Finite Group Databasebeta

← #46822431

Subgroups — #46822431

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×#22575477C211yesmaximal, characteristic, complemented
2C2×#22575472C211yesmaximal, characteristic, complemented
3#31771275C211yesmaximal, characteristic, complemented
4#31771250C211yesmaximal, characteristic, complemented
5#31770992C211yesmaximal, characteristic, complemented
6#31770850C211yesmaximal, characteristic, complemented
7#31770840C211yesmaximal, characteristic, complemented
8#31770828C211yesmaximal, characteristic, complemented
9#31770768C211yesmaximal, characteristic, complemented
10#31770751C211yesmaximal, characteristic, complemented
11#31770747C211yesmaximal, characteristic, complemented
12#31770737C211yesmaximal, characteristic, complemented
13#31770705C211yesmaximal, characteristic, complemented
14#31770675C211yesmaximal, characteristic, complemented
15#459×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×#16864401C2211yescharacteristic
18C2×#16864401C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575477—24no—
21#22575472—24no—
22C2×Dic1076C2211yescharacteristic, complemented
23C2×#16864391C2211yescharacteristic
24C2×#16864390C2211yescharacteristic
25C2×#16864390C2211yescharacteristic, complemented
26C2×#16864390C2211yescharacteristic, complemented
27C2×#16864387C2211yescharacteristic
28C2×#16864387C2211yescharacteristic
29C2×#16864385C2211yescharacteristic
30C2×C8×D269C2211yescharacteristic, complemented
31#22575449—21no—
32#22575441C2211yescharacteristic
33#22575437—21no—
34#22575433C2211yescharacteristic
35#22575428C2211yescharacteristic
36#22575427—21no—
37#22575425—21no—
38#22575425—21no—
39#22575424C2211yescharacteristic
40#22575424C2211yescharacteristic, complemented
41#22575423—21no—
42C4×Dic538C2211yescharacteristic, complemented
43C4×D1076C2211yescharacteristic
44D4×Dic269—21no—
45C2×#16864379—21no—
46#22575386C2211yescharacteristic, complemented
47#22575378C2211yescharacteristic, complemented
48#22575376C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic
50#22575374C2211yescharacteristic
51#22575365C2211yescharacteristic
52#22575365C2211yescharacteristic, complemented
53#22575364C2211yescharacteristic, complemented
54C2×D8×C269C2211yescharacteristic
55C2×D8×C269C2211yescharacteristic
56#127×C269C2211yescharacteristic, complemented
57C4×D4×C269C2211yescharacteristic
58C4⋊C8×C269C2211yescharacteristic, complemented
59#102×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864401—22no—
63#16864401—22no—
64#16864401—22no—
65#16864401—41no—
66#16864401—41no—
67#16864401—41no—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70C2×D1076—21no—
71C2×D1076D411yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269—21no—
77C2×C22⋊D269—21no—
78C2×C22⋊D269—21no—
79D4×D269—22no—
80D4×D269—41no—
81C22×Dic269—21no—
82C22×Dic269—21no—
83C22×Dic269—21no—
84C2×C4×D269C2311yescharacteristic
85C2×C4×D269C2311yescharacteristic
86Dic1076—22no—
87#16864391—22no—
88#16864390—22no—
89#16864390—41no—
90#16864390D414yescomplemented
91#16864387—22no—
92#16864387—41no—
93#16864385—22no—
94C2×#14524130C2311yescharacteristic
95C2×#14524130C2311yescharacteristic
96C8×D269—22no—
97C4⋊Dic269—21no—
98C4⋊Dic269D411yescharacteristic
99C4⋊Dic269C2311yescharacteristic
100#16864381—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379D411yescharacteristic
108#16864379D411yescharacteristic
109C4×Dic269—21no—
110C2×D4×C269—21no—
111C2×D4×C269—21no—
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115C22×C4×C269—21no—
116D8×C269—22no—
117D8×C269—22no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269C2311yescharacteristic
121C22⋊C4×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—42no—
125C22×D269—42no—
126C22×D269—81no—
127C22×D269C2×D411yescharacteristic
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131Dic538—21no—
132Dic538C2×D412yes—
133D1076—21no—
134D1076—21no—
135D1076—41no—
136D1076—81no—
137C22⋊D269—42no—
138C22⋊D269—42no—
139C22⋊D269—42no—
140C22⋊D269—42no—
141C22⋊D269—81no—
142C22⋊D269—81no—
143C2×Dic269—21no—
144C2×Dic269—21no—
145C2×Dic269—21no—
146C2×Dic269—42no—
147C2×Dic269—42no—
148C2×Dic269—42no—
149C2×Dic269C2×D411yescharacteristic
150C2×Dic269C2×D411yescharacteristic
151C4×D269—21no—
152C4×D269—21no—
153C4×D269—21no—
154C4×D269—21no—
155#14524130—21no—
156#14524130C2×D412yes—
157C23×C269—21no—
158C23×C269—21no—
159C23×C269—21no—
160D4×C269—21no—
161D4×C269—21no—
162D4×C269—21no—
163D4×C269—21no—
164D4×C269—21no—
165D4×C269—42no—
166D4×C269—81no—
167D4×C269C2×D412yes—
168C2×C4×C269—21no—
169C2×C4×C269—81no—
170C2×C4×C269C2×D411yescharacteristic
171C2×C4×C269C2×D411yescharacteristic
172C2×C4×C269C2411yescharacteristic
173C8×C269—21no—
174C8×C269—21no—
175C22×C269—22no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269—41no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269C22×D411yescharacteristic
185C2×D269—21no—
186C2×D269—21no—
187C2×D269—21no—
188C2×D269—21no—
189C2×D269—21no—
190C2×D269—21no—
191C2×D269—41no—
192C2×D269—41no—
193C2×D269—41no—
194C2×D269—84no—
195C4×C269—21no—
196C4×C269—21no—
197C4×C269—41no—
198C4×C269C22×D411yescharacteristic
199C4×C269H(5,𝔽2)11yescharacteristic
200Dic269—21no—
201Dic269—21no—
202Dic269—22no—
203Dic269—41no—
204Dic269—41no—
205C2×C269—42no—
206C2×C269—81no—
207C2×C269—81no—
208C2×C269D4211yescharacteristic
209C2×C269#57511yescharacteristic
210C2×C269#57611yescharacteristic
211D269—41no—
212D269—41no—
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#459D26911yescharacteristic, complemented
225#452—2691no—
226#449—2691no—
227#449—2691no—
228#447—2691no—
229#440—2691no—
230#435—2691no—
231C2×#41—2691no—
232C2×#41—2691no—
233C2×#41—2691no—
234C22×D4—2691no—
235AGL(1,ℤ/8)—5384no—
236#135—5384no—
237C2×Q16—2691no—
238C2×SD16—2691no—
239C2×SD16—2691no—
240C2×SD16—2691no—
241C2×SD16—2691no—
242C2×D8—2691no—
243C2×D8—2691no—
244C2×D8C2×D26911yescharacteristic
245C2×D8C2×D26911yescharacteristic
246C2×M16—2691no—
247C22×C8—2691no—
248#127—5381no—
249#127C2×D26911yescharacteristic, complemented
250#124—2691no—
251#123—5381no—
252C22⋊Q8—2691no—
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—5381no—
263C4×D4C2×D26911yescharacteristic
264C2×C22⋊C4—5381no—
265#107—2691no—
266C4⋊C8C2×D26911yescharacteristic, complemented
267#103—2691no—
268#103—2691no—
269#102—2691no—
270#102—2691no—
271#102C2×D26911yescharacteristic, complemented
272#102C2×D26911yescharacteristic, complemented
273C22⋊C8—2691no—
274C22⋊C8—2691no—
275#97—2691no—
276C24—5381no—
277#41—5382no—
278#41—5382no—
279#41—5382no—
280#41—10761no—
281#41—10761no—
282#41—10761no—
283C2×Q8—2691no—
284C2×Q8—2691no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2691no—
288C2×D4—2691no—
289C2×D4—5381no—
290C2×D4—5381no—
291C2×D4—5382no—
292C2×D4—5381no—
293C2×D4—5381no—
294C2×D4—5381no—
295C2×D4—5381no—
296C2×D4—5381no—
297C2×D4—10761no—
298C2×D4C22×D26911yescharacteristic
299C2×D4C22×D26911yescharacteristic
300C2×D4C22×D26911yescharacteristic
301C22×C4—21no—
302C22×C4—2691no—
303C22×C4—2691no—
304C22×C4—5381no—
305C22×C4—5381no—
306C22×C4—5381no—
307Q16—5382no—
308SD16—2694no—
309SD16—5382no—
310SD16—5382no—
311SD16—10761no—
312D8—22no—
313D8—22no—
314D8—5382no—
315D8—10761no—
316M16—5382no—
317C2×C8—2691no—
318C2×C8—2691no—
319C2×C8—5382no—
320C2×C8C22×D26911yescharacteristic
321C2×C8C22×D26911yescharacteristic
322C4⋊C4—2691no—
323C4⋊C4—2691no—
324C4⋊C4—5381no—
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—5381no—
337C42C22×D26911yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2691no—
342C23—2691no—
343C23—5381no—
344C23—10762no—
345C23—10762no—
346C23—21521no—
347Q8—2692no—
348Q8—5381no—
349Q8—5381no—
350Q8—5381no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—5381no—
359D4—5381no—
360D4—10761no—
361D4—10762no—
362D4—10762no—
363D4—10762no—
364D4—10762no—
365D4—21521no—
366D4—21521no—
367D4—21521no—
368D4D4×D26912yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—2691no—
372C2×C4—2691no—
373C2×C4—5381no—
374C2×C4—5381no—
375C2×C4—5381no—
376C2×C4—5381no—
377C2×C4—5381no—
378C2×C4—5381no—
379C2×C4—5381no—
380C2×C4—10762no—
381C2×C4—10762no—
382C2×C4—10762no—
383C2×C4D4×D26911yescharacteristic
384C2×C4D4×D26911yescharacteristic
385C2×C4C23×D26911yescharacteristic
386C8—21no—
387C8—21no—
388C8—2692no—
389C8—5381no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5381no—
400C22—5381no—
401C22—5381no—
402C22—5381no—
403C22—5381no—
404C22—5381no—
405C22—10761no—
406C22—10761no—
407C22—10761no—
408C22—21524no—
409C22C2×D4×D26911yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5381no—
414C4—5381no—
415C4—5382no—
416C4—10761no—
417C4—10761no—
418C4C2×D4×D26911yescharacteristic
419C4#2257549911yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—10761no—
424C2—10761no—
425C2—21521no—
426C2#3177123911yescharacteristic
427C2#3177140111yescharacteristic
428C2#3177140611yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#459×C269C2
socleC22×C269C22×D4
hypercenter#459D269
hypocenterC269#3268
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C2×#22575472C2semidirect
3. #31771275C2semidirect
4. #31771250C2semidirect
5. #31770992C2semidirect
6. #31770850C2semidirect
7. #31770840C2semidirect
8. #31770828C2semidirect
9. #31770768C2semidirect
10. #31770751C2semidirect
11. #31770747C2semidirect
12. #31770737C2semidirect
13. #31770705C2semidirect
14. #31770675C2semidirect
15. #459×C269C2semidirect
18. C2×#16864401C22semidirect
22. C2×Dic1076C22semidirect
25. C2×#16864390C22semidirect
26. C2×#16864390C22semidirect
30. C2×C8×D269C22semidirect
40. #22575424C22semidirect
42. C4×Dic538C22semidirect
46. #22575386C22semidirect
47. #22575378C22semidirect
48. #22575376C22semidirect
52. #22575365C22semidirect
53. #22575364C22semidirect
56. #127×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #102×C269C22semidirect
60. #102×C269C22semidirect
90. #16864390D4semidirect
214. C269#3268semidirect
224. #459D269semidirect
249. #127C2×D269semidirect
266. C4⋊C8C2×D269semidirect
271. #102C2×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×#225754772
C2×#225754722
#317712752
#317712502
#317709922
#317708502
#317708402
#317708282
#317707682
#317707512
#317707472
#317707372
#317707052
#317706752
#459×C2692
#3268269