Finite Group Databasebeta

← #46822413

Subgroups — #46822413

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575478C211yesmaximal, characteristic, complemented
2C2×#22575474C211yesmaximal, characteristic, complemented
3#31771254C211yesmaximal, characteristic, complemented
4#31771246C211yesmaximal, characteristic, complemented
5#31770998C211yesmaximal, characteristic, complemented
6#31770861C211yesmaximal, characteristic, complemented
7#31770837C211yesmaximal, characteristic, complemented
8#31770822C211yesmaximal, characteristic, complemented
9#31770809C211yesmaximal, characteristic, complemented
10#31770766C211yesmaximal, characteristic, complemented
11#31770758C211yesmaximal, characteristic, complemented
12#31770751C211yesmaximal, characteristic, complemented
13#31770714C211yesmaximal, characteristic, complemented
14#31770679C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864401C2211yescharacteristic, complemented
18C22×D1076C2211yescharacteristic
19#22575478—24no—
20#22575474—24no—
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864391C2211yescharacteristic
23C2×#16864391C2211yescharacteristic
24C2×#16864389C2211yescharacteristic
25C2×#16864389C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864387C2211yescharacteristic
28C2×#16864386C2211yescharacteristic
29C2×C8×D269C2211yescharacteristic, complemented
30#22575451C2211yescharacteristic, complemented
31#22575442C2211yescharacteristic
32#22575438—21no—
33#22575436—21no—
34#22575434—21no—
35#22575432C2211yescharacteristic
36#22575429C2211yescharacteristic, complemented
37#22575428—21no—
38#22575428C2211yescharacteristic
39#22575425—21no—
40#22575423—21no—
41D4×Dic269C2211yescharacteristic, complemented
42#22575415C2211yescharacteristic
43C4×C22⋊D269C2211yescharacteristic
44C2×#16864381—21no—
45C22⋊C4×D269—21no—
46#22575384C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575376C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic
51#22575375C2211yescharacteristic
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—41no—
64#16864401—22no—
65#16864401—41no—
66C2×Dic538C2311yescharacteristic
67C2×D1076—22no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076—41no—
72C2×D1076C2311yescharacteristic
73C2×D1076C2311yescharacteristic
74C2×C22⋊D269—21no—
75C2×C22⋊D269D411yescharacteristic
76C22×Dic269—21no—
77C22×Dic269—21no—
78C2×C4×D269—21no—
79C2×C4×D269—21no—
80C2×C4×D269C2311yescharacteristic
81#16864392—22no—
82#16864391—22no—
83#16864391—41no—
84#16864389—22no—
85#16864389—22no—
86#16864387—22no—
87#16864387D414yescomplemented
88#16864386—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C8×D269—22no—
92C4⋊Dic269C2311yescharacteristic
93#16864381—21no—
94#16864381—21no—
95#16864381—21no—
96#16864381—21no—
97#16864381D411yescharacteristic
98C22⋊Dic269—21no—
99C22⋊Dic269—21no—
100C22⋊Dic269D411yescharacteristic
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379D411yescharacteristic
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—41no—
117SD16×C269—22no—
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—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131D1076—21no—
132D1076—21no—
133D1076—21no—
134D1076—42no—
135D1076—42no—
136D1076—81no—
137D1076C2×D412yes—
138C22⋊D269—41no—
139C22⋊D269—42no—
140C2×Dic269—21no—
141C2×Dic269—21no—
142C2×Dic269—42no—
143C2×Dic269—42no—
144C2×Dic269C2×D411yescharacteristic
145C2×Dic269C2×D411yescharacteristic
146C2×Dic269C2×D411yescharacteristic
147C4×D269—21no—
148C4×D269—21no—
149C4×D269—42no—
150C4×D269—81no—
151#14524130—21no—
152#14524130C2×D412yes—
153C23×C269—21no—
154C23×C269C2×D411yescharacteristic
155Q8×C269—21no—
156Q8×C269—21no—
157Q8×C269—21no—
158Q8×C269—21no—
159D4×C269—21no—
160D4×C269—42no—
161D4×C269C2×D412yes—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—42no—
168C2×C4×C269C2×D411yescharacteristic
169C2×C4×C269C2411yescharacteristic
170C8×C269—21no—
171C8×C269—21no—
172C22×C269—21no—
173C22×C269—22no—
174C22×C269—21no—
175C22×C269—21no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269C22×D411yescharacteristic
179C2×D269—21no—
180C2×D269—22no—
181C2×D269—21no—
182C2×D269—21no—
183C2×D269—41no—
184C2×D269—41no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—41no—
191C4×C269—41no—
192C4×C269—41no—
193C4×C269C22×D411yescharacteristic
194C4×C269H(5,𝔽2)11yescharacteristic
195Dic269—21no—
196Dic269—21no—
197Dic269—21no—
198Dic269—41no—
199Dic269—41no—
200C2×C269—41no—
201C2×C269—42no—
202C2×C269D4211yescharacteristic
203C2×C269#57511yescharacteristic
204C2×C269#57711yescharacteristic
205D269—41no—
206D269—42no—
207D269—81no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135—2691no—
212#547—2691no—
213#546—2691no—
214#497—2691no—
215#470—2691no—
216#465—2691no—
217#463—2691no—
218#461—2691no—
219#452—2691no—
220#452D26911yescharacteristic, complemented
221#450—2691no—
222#449—2691no—
223#442—2691no—
224#435—2691no—
225C2×#41—2691no—
226C2×#41—2691no—
227C2×#41C2×D26911yescharacteristic
228C22×D4—2691no—
229AGL(1,ℤ/8)—5384no—
230#135—5384no—
231C2×Q16—2691no—
232C2×Q16C2×D26911yescharacteristic, complemented
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16—2691no—
237C2×SD16C2×D26911yescharacteristic, complemented
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16—2691no—
241C22×C8—2691no—
242C4⋊Q8—2691no—
243#124—2691no—
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8—2691no—
249C22⋊Q8C2×D26911yescharacteristic
250#121—2691no—
251#121—5381no—
252#121—5381no—
253#120—5381no—
254C4×D4—2691no—
255C4×D4—2691no—
256C4×D4—2691no—
257C2×C4⋊C4—5381no—
258C2×C22⋊C4—5381no—
259#106—2691no—
260C4⋊C8—2691no—
261#103—2691no—
262#103—2691no—
263#103C2×D26911yescharacteristic, complemented
264#102—2691no—
265#102—2691no—
266#102C2×D26911yescharacteristic, complemented
267C22⋊C8—2691no—
268C22⋊C8C2×D26911yescharacteristic
269#97—2691no—
270C24—5381no—
271#41—22no—
272#41—41no—
273#41—5382no—
274#41—5382no—
275#41—10761no—
276#41—10761no—
277C2×Q8—2691no—
278C2×Q8C22×D26911yescharacteristic
279C2×Q8C22×D26911yescharacteristic
280C2×D4—21no—
281C2×D4—2691no—
282C2×D4—2691no—
283C2×D4—2691no—
284C2×D4—5382no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4—5381no—
288C2×D4—5381no—
289C2×D4—10761no—
290C2×D4C22×D26911yescharacteristic
291C22×C4—21no—
292C22×C4—2691no—
293C22×C4—5381no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4—5381no—
297C22×C4C22×D26911yescharacteristic
298Q16—41no—
299Q16—5382no—
300SD16—22no—
301SD16—5382no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—2694no—
306D8—5382no—
307M16—5382no—
308C2×C8—2691no—
309C2×C8—2691no—
310C2×C8—5382no—
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2691no—
315C4⋊C4—2691no—
316C4⋊C4—5381no—
317C4⋊C4—5381no—
318C4⋊C4—5381no—
319C4⋊C4—5381no—
320C4⋊C4C22×D26911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2691no—
323C22⋊C4—2691no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C22⋊C4—5381no—
327C22⋊C4—5381no—
328C22⋊C4—5381no—
329C22⋊C4—5381no—
330C22⋊C4—5381no—
331C42—2691no—
332C23—21no—
333C23—2691no—
334C23—5381no—
335C23—5381no—
336C23—10762no—
337C23—10762no—
338C23—21521no—
339C23D4×D26911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5381no—
345Q8—5381no—
346D4—21no—
347D4—42no—
348D4—2692no—
349D4—5381no—
350D4—5381no—
351D4—5381no—
352D4—10762no—
353D4—10762no—
354D4—10761no—
355D4—10762no—
356D4—21521no—
357D4D4×D26912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—2691no—
365C2×C4—2691no—
366C2×C4—2691no—
367C2×C4—5381no—
368C2×C4—5381no—
369C2×C4—5381no—
370C2×C4—5381no—
371C2×C4—10762no—
372C2×C4—10762no—
373C2×C4—10762no—
374C2×C4—21521no—
375C2×C4D4×D26911yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—21no—
378C8—21no—
379C8—2692no—
380C8—5381no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—5381no—
388C22—5381no—
389C22—5381no—
390C22—5382no—
391C22—10761no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C2×D4×D26911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5381no—
403C4—5381no—
404C4—5381no—
405C4—10761no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549811yescharacteristic
409C2—41no—
410C2—42no—
411C2—10761no—
412C2—10762no—
413C2—21521no—
414C2#3177123711yescharacteristic
415C2#3177140411yescharacteristic
416C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
hypercenter#452D269
hypocenterC269#3267
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×#22575474C2semidirect
3. #31771254C2semidirect
4. #31771246C2semidirect
5. #31770998C2semidirect
6. #31770861C2semidirect
7. #31770837C2semidirect
8. #31770822C2semidirect
9. #31770809C2semidirect
10. #31770766C2semidirect
11. #31770758C2semidirect
12. #31770751C2semidirect
13. #31770714C2semidirect
14. #31770679C2semidirect
15. #452×C269C2semidirect
17. C2×#16864401C22semidirect
21. C2×#16864392C22semidirect
29. C2×C8×D269C22semidirect
30. #22575451C22semidirect
36. #22575429C22semidirect
41. D4×Dic269C22semidirect
46. #22575384C22semidirect
47. #22575382C22semidirect
48. #22575377C22semidirect
49. #22575376C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
55. C2×Q16×C269C22semidirect
56. C2×SD16×C269C22semidirect
58. #103×C269C22semidirect
59. #102×C269C22semidirect
87. #16864387D4semidirect
208. C269#3267semidirect
220. #452D269semidirect
232. C2×Q16C2×D269semidirect
237. C2×SD16C2×D269semidirect
263. #103C2×D269semidirect
266. #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×#225754742
#317712542
#317712462
#317709982
#317708612
#317708372
#317708222
#317708092
#317707662
#317707582
#317707512
#317707142
#317706792
#452×C2692
#3267269