Finite Group Databasebeta

← #46822428

Subgroups — #46822428

Order 34432.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575478C211yesmaximal, characteristic, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3#31771274C211yesmaximal, characteristic, complemented
4#31771246C211yesmaximal, characteristic, complemented
5#31770991C211yesmaximal, characteristic, complemented
6#31770853C211yesmaximal, characteristic, complemented
7#31770840C211yesmaximal, characteristic, complemented
8#31770829C211yesmaximal, characteristic, complemented
9#31770799C211yesmaximal, characteristic, complemented
10#31770753C211yesmaximal, characteristic, complemented
11#31770749C211yesmaximal, characteristic, complemented
12#31770741C211yesmaximal, characteristic, complemented
13#31770708C211yesmaximal, characteristic, complemented
14#31770679C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic, complemented
18C22×D1076C2211yescharacteristic
19#22575478—24no—
20#22575476—24no—
21C2×#16864389C2211yescharacteristic
22C2×#16864389C2211yescharacteristic
23C2×#16864389C2211yescharacteristic, complemented
24C2×D2152C2211yescharacteristic
25C2×D2152C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864387C2211yescharacteristic
28C2×#16864386C2211yescharacteristic
29C2×C8×D269C2211yescharacteristic, complemented
30#22575449—21no—
31#22575449C2211yescharacteristic, complemented
32#22575442C2211yescharacteristic
33#22575438—21no—
34#22575428—21no—
35#22575428C2211yescharacteristic
36#22575427—21no—
37#22575427C2211yescharacteristic
38#22575424—21no—
39#22575424C2211yescharacteristic, complemented
40#22575423—21no—
41Q8×Dic269C2211yescharacteristic, complemented
42#22575415C2211yescharacteristic
43C4×C22⋊D269—21no—
44C4×C22⋊D269C2211yescharacteristic
45C22⋊C4×D269—21no—
46#22575386C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic
51#22575373C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C2×#41×C269C2211yescharacteristic
55C2×Q16×C269C2211yescharacteristic, complemented
56C2×SD16×C269C2211yescharacteristic, complemented
57C22⋊Q8×C269C2211yescharacteristic
58#103×C269C2211yescharacteristic, complemented
59#102×C269C2211yescharacteristic, complemented
60C22⋊C8×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403—22no—
64#16864403—41no—
65#16864403—41no—
66C2×D1076—22no—
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076—41no—
72C2×D1076C2311yescharacteristic
73C2×D1076C2311yescharacteristic
74C2×D1076C2311yescharacteristic
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77C2×C22⋊D269—21no—
78C2×C22⋊D269—21no—
79C2×C22⋊D269D411yescharacteristic
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269C2311yescharacteristic
84#16864389—22no—
85#16864389—22no—
86#16864389D414yescomplemented
87D2152—22no—
88D2152—41no—
89#16864387—22no—
90#16864387—22no—
91#16864386—22no—
92C2×#14524130C2311yescharacteristic
93C2×#14524130C2311yescharacteristic
94C8×D269—22no—
95C4⋊Dic269—21no—
96C4⋊Dic269C2311yescharacteristic
97#16864381D411yescharacteristic
98C22⋊Dic269—21no—
99C22⋊Dic269D411yescharacteristic
100#16864379—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379D411yescharacteristic
106C4×Dic269—21no—
107C4×Dic269C2311yescharacteristic
108#41×C269—22no—
109#41×C269—41no—
110C2×Q8×C269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C2×D4×C269—21no—
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—22no—
117SD16×C269—41no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269—21no—
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—81no—
129C22×D269C2×D411yescharacteristic
130D1076—21no—
131D1076—21no—
132D1076—21no—
133D1076—21no—
134D1076—21no—
135D1076—42no—
136D1076—42no—
137D1076—42no—
138D1076—81no—
139D1076C2×D412yes—
140C22⋊D269—41no—
141C22⋊D269—42no—
142C22⋊D269—81no—
143C22⋊D269—81no—
144C22⋊D269—81no—
145C2×Dic269—21no—
146C2×Dic269C2×D411yescharacteristic
147C2×Dic269C2×D411yescharacteristic
148C2×Dic269C2×D411yescharacteristic
149C4×D269—21no—
150C4×D269—21no—
151C4×D269—42no—
152C4×D269—42no—
153C4×D269—81no—
154#14524130—21no—
155#14524130C2×D412yes—
156C23×C269—21no—
157C23×C269C2×D411yescharacteristic
158Q8×C269—21no—
159Q8×C269—21no—
160Q8×C269—21no—
161Q8×C269C2×D412yes—
162D4×C269—21no—
163D4×C269—21no—
164D4×C269—42no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269—42no—
171C2×C4×C269C2×D411yescharacteristic
172C2×C4×C269C2411yescharacteristic
173C8×C269—21no—
174C8×C269—21no—
175C22×C269—21no—
176C22×C269—21no—
177C22×C269—21no—
178C22×C269—41no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269C22×D411yescharacteristic
182C2×D269—21no—
183C2×D269—22no—
184C2×D269—21no—
185C2×D269—21no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—41no—
189C2×D269—41no—
190C2×D269—41no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—84no—
195C4×C269—21no—
196C4×C269—22no—
197C4×C269—41no—
198C4×C269—41no—
199C4×C269C22×D411yescharacteristic
200C4×C269H(5,𝔽2)11yescharacteristic
201Dic269—21no—
202Dic269—21no—
203Dic269—21no—
204Dic269—41no—
205C2×C269—41no—
206C2×C269—81no—
207C2×C269D4211yescharacteristic
208C2×C269#57511yescharacteristic
209C2×C269#57611yescharacteristic
210D269—41no—
211D269—42no—
212D269—81no—
213D269—81no—
214C269#326811yescharacteristic, complemented, Sylow
215#3268—2691nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2691no—
217C2×#135—2691no—
218#550—2691no—
219#546—2691no—
220#496—2691no—
221#468—2691no—
222#466—2691no—
223#464—2691no—
224#459—2691no—
225#452D26911yescharacteristic, complemented
226#449—2691no—
227#449—2691no—
228#447—2691no—
229#440—2691no—
230#435—2691no—
231C2×#41—2691no—
232C2×#41—2691no—
233C2×#41C2×D26911yescharacteristic
234C22×D4—2691no—
235AGL(1,ℤ/8)—5384no—
236#135—5384no—
237C2×Q16C2×D26911yescharacteristic, complemented
238C2×SD16—2691no—
239C2×SD16—2691no—
240C2×SD16—2691no—
241C2×SD16C2×D26911yescharacteristic, complemented
242C2×D8—2691no—
243C2×D8—2691no—
244C2×D8—2691no—
245C2×D8—2691no—
246C2×M16—2691no—
247C22×C8—2691no—
248#127—2691no—
249#127—5381no—
250#124—2691no—
251#123—5381no—
252C22⋊Q8C2×D26911yescharacteristic
253#121—2691no—
254#121—2691no—
255#121—2691no—
256#121—5381no—
257#121—5381no—
258#121—5381no—
259#120—5381no—
260C4×Q8—2691no—
261C4×D4—2691no—
262C4×D4—2691no—
263C4×D4—5381no—
264C2×C22⋊C4—5381no—
265#107—2691no—
266C4⋊C8—2691no—
267#103—2691no—
268#103C2×D26911yescharacteristic, complemented
269#102—2691no—
270#102—2691no—
271#102—2691no—
272#102C2×D26911yescharacteristic, complemented
273C22⋊C8—2691no—
274C22⋊C8C2×D26911yescharacteristic
275#97—2691no—
276C24—5381no—
277#41—22no—
278#41—41no—
279#41—5382no—
280#41—5382no—
281#41—10761no—
282#41—10761no—
283C2×Q8C22×D26911yescharacteristic
284C2×Q8C22×D26911yescharacteristic
285C2×D4—21no—
286C2×D4—2691no—
287C2×D4—2691no—
288C2×D4—2691no—
289C2×D4—2691no—
290C2×D4—5382no—
291C2×D4—5381no—
292C2×D4—5381no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—10761no—
300C2×D4C22×D26911yescharacteristic
301C22×C4—21no—
302C22×C4—2691no—
303C22×C4—5381no—
304C22×C4—5381no—
305C22×C4—5381no—
306C22×C4C22×D26911yescharacteristic
307Q16—22no—
308SD16—41no—
309SD16—2694no—
310SD16—5382no—
311SD16—5382no—
312D8—5382no—
313D8—5382no—
314D8—5382no—
315D8—10761no—
316M16—5382no—
317C2×C8—2691no—
318C2×C8—2691no—
319C2×C8—5382no—
320C2×C8C22×D26911yescharacteristic
321C2×C8C22×D26911yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2691no—
324C4⋊C4—2691no—
325C4⋊C4—5381no—
326C4⋊C4C22×D26911yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2691no—
329C22⋊C4—2691no—
330C22⋊C4—5381no—
331C22⋊C4—5381no—
332C22⋊C4—5381no—
333C22⋊C4—5381no—
334C22⋊C4—5381no—
335C22⋊C4—5381no—
336C42—2691no—
337C42—5381no—
338C23—21no—
339C23—2691no—
340C23—5381no—
341C23—5381no—
342C23—5381no—
343C23—10762no—
344C23—10762no—
345C23—21521no—
346C23D4×D26911yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8D4×D26912yes—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—2692no—
355D4—5381no—
356D4—5381no—
357D4—5381no—
358D4—5381no—
359D4—5381no—
360D4—10762no—
361D4—10762no—
362D4—10762no—
363D4—10761no—
364D4—10762no—
365D4—21521no—
366D4—21521no—
367D4—21521no—
368D4—21521no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—2691no—
376C2×C4—2691no—
377C2×C4—2691no—
378C2×C4—5381no—
379C2×C4—5381no—
380C2×C4—5381no—
381C2×C4—10762no—
382C2×C4—10762no—
383C2×C4—21521no—
384C2×C4D4×D26911yescharacteristic
385C2×C4C23×D26911yescharacteristic
386C8—21no—
387C8—21no—
388C8—2692no—
389C8—5381no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—5381no—
397C22—5381no—
398C22—5381no—
399C22—5382no—
400C22—10761no—
401C22—10761no—
402C22—10761no—
403C22—10761no—
404C22—10761no—
405C22—10761no—
406C22—10761no—
407C22—10761no—
408C22—21524no—
409C22C2×D4×D26911yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—5381no—
415C4—5381no—
416C4—5381no—
417C4—10761no—
418C4C2×D4×D26911yescharacteristic
419C4#2257549811yescharacteristic
420C2—41no—
421C2—81no—
422C2—10761no—
423C2—10762no—
424C2—21521no—
425C2—21521no—
426C2#3177123711yescharacteristic
427C2#3177140411yescharacteristic
428C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
hypercenter#452D269
hypocenterC269#3268
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C2×#22575476C2semidirect
3. #31771274C2semidirect
4. #31771246C2semidirect
5. #31770991C2semidirect
6. #31770853C2semidirect
7. #31770840C2semidirect
8. #31770829C2semidirect
9. #31770799C2semidirect
10. #31770753C2semidirect
11. #31770749C2semidirect
12. #31770741C2semidirect
13. #31770708C2semidirect
14. #31770679C2semidirect
15. #452×C269C2semidirect
17. C2×#16864403C22semidirect
23. C2×#16864389C22semidirect
29. C2×C8×D269C22semidirect
31. #22575449C22semidirect
39. #22575424C22semidirect
41. Q8×Dic269C22semidirect
46. #22575386C22semidirect
47. #22575382C22semidirect
48. #22575377C22semidirect
51. #22575373C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
55. C2×Q16×C269C22semidirect
56. C2×SD16×C269C22semidirect
58. #103×C269C22semidirect
59. #102×C269C22semidirect
86. #16864389D4semidirect
214. C269#3268semidirect
225. #452D269semidirect
237. C2×Q16C2×D269semidirect
241. C2×SD16C2×D269semidirect
268. #103C2×D269semidirect
272. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C2×#225754762
#317712742
#317712462
#317709912
#317708532
#317708402
#317708292
#317707992
#317707532
#317707492
#317707412
#317707082
#317706792
#452×C2692
#3268269