Finite Group Databasebeta

← #46847939

Subgroups — #46847939

Order 34688.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576193C211yesmaximal, characteristic, complemented
2C22×Dic1084C211yesmaximal, characteristic, complemented
3D4×Dic542C211yesmaximal, characteristic, complemented
4#31774556C211yesmaximal, characteristic
5#31774291C211yesmaximal, characteristic, complemented
6#31774155C211yesmaximal, characteristic, complemented
7#31774126C211yesmaximal, characteristic, complemented
8#31774120C211yesmaximal, characteristic, complemented
9#31774108C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774055C211yesmaximal, characteristic, complemented
12#31774055C211yesmaximal, characteristic, complemented
13#31774038C211yesmaximal, characteristic, complemented
14#31774005C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C22×Dic542C2211yescharacteristic
19C22×Dic542C2211yescharacteristic
20#22576193—24no—
21C2×Dic1084—24no—
22C2×Dic1084C2211yescharacteristic
23C2×Dic1084C2211yescharacteristic
24C2×Dic1084C2211yescharacteristic
25C2×Dic1084C2211yescharacteristic
26C2×Dic1084C2211yescharacteristic
27C2×Dic1084C2211yescharacteristic
28C2×#16864643C2211yescharacteristic
29C2×#16864643C2211yescharacteristic
30C2×#16864643C2211yescharacteristic
31#22576160C2211yescharacteristic
32#22576159—21no—
33#22576148C2211yescharacteristic
34#22576146—21no—
35#22576139C2211yescharacteristic
36#22576139C2211yescharacteristic
37#22576139C2211yescharacteristic
38#22576138—21no—
39#22576137—21no—
40#22576137—21no—
41#22576134C2211yescharacteristic
42#22576132—21no—
43C4×Dic542C2211yescharacteristic
44C4×Dic542C2211yescharacteristic
45D4×Dic271—21no—
46(C22⋊D271)⋊C4—21no—
47#22576092C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576085C2211yescharacteristic
50#22576085C2211yescharacteristic
51#22576085C2211yescharacteristic
52#22576085C2211yescharacteristic
53#22576082C2211yescharacteristic
54C2×M16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C22⋊C8×C271C2211yescharacteristic
60#97×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—41no—
63C2×Dic542—22no—
64C2×Dic542—22no—
65C2×Dic542—22no—
66C2×Dic542—21no—
67C2×Dic542—21no—
68C2×Dic542—21no—
69C2×Dic542—41no—
70C2×Dic542—41no—
71C2×Dic542—41no—
72C2×Dic542C2311yescharacteristic
73C2×Dic542C2311yescharacteristic
74C2×Dic542C2311yescharacteristic
75C2×Dic542C2311yescharacteristic
76C2×Dic542C2311yescharacteristic
77C2×D1084C2311yescharacteristic
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C22×Dic271—21no—
81C22×Dic271—21no—
82C22×Dic271—21no—
83C2×C4×D271—21no—
84Dic1084—22no—
85Dic1084—22no—
86Dic1084—22no—
87Dic1084—22no—
88Dic1084—41no—
89Dic1084D414yescomplemented
90#16864643—22no—
91#16864643—22no—
92#16864643—41no—
93C4⋊Dic271—21no—
94C4⋊Dic271—21no—
95C4⋊Dic271C2311yescharacteristic
96C4⋊Dic271C2311yescharacteristic
97Dic271⋊C4—21no—
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—
108C4×Dic271—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—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—42no—
133Dic542—42no—
134Dic542—42no—
135Dic542C2×D412yes—
136Dic542C2×D412yes—
137D1084—21no—
138D1084—21no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—42no—
149C2×Dic271—42no—
150C2×Dic271—42no—
151C4×D271—42no—
152C23×C271C2×D411yescharacteristic
153C23×C271C2×D411yescharacteristic
154D4×C271—22no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271C2×D411yescharacteristic
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2×D411yescharacteristic
162C2×C4×C271C2×D411yescharacteristic
163C2×C4×C271C2411yescharacteristic
164C8×C271—21no—
165C8×C271—21no—
166C8×C271—21no—
167C8×C271C2×D412yes—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271#13712yescomplemented
174C22×C271C22×D411yescharacteristic
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—41no—
178C4×C271—21no—
179C4×C271—21no—
180C4×C271—21no—
181C4×C271—21no—
182C4×C271C22×D411yescharacteristic
183C4×C271H(5,𝔽2)11yescharacteristic
184Dic271—22no—
185Dic271—22no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—22no—
192C2×C271—41no—
193C2×C271D4211yescharacteristic
194C2×C271C2×#13711yescharacteristic
195C2×C271#57811yescharacteristic
196D271—81no—
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137—2711no—
200C22×Q16—2711no—
201D4×Q8—2711no—
202#548—2711no—
203#497—2711no—
204#470—2711no—
205#465—2711no—
206#464—2711no—
207#462—2711no—
208#452—2711no—
209#451—2711no—
210#451—2711no—
211#448—2711no—
212#441—2711no—
213#435D27111yescharacteristic, complemented
214C2×#41—2711no—
215C22×Q8—2711no—
216C22×Q8—2711no—
217C22×Q8—2711no—
218#137—5424no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—2711no—
224C2×Q16—2711no—
225C2×Q16—5424no—
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×M16C2×D27111yescharacteristic
230C22×C8C2×D27111yescharacteristic
231C4⋊Q8—2711no—
232C4⋊Q8—5421no—
233#124—2711no—
234#124—5421no—
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8—5421no—
241#121—2711no—
242#121—5421no—
243C4×Q8—2711no—
244C4×Q8—2711no—
245C4×D4—5421no—
246C4×D4—5421no—
247C4×D4C2×D27111yescharacteristic
248#106—2711no—
249C4⋊C8C2×D27111yescharacteristic
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#102—2711no—
256C22⋊C8C2×D27111yescharacteristic
257C22⋊C8C2×D27111yescharacteristic
258#97C2×D27111yescharacteristic
259#41—5422no—
260#41—10841no—
261C2×Q8—2711no—
262C2×Q8—2711no—
263C2×Q8—2711no—
264C2×Q8—2711no—
265C2×Q8—2711no—
266C2×Q8—5422no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8—5421no—
270C2×Q8—5421no—
271C2×Q8—5421no—
272C2×Q8—10841no—
273C2×Q8—10841no—
274C2×Q8—10841no—
275C2×D4—2711no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4D108411yescharacteristic
279C22×C4—5421no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4C22×D27111yescharacteristic
284C22×C4C22×D27111yescharacteristic
285Q16—2714no—
286Q16—5422no—
287Q16—5422no—
288Q16—5422no—
289Q16—5422no—
290Q16—10841no—
291SD16—5422no—
292SD16—5422no—
293SD16—10841no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D27111yescharacteristic
297C2×C8C22×D27111yescharacteristic
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—2711no—
301C4⋊C4—2711no—
302C4⋊C4—5421no—
303C4⋊C4—5421no—
304C4⋊C4—5421no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4D108411yescharacteristic
310C22⋊C4—5421no—
311C22⋊C4—5421no—
312C22⋊C4—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4D108411yescharacteristic
317C22⋊C4D108411yescharacteristic
318C42—5421no—
319C42—5421no—
320C42C22×D27111yescharacteristic
321C23—5421no—
322C23C2×D108411yescharacteristic
323C23C2×D108411yescharacteristic
324Q8—2712no—
325Q8—2712no—
326Q8—5421no—
327Q8—5421no—
328Q8—5421no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8—10842no—
336Q8—10842no—
337D4—22no—
338D4—5421no—
339D4—5421no—
340D4—10842no—
341D4—10842no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—10842no—
354C2×C4—10842no—
355C2×C4—10842no—
356C2×C4—10842no—
357C2×C4C2×D108411yescharacteristic
358C2×C4C2×D108411yescharacteristic
359C2×C4C2×D108411yescharacteristic
360C2×C4C2×D108411yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D27112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—10841no—
372C22—10841no—
373C22—10841no—
374C22#2257619312yescomplemented
375C22C22×D108411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—5422no—
381C4—5422no—
382C4—10841no—
383C4—10841no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620611yescharacteristic
389C2—22no—
390C2—41no—
391C2—21681no—
392C2D4×D108411yescharacteristic
393C2C2×#2257619311yescharacteristic
394C2#3177471411yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
prime radical#31774556C2
hypercenter#435D271
hypocenterC271#3270
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×Dic1084C2semidirect
3. D4×Dic542C2semidirect
5. #31774291C2semidirect
6. #31774155C2semidirect
7. #31774126C2semidirect
8. #31774120C2semidirect
9. #31774108C2semidirect
10. #31774061C2semidirect
11. #31774055C2semidirect
12. #31774055C2semidirect
13. #31774038C2semidirect
14. #31774005C2semidirect
15. #435×C271C2semidirect
89. Dic1084D4semidirect
173. C22×C271#137semidirect
197. C271#3270semidirect
213. #435D271semidirect
374. 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×Dic10842
D4×Dic5422
#317745562
#317742912
#317741552
#317741262
#317741202
#317741082
#317740612
#317740552
#317740382
#317740052
#435×C2712
#3270271