Finite Group Databasebeta

← #46845904

Subgroups — #46845904

Order 34688.

504 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×#22576184C211yesmaximal, characteristic, complemented
2C2×#22576181C211yesmaximal, characteristic, complemented
3#31774465C211yesmaximal, characteristic, complemented
4#31774376C211yesmaximal, characteristic, complemented
5#31774122C211yesmaximal, characteristic, complemented
6#31774116C211yesmaximal, characteristic, complemented
7#31774114C211yesmaximal, characteristic, complemented
8#31774098C211yesmaximal, characteristic, complemented
9#31774096C211yesmaximal, characteristic, complemented
10#31774092C211yesmaximal, characteristic, complemented
11#31774090C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576184—24no—
20#22576181—24no—
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864643C2211yescharacteristic
23C2×#16864643C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic, complemented
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576157C2211yescharacteristic
30#22576156C2211yescharacteristic
31#22576149C2211yescharacteristic
32#22576133—21no—
33#22576133—21no—
34#22576130—21no—
35#22576130—21no—
36C4×Dic542C2211yescharacteristic, complemented
37#22576125C2211yescharacteristic, complemented
38D4×Dic271C2211yescharacteristic, complemented
39C4×C22⋊D271—21no—
40(C22⋊D271)⋊C4—21no—
41#22576117C2211yescharacteristic
42#22576116—21no—
43C2×Dic271⋊C4—21no—
44C4⋊C4×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46Dic271⋊C8C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic
52#22576080C2211yescharacteristic
53#22576080C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64#16864653—22no—
65#16864653D414yescomplemented
66C2×Dic542C2311yescharacteristic
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76D4×D271—22no—
77D4×D271—22no—
78C22×Dic271—21no—
79C22×Dic271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83#16864644—22no—
84#16864643—22no—
85#16864643—22no—
86#16864642—22no—
87#16864639—22no—
88#16864639—22no—
89#16864637—22no—
90C2×C271⋊C8C2311yescharacteristic
91C2×C271⋊C8C2311yescharacteristic
92C8×D271—22no—
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4D411yescharacteristic
98Dic271⋊C4D411yescharacteristic
99C22⋊Dic271—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4D411yescharacteristic
103(C2×D271)⋊C4D411yescharacteristic
104C4×Dic271—21no—
105C4×Dic271—21no—
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×D271—21no—
122C22×D271—42no—
123C22×D271—81no—
124C22×D271—81no—
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542C2×D412yes—
128D1084—21no—
129D1084—21no—
130D1084—21no—
131D1084—21no—
132D1084—42no—
133D1084—81no—
134D1084—81no—
135C22⋊D271—24no—
136C22⋊D271—42no—
137C22⋊D271—81no—
138C22⋊D271—81no—
139C2×Dic271—21no—
140C2×Dic271—24no—
141C2×Dic271—21no—
142C2×Dic271—81no—
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C2×Dic271C2×D411yescharacteristic
146C4×D271—42no—
147C4×D271—81no—
148C4×D271C2×D412yes—
149C4×D271#4112yes—
150C271⋊C8—21no—
151C271⋊C8—21no—
152C23×C271—21no—
153C23×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271C2×D412yes—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—81no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271—21no—
168C22×C271—22no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271C22×D411yescharacteristic
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—41no—
178C2×D271—41no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—84no—
184C2×D271#12112yes—
185C4×C271—21no—
186C4×C271—21no—
187C4×C271—41no—
188C4×C271—41no—
189C4×C271C22×D411yescharacteristic
190C4×C271C2×#4111yescharacteristic
191Dic271—22no—
192Dic271—21no—
193Dic271—21no—
194Dic271—41no—
195Dic271#12112yes—
196C2×C271—42no—
197C2×C271—81no—
198C2×C271C2×#12111yescharacteristic
199C2×C271#57611yescharacteristic
200C2×C271#57711yescharacteristic
201D271—22no—
202D271—81no—
203D271—81no—
204C271#304511yescharacteristic, complemented, Sylow
205#3045—2711nomaximal, Sylow
206C2×AGL(1,ℤ/8)—2711no—
207C2×#135—2711no—
208#535—2711no—
209#520—2711no—
210#464—2711no—
211#464—2711no—
212#463—2711no—
213#463D27111yescharacteristic, complemented
214#460—2711no—
215#460—2711no—
216#459—2711no—
217#459—2711no—
218#424—2711no—
219#417—2711no—
220C2×#102—2711no—
221C2×#41—2711no—
222C2×#41—2711no—
223C22×D4—2711no—
224AGL(1,ℤ/8)—5424no—
225#135—5424no—
226C2×Q16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×SD16—2711no—
230C2×SD16C2×D27111yescharacteristic, complemented
231C2×D8—2711no—
232C2×D8—2711no—
233C2×D8C2×D27111yescharacteristic
234C2×M16—2711no—
235C22×C8—2711no—
236#127—2711no—
237#127—2711no—
238#124—2711no—
239#124C2×D27111yescharacteristic
240#121—5421no—
241#121—5421no—
242#120—5421no—
243#120—5421no—
244C4×Q8—2711no—
245C4×Q8—2711no—
246C4×D4—2711no—
247C4×D4—5421no—
248C4×D4—5421no—
249C4×D4C2×D27111yescharacteristic, complemented
250#117—2711no—
251#117—5421no—
252C2×C4⋊C4—2711no—
253C2×C4⋊C4—5421no—
254C4⋊C8—2711no—
255C4⋊C8—2711no—
256C4⋊C8—2711no—
257C4⋊C8C2×D27111yescharacteristic, complemented
258#103—2711no—
259#103C2×D27111yescharacteristic, complemented
260#102—2711no—
261#102—2711no—
262#102—2711no—
263#102—2711no—
264#102—2711no—
265#102C2×D27111yescharacteristic
266C24—5421no—
267#41—2714no—
268#41—5422no—
269#41—5422no—
270#41—5422no—
271C2×Q8—2711no—
272C2×Q8C22×D27111yescharacteristic
273C2×D4—2711no—
274C2×D4—2711no—
275C2×D4—5421no—
276C2×D4—5422no—
277C2×D4—5422no—
278C2×D4—5421no—
279C2×D4—5421no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4C22×D27111yescharacteristic
285C2×D4C22×D27111yescharacteristic
286C22×C4—21no—
287C22×C4—2711no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4—5421no—
292Q16—5422no—
293SD16—22no—
294SD16—5422no—
295SD16—5422no—
296SD16—5422no—
297D8—22no—
298D8—5422no—
299D8—5422no—
300M16—5422no—
301C2×C8—2711no—
302C2×C8—2711no—
303C2×C8—5422no—
304C2×C8C22×D27111yescharacteristic
305C2×C8C22×D27111yescharacteristic
306C4⋊C4—2711no—
307C4⋊C4—2711no—
308C4⋊C4—2711no—
309C4⋊C4—5421no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4C22×D27111yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—2711no—
317C22⋊C4—2711no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C42—2711no—
322C42—5421no—
323C42—5421no—
324C42C22×D27111yescharacteristic
325C23—21no—
326C23—21no—
327C23—2711no—
328C23—5421no—
329C23—5421no—
330C23—10842no—
331C23—21681no—
332C23—21681no—
333Q8—21no—
334Q8—21no—
335Q8—2712no—
336Q8—5421no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—5424no—
341D4—5421no—
342D4—5421no—
343D4—5421no—
344D4—5421no—
345D4—10842no—
346D4—10842no—
347D4—21681no—
348D4—21681no—
349D4—21681no—
350D4—21681no—
351D4D4×D27112yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—81no—
355C2×C4—2711no—
356C2×C4—2712no—
357C2×C4—2712no—
358C2×C4—2711no—
359C2×C4—2711no—
360C2×C4—5424no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—10842no—
364C2×C4—21681no—
365C2×C4—21681no—
366C2×C4D4×D27111yescharacteristic
367C2×C4D4×D27111yescharacteristic
368C2×C4C23×D27111yescharacteristic
369C8—21no—
370C8—21no—
371C8—5421no—
372C8—5421no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2712no—
380C22—5421no—
381C22—5421no—
382C22—10841no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22—10841no—
388C22—21684no—
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—2712no—
395C4—5422no—
396C4—5421no—
397C4—5421no—
398C4—10841no—
399C4C2×D4×D27111yescharacteristic
400C4#41×D27111yescharacteristic
401C2—42no—
402C2—81no—
403C2—5422no—
404C2—21681no—
405C2—21681no—
406C2#121×D27111yescharacteristic
407C2#3177470111yescharacteristic
408C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3045
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×#22576184C2semidirect
2. C2×#22576181C2semidirect
3. #31774465C2semidirect
4. #31774376C2semidirect
5. #31774122C2semidirect
6. #31774116C2semidirect
7. #31774114C2semidirect
8. #31774098C2semidirect
9. #31774096C2semidirect
10. #31774092C2semidirect
11. #31774090C2semidirect
12. #31773922C2semidirect
13. #31773897C2semidirect
14. #102×D271C2semidirect
15. #463×C271C2semidirect
21. C2×#16864644C22semidirect
23. C2×#16864643C22semidirect
26. C2×#16864639C22semidirect
36. C4×Dic542C22semidirect
37. #22576125C22semidirect
38. D4×Dic271C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. Dic271⋊C8C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576084C22semidirect
50. #22576082C22semidirect
53. #22576080C22semidirect
54. C2×SD16×C271C22semidirect
57. C4×D4×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
65. #16864653D4semidirect
204. C271#3045semidirect
213. #463D271semidirect
230. C2×SD16C2×D271semidirect
249. C4×D4C2×D271semidirect
257. C4⋊C8C2×D271semidirect
259. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×#225761812
#317744652
#317743762
#317741222
#317741162
#317741142
#317740982
#317740962
#317740922
#317740902
#317739222
#317738972
#102×D2712
#463×C2712
#3045271