no—
110C2×D4×C271D411yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113M16×C271—22no—
114C2×C8×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128Dic542—21no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—21no—
133Dic542—21no—
134Dic542—21no—
135Dic542—42no—
136Dic542—42no—
137Dic542C2×D412yes—
138D1084—21no—
139D1084—21no—
140D1084—21no—
141D1084—42no—
142D1084C2×D412yes—
143C22⋊D271—42no—
144C22⋊D271—81no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—42no—
152C2×Dic271—42no—
153C4×D271—42no—
154C4×D271—81no—
155C23×C271C2×D411yescharacteristic
156C23×C271C2×D411yescharacteristic
157D4×C271—22no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271C2×D411yescharacteristic
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2411yescharacteristic
167C8×C271—21no—
168C8×C271—21no—
169C8×C271—21no—
170C8×C271C2×D412yes—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271#13712yescomplemented
177C22×C271C22×D411yescharacteristic
178C2×D271—22no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—84no—
185C4×C271—21no—
186C4×C271—21no—
187C4×C271—21no—
188C4×C271—21no—
189C4×C271C22×D411yescharacteristic
190C4×C271H(5,𝔽2)11yescharacteristic
191Dic271—22no—
192Dic271—41no—
193Dic271—41no—
194Dic271—41no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—22no—
198C2×C271—41no—
199C2×C271D4211yescharacteristic
200C2×C271C2×#13711yescharacteristic
201C2×C271#57711yescharacteristic
202D271—42no—
203D271—81no—
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8—2711no—
209#546—2711no—
210#497—2711no—
211#468—2711no—
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450—2711no—
218#448—2711no—
219#440—2711no—
220#435D27111yescharacteristic, complemented
221C2×#41—2711no—
222C22×Q8—2711no—
223C22×Q8—2711no—
224C22×D4—2711no—
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16—2711no—
228C2×Q16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—5424no—
236C2×M16C2×D27111yescharacteristic, complemented
237C22×C8C2×D27111yescharacteristic
238C4⋊Q8—2711no—
239C4⋊Q8—5421no—
240#124—2711no—
241#123—5421no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8—2711no—
245C22⋊Q8—5421no—
246C22⋊Q8—5421no—
247#121—2711no—
248#121—5421no—
249#120—5421no—
250C4×Q8—2711no—
251C4×D4—2711no—
252C4×D4—5421no—
253C4×D4C2×D27111yescharacteristic
254C2×C22⋊C4—5421no—
255#107—2711no—
256C4⋊C8C2×D27111yescharacteristic, complemented
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103—2711no—
261#102—2711no—
262#102—2711no—
263C22⋊C8C2×D27111yescharacteristic
264C22⋊C8C2×D27111yescharacteristic, complemented
265#97C2×D27111yescharacteristic, complemented
266C24—5421no—
267#41—5422no—
268#41—10841no—
269C2×Q8—2711no—
270C2×Q8—2711no—
271C2×Q8—2711no—
272C2×Q8—2711no—
273C2×Q8—5422no—
274C2×Q8—5422no—
275C2×Q8—5421no—
276C2×Q8—5421no—
277C2×Q8—10841no—
278C2×Q8—10841no—
279C2×D4—2711no—
280C2×D4—2711no—
281C2×D4—5422no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4—10841no—
286C2×D4D108411yescharacteristic
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4C22×D27111yescharacteristic
292C22×C4C22×D27111yescharacteristic
293Q16—5422no—
294Q16—5422no—
295Q16—10841no—
296SD16—2714no—
297SD16—5422no—
298SD16—5422no—
299SD16—5422no—
300SD16—5422no—
301SD16—10841no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D27111yescharacteristic
305C2×C8C22×D27111yescharacteristic
306C2×C8C22×D27111yescharacteristic
307C2×C8C22×D27111yescharacteristic
308C4⋊C4—2711no—
309C4⋊C4—2711no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4—5421no—
316C4⋊C4D108411yescharacteristic
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4D108411yescharacteristic
325C22⋊C4D108411yescharacteristic
326C42—5421no—
327C42C22×D27111yescharacteristic
328C23—5421no—
329C23—5421no—
330C23—10842no—
331C23—10842no—
332C23—21681no—
333C23C2×D108411yescharacteristic
334C23C2×D108411yescharacteristic
335Q8—2712no—
336Q8—5421no—
337Q8—5421no—
338Q8—5421no—
339Q8—5421no—
340Q8—5421no—
341Q8—5421no—
342Q8—5421no—
343Q8—10842no—
344Q8—10842no—
345D4—22no—
346D4—2712no—
347D4—5421no—
348D4—5421no—
349D4—5421no—
350D4—10842no—
351D4—10842no—
352D4—21681no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—10842no—
364C2×C4—10842no—
365C2×C4—10842no—
366C2×C4—21681no—
367C2×C4C2×D108411yescharacteristic
368C2×C4C2×D108411yescharacteristic
369C2×C4C2×D108411yescharacteristic
370C2×C4C2×D108411yescharacteristic
371C2×C4C23×D27111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D27112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—5422no—
382C22—10841no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22—21684no—
388C22#2257619312yescomplemented
389C22C22×D108411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—5422no—
395C4—10841no—
396C4—10841no—
397C4—10841no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620611yescharacteristic
402C2—22no—
403C2—41no—
404C2—10842no—
405C2—21681no—
406C2D4×D108411yescharacteristic
407C2C2×#2257619311yescharacteristic
408C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
prime radical#31774543C2
hypercenter#435D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
2. C22×#16864643C2semidirect
3. D4×Dic542C2semidirect
5. #31774291C2semidirect
6. #31774143C2semidirect
7. #31774138C2semidirect
8. #31774120C2semidirect
9. #31774103C2semidirect
10. #31774061C2semidirect
11. #31774055C2semidirect
12. #31774050C2semidirect
13. #31774038C2semidirect
14. #31773999C2semidirect
15. #435×C271C2semidirect
18. C22×Dic542C22semidirect
21. C2×Dic1084C22semidirect
22. C2×Dic1084C22semidirect
23. C2×Dic1084C22semidirect
31. #22576160C22semidirect
37. #22576139C22semidirect
43. C4×Dic542C22semidirect
47. #22576094C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
50. #22576085C22semidirect
51. #22576085C22semidirect
54. C2×M16×C271C22semidirect
57. C4⋊C8×C271C22semidirect
59. C22⋊C8×C271C22semidirect
60. #97×C271C22semidirect
93. #16864643D4semidirect
176. C22×C271#137semidirect
204. C271#3266semidirect
220. #435D271semidirect
236. C2×M16C2×D271semidirect
256. C4⋊C8C2×D271semidirect
264. C22⋊C8C2×D271semidirect
265. #97C2×D271semidirect
388. C22#22576193semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C22×#168646432
D4×Dic5422
#317745432
#317742912
#317741432
#317741382
#317741202
#317741032
#317740612
#317740552
#317740502
#317740382
#317739992
#435×C2712
#3266271