Finite Group Databasebeta

← #46845815

Subgroups — #46845815

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×#22576180C211yesmaximal, characteristic, complemented
2#31774443C211yesmaximal, characteristic, complemented
3#41×Dic271C211yesmaximal, characteristic, complemented
4#31774121C211yesmaximal, characteristic, complemented
5#31774117C211yesmaximal, characteristic, complemented
6#31774114C211yesmaximal, characteristic, complemented
7#31774110C211yesmaximal, characteristic, complemented
8#31774105C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774095C211yesmaximal, characteristic, complemented
11#31774090C211yesmaximal, characteristic, complemented
12C2×Dic271⋊C8C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14#31773893C211yesmaximal, characteristic, complemented
15C2×#135×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17#22576180—24no—
18C2×#16864644C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21C2×#16864639C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic
23#22576158C2211yescharacteristic
24#22576156C2211yescharacteristic
25#22576149C2211yescharacteristic
26#22576147C2211yescharacteristic
27#22576132—21no—
28#22576132—21no—
29Q8×Dic271C2211yescharacteristic
30Q8×Dic271C2211yescharacteristic
31D4×Dic271—21no—
32D4×Dic271—21no—
33D4×Dic271C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35C4×C22⋊D271—21no—
36C4×C22⋊D271—21no—
37#22576119—21no—
38#22576119C2211yescharacteristic
39C2×C4×Dic271—21no—
40C2×C4×Dic271C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44Dic271⋊C8C2211yescharacteristic
45#22576085C2211yescharacteristic
46#22576085C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576084C2211yescharacteristic
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic
51#22576081C2211yescharacteristic
52#22576081C2211yescharacteristic
53C2×#41×C271C2211yescharacteristic
54C2×#41×C271C2211yescharacteristic
55#135×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic
58C2×SD16×C271C2211yescharacteristic
59C2×D8×C271C2211yescharacteristic
60C22×C8×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—22no—
63C2×Dic542C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C22×Dic271—21no—
69C22×Dic271—21no—
70C22×Dic271D411yescharacteristic
71C22×Dic271D411yescharacteristic
72C2×C4×D271—21no—
73#16864644—22no—
74#16864642—22no—
75#16864641—22no—
76#16864639—22no—
77C2×C271⋊C8—22no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271—21no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271D411yescharacteristic
97C4×Dic271D411yescharacteristic
98C4×Dic271C2311yescharacteristic
99C4×Dic271C2311yescharacteristic
100#41×C271—22no—
101#41×C271—22no—
102#41×C271—22no—
103#41×C271D414yescomplemented
104C2×Q8×C271C2311yescharacteristic
105C2×Q8×C271C2311yescharacteristic
106C2×D4×C271—21no—
107C2×D4×C271—21no—
108C2×D4×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271—21no—
111C22×C4×C271—21no—
112C22×C4×C271C2311yescharacteristic
113Q16×C271—22no—
114SD16×C271—22no—
115SD16×C271—22no—
116D8×C271—22no—
117C2×C8×C271—22no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C22×D271—21no—
121Dic542—21no—
122Dic542—21no—
123D1084—21no—
124D1084—21no—
125C22⋊D271—42no—
126C22⋊D271—42no—
127C22⋊D271—42no—
128C2×Dic271—22no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—41no—
133C2×Dic271—42no—
134C2×Dic271—42no—
135C2×Dic271C2×D411yescharacteristic
136C2×Dic271C2×D411yescharacteristic
137C2×Dic271C2×D411yescharacteristic
138C2×Dic271C2×D411yescharacteristic
139C4×D271—42no—
140C271⋊C8—21no—
141C271⋊C8—21no—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271C2×D411yescharacteristic
145Q8×C271—21no—
146Q8×C271—21no—
147Q8×C271—21no—
148Q8×C271C2×D412yes—
149D4×C271—24no—
150D4×C271—21no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—42no—
154D4×C271C2×D412yes—
155C2×C4×C271—21no—
156C2×C4×C271—24no—
157C2×C4×C271—21no—
158C2×C4×C271—42no—
159C2×C4×C271C2×D411yescharacteristic
160C2×C4×C271C2×D412yes—
161C2×C4×C271#4112yes—
162C2×C4×C271C2411yescharacteristic
163C8×C271—21no—
164C8×C271—21no—
165C22×C271—21no—
166C22×C271—21no—
167C22×C271—22no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271#12112yes—
174C22×C271C22×D411yescharacteristic
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—41no—
178C4×C271—22no—
179C4×C271—41no—
180C4×C271#12112yes—
181C4×C271C22×D411yescharacteristic
182C4×C271C2×#4111yescharacteristic
183Dic271—21no—
184Dic271—21no—
185Dic271—21no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189Dic271#13512yescomplemented
190C2×C271—22no—
191C2×C271—42no—
192C2×C271—81no—
193C2×C271C2×#12111yescharacteristic
194C2×C271C2×#13511yescharacteristic
195C2×C271#57511yescharacteristic
196D271—81no—
197C271#303511yescharacteristic, complemented, Sylow
198#3035—2711nomaximal, Sylow
199C2×#135—2711no—
200C2×#135D27111yescharacteristic, complemented
201#532—2711no—
202C4×#41—2711no—
203#464—2711no—
204#464—2711no—
205#463—2711no—
206#463—2711no—
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×#41C2×D27111yescharacteristic
216C2×#41C2×D27111yescharacteristic
217#135—24no—
218#135—5424no—
219C2×Q16—2711no—
220C2×Q16C2×D27111yescharacteristic
221C2×SD16—2711no—
222C2×SD16—2711no—
223C2×SD16C2×D27111yescharacteristic
224C2×SD16C2×D27111yescharacteristic
225C2×D8—2711no—
226C2×D8C2×D27111yescharacteristic
227C22×C8—2711no—
228C22×C8C2×D27111yescharacteristic
229C4⋊Q8—2711no—
230#127—2711no—
231#124—2711no—
232#124—2711no—
233#121—5421no—
234#121—5421no—
235C4×Q8—2711no—
236C4×Q8—2711no—
237C4×D4—2711no—
238C4×D4—2711no—
239C4×D4—5421no—
240C4×D4—5421no—
241C4×D4—5421no—
242C4×D4—5421no—
243#117—2711no—
244#117—5421no—
245C2×C42—2711no—
246C2×C42—5421no—
247C4⋊C8—2711no—
248C4⋊C8—2711no—
249C4⋊C8—2711no—
250C4⋊C8—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#102—2711no—
256#102—2711no—
257#102—2711no—
258#102—2711no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—5422no—
263#41—5422no—
264#41C22⋊D27114yescomplemented
265C2×Q8—2711no—
266C2×Q8C22×D27111yescharacteristic
267C2×Q8C22×D27111yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—2711no—
271C2×D4—5421no—
272C2×D4—5421no—
273C2×D4—5421no—
274C2×D4C22×D27111yescharacteristic
275C2×D4C22×D27111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2711no—
279C22×C4—2711no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4C22×D27111yescharacteristic
284Q16—22no—
285Q16—5422no—
286SD16—22no—
287SD16—22no—
288SD16—5422no—
289SD16—5422no—
290D8—22no—
291D8—5422no—
292C2×C8—22no—
293C2×C8—2711no—
294C2×C8—2711no—
295C2×C8—5422no—
296C2×C8C22×D27111yescharacteristic
297C2×C8C22×D27111yescharacteristic
298C4⋊C4—2711no—
299C4⋊C4—2711no—
300C4⋊C4—5421no—
301C4⋊C4—5421no—
302C4⋊C4—5421no—
303C4⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C22⋊C4—5421no—
309C22⋊C4—5421no—
310C22⋊C4—5421no—
311C42—2711no—
312C42—2711no—
313C42—2711no—
314C42—2711no—
315C42—5421no—
316C42—5421no—
317C42—5421no—
318C23—21no—
319C23—21no—
320C23—5421no—
321C23D4×D27111yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—5421no—
326Q8—5421no—
327Q8C2×C22⋊D27112yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—5421no—
334D4—5421no—
335D4—10842no—
336D4—10842no—
337D4—10842no—
338D4C2×C22⋊D27112yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—2711no—
344C2×C4—2711no—
345C2×C4—2711no—
346C2×C4—2711no—
347C2×C4—5422no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—10842no—
352C2×C4—10841no—
353C2×C4—10842no—
354C2×C4—10842no—
355C2×C4D4×D27111yescharacteristic
356C2×C4C2×C22⋊D27112yes—
357C2×C4#1686465312yes—
358C2×C4C23×D27111yescharacteristic
359C8—21no—
360C8—21no—
361C8—5421no—
362C8—5421no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—10841no—
372C22—10841no—
373C22—10841no—
374C22#2257613212yes—
375C22C2×D4×D27111yescharacteristic
376C4—22no—
377C4—41no—
378C4—2712no—
379C4—5421no—
380C4—5421no—
381C4—5421no—
382C4—10841no—
383C4—10841no—
384C4—10841no—
385C4#2257613212yes—
386C4C22×C22⋊D27111yescharacteristic
387C4C2×#1686465311yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—21681no—
392C2C2×#2257613211yescharacteristic
393C2#135×D27111yescharacteristic
394C2#3177469411yescharacteristic
subgroupG/H
centerC2×C4D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#135×C271C2
socleC22×C271C22×D4
hypercenterC2×#135D271
hypocenterC271#3035
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22576180C2semidirect
2. #31774443C2semidirect
3. #41×Dic271C2semidirect
4. #31774121C2semidirect
5. #31774117C2semidirect
6. #31774114C2semidirect
7. #31774110C2semidirect
8. #31774105C2semidirect
9. #31774100C2semidirect
10. #31774095C2semidirect
11. #31774090C2semidirect
12. C2×Dic271⋊C8C2semidirect
13. #31773894C2semidirect
14. #31773893C2semidirect
15. C2×#135×C271C2semidirect
103. #41×C271D4semidirect
189. Dic271#135semidirect
197. C271#3035semidirect
200. C2×#135D271semidirect
264. #41C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761802
#317744432
#41×Dic2712
#317741212
#317741172
#317741142
#317741102
#317741052
#317741002
#317740952
#317740902
C2×Dic271⋊C82
#317738942
#317738932
C2×#135×C2712
#3035271