Finite Group Databasebeta

← #46820427

Subgroups — #46820427

Order 34432.

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×#22575480C211yesmaximal, characteristic, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3#31771173C211yesmaximal, characteristic, complemented
4#31771087C211yesmaximal, characteristic, complemented
5#31770829C211yesmaximal, characteristic, complemented
6#31770823C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770805C211yesmaximal, characteristic, complemented
9#31770803C211yesmaximal, characteristic, complemented
10#31770801C211yesmaximal, characteristic, complemented
11#31770799C211yesmaximal, characteristic, complemented
12#31770630C211yesmaximal, characteristic, complemented
13#31770605C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic
18C2×D4×D269C2211yescharacteristic
19#22575480—24no—
20#22575476—24no—
21C2×#16864389C2211yescharacteristic
22C2×#16864389C2211yescharacteristic, complemented
23C2×#16864389C2211yescharacteristic, complemented
24C2×D2152C2211yescharacteristic
25C2×D2152C2211yescharacteristic, complemented
26C2×#16864387C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C2×C8×D269C2211yescharacteristic
29#22575450C2211yescharacteristic
30#22575449C2211yescharacteristic
31#22575442C2211yescharacteristic
32#22575426—21no—
33#22575426—21no—
34#22575423—21no—
35#22575423—21no—
36Q8×Dic269C2211yescharacteristic, complemented
37C4×D1076—21no—
38C4×D1076C2211yescharacteristic, complemented
39#22575415—21no—
40#22575415C2211yescharacteristic, complemented
41#22575410—21no—
42#22575410C2211yescharacteristic
43C4⋊C4×D269—21no—
44C4⋊C4×D269C2211yescharacteristic
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575382C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic
52#22575373C2211yescharacteristic
53#22575373C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic
56#124×C269C2211yescharacteristic
57C4×Q8×C269C2211yescharacteristic, complemented
58C4⋊C8×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403—22no—
64#16864403—22no—
65#16864403D414yescomplemented
66C2×D1076—21no—
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076C2311yescharacteristic
71C2×D1076C2311yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—22no—
77D4×D269—22no—
78C2×C4×D269—21no—
79C2×C4×D269—21no—
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269C2311yescharacteristic
84#16864389—22no—
85#16864389—22no—
86#16864389—22no—
87D2152—22no—
88D2152—22no—
89#16864387—22no—
90#16864385—22no—
91C2×#14524130C2311yescharacteristic
92C2×#14524130C2311yescharacteristic
93C8×D269—22no—
94C4⋊Dic269—21no—
95C4⋊Dic269C2311yescharacteristic
96#16864381—21no—
97#16864381D411yescharacteristic
98#16864381D411yescharacteristic
99#16864379—21no—
100#16864379—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379D411yescharacteristic
104#16864379D411yescharacteristic
105C4×Dic269—21no—
106C4×Dic269C2311yescharacteristic
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2311yescharacteristic
110Q16×C269—22no—
111SD16×C269—22no—
112C2×C8×C269C2311yescharacteristic
113C2×C8×C269C2311yescharacteristic
114C4⋊C4×C269—21no—
115C4⋊C4×C269C2311yescharacteristic
116C22⋊C4×C269—21no—
117C22⋊C4×C269—21no—
118C42×C269—21no—
119C42×C269C2311yescharacteristic
120C22×D269—21no—
121C22×D269—21no—
122C22×D269—21no—
123C22×D269—42no—
124C22×D269—81no—
125C22×D269—81no—
126C22×D269C2×D411yescharacteristic
127D1076—24no—
128D1076—21no—
129D1076—21no—
130D1076—21no—
131D1076—21no—
132D1076—21no—
133D1076—42no—
134D1076—81no—
135D1076—81no—
136D1076C2×D412yes—
137C22⋊D269—42no—
138C22⋊D269—81no—
139C22⋊D269—81no—
140C2×Dic269—21no—
141C2×Dic269C2×D411yescharacteristic
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C4×D269—24no—
145C4×D269—42no—
146C4×D269—81no—
147C4×D269—81no—
148C4×D269—81no—
149C4×D269C2×D412yes—
150C4×D269#4112yes—
151#14524130—21no—
152#14524130—21no—
153C23×C269—21no—
154Q8×C269—21no—
155Q8×C269—21no—
156Q8×C269—21no—
157Q8×C269C2×D412yes—
158D4×C269—21no—
159D4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269C2×D411yescharacteristic
164C2×C4×C269C2×D411yescharacteristic
165C2×C4×C269C2411yescharacteristic
166C8×C269—21no—
167C8×C269—21no—
168C22×C269—41no—
169C22×C269—41no—
170C22×C269—41no—
171C22×C269C22×D411yescharacteristic
172C2×D269—21no—
173C2×D269—21no—
174C2×D269—22no—
175C2×D269—41no—
176C2×D269—41no—
177C2×D269—41no—
178C2×D269—41no—
179C2×D269—41no—
180C2×D269—41no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—84no—
184C2×D269#12112yes—
185C4×C269—22no—
186C4×C269—21no—
187C4×C269—21no—
188C4×C269—41no—
189C4×C269—41no—
190C4×C269C22×D411yescharacteristic
191C4×C269C2×#4111yescharacteristic
192Dic269—21no—
193Dic269—21no—
194Dic269—41no—
195Dic269#12112yes—
196C2×C269—81no—
197C2×C269C2×#12111yescharacteristic
198C2×C269#57611yescharacteristic
199C2×C269#57711yescharacteristic
200D269—22no—
201D269—42no—
202D269—81no—
203D269—81no—
204C269#304511yescharacteristic, complemented, Sylow
205#3045—2691nomaximal, Sylow
206C2×AGL(1,ℤ/8)—2691no—
207C2×#135—2691no—
208#535—2691no—
209#520—2691no—
210#464—2691no—
211#464D26911yescharacteristic, complemented
212#463—2691no—
213#463—2691no—
214#460—2691no—
215#460—2691no—
216#459—2691no—
217#459—2691no—
218#424—2691no—
219#417—2691no—
220C2×#102—2691no—
221C2×#41—2691no—
222C2×#41—2691no—
223C22×D4—2691no—
224AGL(1,ℤ/8)—5384no—
225#135—5384no—
226C2×Q16C2×D26911yescharacteristic, complemented
227C2×SD16—2691no—
228C2×SD16—2691no—
229C2×SD16—2691no—
230C2×SD16C2×D26911yescharacteristic
231C2×D8—2691no—
232C2×D8—2691no—
233C2×D8—2691no—
234C2×M16—2691no—
235C22×C8—2691no—
236#127—2691no—
237#127—2691no—
238#124—2691no—
239#124C2×D26911yescharacteristic
240#121—5381no—
241#121—5381no—
242#120—5381no—
243#120—5381no—
244C4×Q8—2691no—
245C4×Q8C2×D26911yescharacteristic, complemented
246C4×D4—2691no—
247C4×D4—2691no—
248C4×D4—5381no—
249C4×D4—5381no—
250#117—2691no—
251#117—5381no—
252C2×C4⋊C4—2691no—
253C2×C4⋊C4—5381no—
254C4⋊C8—2691no—
255C4⋊C8—2691no—
256C4⋊C8—2691no—
257C4⋊C8C2×D26911yescharacteristic, complemented
258#103—2691no—
259#103C2×D26911yescharacteristic, complemented
260#102—2691no—
261#102—2691no—
262#102—2691no—
263#102—2691no—
264#102—2691no—
265#102C2×D26911yescharacteristic
266C24—5381no—
267#41—2694no—
268#41—5382no—
269#41—5382no—
270#41—5382no—
271C2×Q8C22×D26911yescharacteristic
272C2×Q8C22×D26911yescharacteristic
273C2×D4—2691no—
274C2×D4—2691no—
275C2×D4—2691no—
276C2×D4—5381no—
277C2×D4—5381no—
278C2×D4—5382no—
279C2×D4—5382no—
280C2×D4—5381no—
281C2×D4—5381no—
282C2×D4—5381no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4C22×D26911yescharacteristic
286C22×C4—2691no—
287C22×C4—5381no—
288C22×C4—5381no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4—5381no—
292Q16—22no—
293SD16—22no—
294SD16—5382no—
295SD16—5382no—
296SD16—5382no—
297D8—5382no—
298D8—5382no—
299D8—5382no—
300M16—5382no—
301C2×C8—2691no—
302C2×C8—2691no—
303C2×C8—5382no—
304C2×C8C22×D26911yescharacteristic
305C2×C8C22×D26911yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—2691no—
308C4⋊C4—2691no—
309C4⋊C4—2691no—
310C4⋊C4—5381no—
311C4⋊C4—5381no—
312C4⋊C4C22×D26911yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—2691no—
316C22⋊C4—2691no—
317C22⋊C4—5381no—
318C22⋊C4—5381no—
319C22⋊C4—5381no—
320C22⋊C4—5381no—
321C42—21no—
322C42—2691no—
323C42—5381no—
324C42C22×D26911yescharacteristic
325C23—21no—
326C23—2691no—
327C23—5381no—
328C23—5381no—
329C23—5381no—
330C23—10762no—
331C23—21521no—
332C23—21521no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8D4×D26912yes—
337D4—21no—
338D4—21no—
339D4—2692no—
340D4—5384no—
341D4—5381no—
342D4—5381no—
343D4—5381no—
344D4—5381no—
345D4—5381no—
346D4—10762no—
347D4—10762no—
348D4—21521no—
349D4—21521no—
350D4—21521no—
351D4—21521no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2691no—
356C2×C4—2692no—
357C2×C4—2692no—
358C2×C4—2691no—
359C2×C4—2691no—
360C2×C4—5384no—
361C2×C4—5381no—
362C2×C4—10762no—
363C2×C4—21521no—
364C2×C4—21521no—
365C2×C4—21521no—
366C2×C4D4×D26911yescharacteristic
367C2×C4D4×D26911yescharacteristic
368C2×C4C23×D26911yescharacteristic
369C8—21no—
370C8—21no—
371C8—5381no—
372C8—5381no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—2692no—
377C22—5381no—
378C22—5381no—
379C22—5382no—
380C22—10761no—
381C22—10761no—
382C22—10761no—
383C22—10761no—
384C22—10761no—
385C22—10761no—
386C22—10761no—
387C22—10761no—
388C22—21524no—
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—2692no—
396C4—5381no—
397C4—5381no—
398C4—10761no—
399C4C2×D4×D26911yescharacteristic
400C4#41×D26911yescharacteristic
401C2—81no—
402C2—5382no—
403C2—10762no—
404C2—21521no—
405C2—21521no—
406C2#121×D26911yescharacteristic
407C2#3177140711yescharacteristic
408C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
hypercenter#464D269
hypocenterC269#3045
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×#22575476C2semidirect
3. #31771173C2semidirect
4. #31771087C2semidirect
5. #31770829C2semidirect
6. #31770823C2semidirect
7. #31770817C2semidirect
8. #31770805C2semidirect
9. #31770803C2semidirect
10. #31770801C2semidirect
11. #31770799C2semidirect
12. #31770630C2semidirect
13. #31770605C2semidirect
14. #102×D269C2semidirect
15. #464×C269C2semidirect
22. C2×#16864389C22semidirect
23. C2×#16864389C22semidirect
25. C2×D2152C22semidirect
36. Q8×Dic269C22semidirect
38. C4×D1076C22semidirect
40. #22575415C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575382C22semidirect
47. #22575382C22semidirect
48. #22575377C22semidirect
50. #22575375C22semidirect
53. #22575373C22semidirect
54. C2×Q16×C269C22semidirect
57. C4×Q8×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #103×C269C22semidirect
65. #16864403D4semidirect
204. C269#3045semidirect
211. #464D269semidirect
226. C2×Q16C2×D269semidirect
245. C4×Q8C2×D269semidirect
257. C4⋊C8C2×D269semidirect
259. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C2×#225754762
#317711732
#317710872
#317708292
#317708232
#317708172
#317708052
#317708032
#317708012
#317707992
#317706302
#317706052
#102×D2692
#464×C2692
#3045269