Finite Group Databasebeta

← #46845816

Subgroups — #46845816

Order 34688.

496 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×#22576181C211yesmaximal, characteristic, complemented
2C2×#22576179C211yesmaximal, characteristic, complemented
3#31774446C211yesmaximal, characteristic, complemented
4C4×#16864653C211yesmaximal, characteristic, complemented
5#31774120C211yesmaximal, characteristic, complemented
6#31774116C211yesmaximal, characteristic, complemented
7#31774112C211yesmaximal, characteristic, complemented
8#31774106C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774096C211yesmaximal, characteristic, complemented
11#31774090C211yesmaximal, characteristic, complemented
12C4⋊C8×D271C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×#16864653C2211yescharacteristic
19#22576181—24no—
20#22576179—24no—
21C2×Dic1084C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864643C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576158C2211yescharacteristic
30#22576156C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576133—21no—
33#22576133—21no—
34C4×Dic542C2211yescharacteristic
35#22576125C2211yescharacteristic
36D4×Dic271C2211yescharacteristic
37C4×C22⋊D271—21no—
38(C22⋊D271)⋊C4—21no—
39(C22⋊D271)⋊C4—21no—
40(C22⋊D271)⋊C4—21no—
41#22576118C2211yescharacteristic
42#22576116—21no—
43C2×C4×Dic271—21no—
44C42×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic
46Dic271⋊C8C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576083C2211yescharacteristic
51#22576082C2211yescharacteristic
52#22576081C2211yescharacteristic
53#22576080C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—22no—
63#16864653—22no—
64#16864653—22no—
65#16864653—22no—
66#16864653D414yescomplemented
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69C2×D1084—21no—
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C22×Dic271—21no—
76C22×Dic271—21no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271D411yescharacteristic
80C2×C4×D271D411yescharacteristic
81C2×C4×D271C2311yescharacteristic
82Dic1084—22no—
83#16864644—22no—
84#16864643—22no—
85#16864642—22no—
86#16864642—22no—
87#16864639—22no—
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C8×D271—22no—
91C8×D271—22no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97C22⋊Dic271—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101C4×Dic271—21no—
102C4×Dic271—21no—
103C4×Dic271—21no—
104C4×Dic271D411yescharacteristic
105C4×Dic271D411yescharacteristic
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271—21no—
111SD16×C271—22no—
112D8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271C2311yescharacteristic
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271C2×D411yescharacteristic
122Dic542—21no—
123Dic542—21no—
124Dic542—21no—
125Dic542C2×D412yes—
126D1084—21no—
127D1084—21no—
128D1084—42no—
129C22⋊D271—24no—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C22⋊D271—42no—
133C2×Dic271—21no—
134C2×Dic271—24no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C2×Dic271—42no—
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—22no—
143C4×D271—41no—
144C4×D271—42no—
145C4×D271C2×D412yes—
146C4×D271#4112yes—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C23×C271—21no—
150C23×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153D4×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271C2×D412yes—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—42no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2×D411yescharacteristic
162C2×C4×C271C2411yescharacteristic
163C8×C271—21no—
164C8×C271—21no—
165C22×C271—22no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C2×D271#12112yes—
178C4×C271—21no—
179C4×C271—21no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271C22×D411yescharacteristic
183C4×C271C2×#4111yescharacteristic
184Dic271—22no—
185Dic271—21no—
186Dic271—41no—
187Dic271—41no—
188Dic271#12112yes—
189Dic271#13512yescomplemented
190C2×C271—42no—
191C2×C271—81no—
192C2×C271C2×#12111yescharacteristic
193C2×C271C2×#13511yescharacteristic
194C2×C271#57511yescharacteristic
195D271—22no—
196D271—81no—
197C271#303511yescharacteristic, complemented, Sylow
198#3035—2711nomaximal, Sylow
199C2×#135—2711no—
200C2×#135—2711no—
201#532—2711no—
202C4×#41—2711no—
203#464—2711no—
204#464—2711no—
205#463—2711no—
206#463D27111yescharacteristic, complemented
207#462—2711no—
208#461—2711no—
209#460—2711no—
210#459—2711no—
211C2×C4⋊C8—2711no—
212#416—2711no—
213#416—2711no—
214C2×#41—2711no—
215C2×#41—2711no—
216C2×#41—2711no—
217#135—5424no—
218#135—5424no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16—2711no—
224C2×SD16C2×D27111yescharacteristic
225C2×D8—2711no—
226C2×D8C2×D27111yescharacteristic
227C22×C8—2711no—
228C22×C8—2711no—
229C4⋊Q8—2711no—
230#127—2711no—
231#124—2711no—
232#124C2×D27111yescharacteristic
233#121—5421no—
234#121—5421no—
235C4×Q8—2711no—
236C4×Q8—2711no—
237C4×D4—2711no—
238C4×D4—5421no—
239C4×D4—5421no—
240C4×D4—5421no—
241C4×D4—5421no—
242C4×D4C2×D27111yescharacteristic
243#117—2711no—
244#117—5421no—
245C2×C42—2711no—
246C2×C42—5421no—
247C4⋊C8—2711no—
248C4⋊C8—2711no—
249C4⋊C8—2711no—
250C4⋊C8C2×D27111yescharacteristic
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103C2×D27111yescharacteristic
255#102—2711no—
256#102—2711no—
257#102—2711no—
258#102C2×D27111yescharacteristic
259#41—2714no—
260#41—5422no—
261#41—5422no—
262#41—5422no—
263#41—5422no—
264#41—5422no—
265C2×Q8—2711no—
266C2×Q8—2711no—
267C2×Q8C22×D27111yescharacteristic
268C2×D4—2711no—
269C2×D4—5421no—
270C2×D4—5421no—
271C2×D4—5421no—
272C2×D4—5421no—
273C2×D4—5421no—
274C2×D4C22×D27111yescharacteristic
275C2×D4C22×D27111yescharacteristic
276C22×C4—21no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—2711no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284Q16—5422no—
285Q16—5422no—
286SD16—22no—
287SD16—5422no—
288SD16—5422no—
289SD16—5422no—
290D8—22no—
291D8—5422no—
292C2×C8—2711no—
293C2×C8—2711no—
294C2×C8—5422no—
295C2×C8—5422no—
296C2×C8C22×D27111yescharacteristic
297C2×C8C22×D27111yescharacteristic
298C4⋊C4—2711no—
299C4⋊C4—5421no—
300C4⋊C4—5421no—
301C4⋊C4—5421no—
302C4⋊C4—5421no—
303C4⋊C4C22×D27111yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—5421no—
308C22⋊C4—5421no—
309C22⋊C4—5421no—
310C22⋊C4—5421no—
311C42—2711no—
312C42—2711no—
313C42—2711no—
314C42—5421no—
315C42—5421no—
316C42—5421no—
317C42C22×D27111yescharacteristic
318C23—21no—
319C23—21no—
320C23—2711no—
321C23—5421no—
322Q8—21no—
323Q8—21no—
324Q8—2712no—
325Q8—5421no—
326Q8—5421no—
327Q8—5421no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—5424no—
332D4—5421no—
333D4—5421no—
334D4—10842no—
335D4—10842no—
336D4—10842no—
337D4—10842no—
338D4D4×D27112yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—2711no—
343C2×C4—2711no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—2711no—
347C2×C4—5422no—
348C2×C4—5424no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—10841no—
353C2×C4—10842no—
354C2×C4—10842no—
355C2×C4—10842no—
356C2×C4D4×D27111yescharacteristic
357C2×C4D4×D27111yescharacteristic
358C2×C4C23×D27111yescharacteristic
359C8—21no—
360C8—21no—
361C8—5421no—
362C8—5421no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—2712no—
370C22—5421no—
371C22—5421no—
372C22—10841no—
373C22—10841no—
374C22—10841no—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—2712no—
381C4—2712no—
382C4—5422no—
383C4—5421no—
384C4—10841no—
385C4—10841no—
386C4C2×D4×D27111yescharacteristic
387C4#41×D27111yescharacteristic
388C2—42no—
389C2—81no—
390C2—5422no—
391C2—21681no—
392C2#121×D27111yescharacteristic
393C2#135×D27111yescharacteristic
394C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3035
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22576181C2semidirect
2. C2×#22576179C2semidirect
3. #31774446C2semidirect
4. C4×#16864653C2semidirect
5. #31774120C2semidirect
6. #31774116C2semidirect
7. #31774112C2semidirect
8. #31774106C2semidirect
9. #31774100C2semidirect
10. #31774096C2semidirect
11. #31774090C2semidirect
12. C4⋊C8×D271C2semidirect
13. #31773892C2semidirect
14. #31773890C2semidirect
15. #463×C271C2semidirect
66. #16864653D4semidirect
189. Dic271#135semidirect
197. C271#3035semidirect
206. #463D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761812
C2×#225761792
#317744462
C4×#168646532
#317741202
#317741162
#317741122
#317741062
#317741002
#317740962
#317740902
C4⋊C8×D2712
#317738922
#317738902
#463×C2712
#3035271