Finite Group Databasebeta

← #46822411

Subgroups — #46822411

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×#22575473C211yesmaximal, characteristic, complemented
3#31771251C211yesmaximal, characteristic, complemented
4#31770997C211yesmaximal, characteristic, complemented
5#31770861C211yesmaximal, characteristic, complemented
6#31770836C211yesmaximal, characteristic, complemented
7#31770820C211yesmaximal, characteristic, complemented
8#31770807C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770765C211yesmaximal, characteristic, complemented
11#31770755C211yesmaximal, characteristic, complemented
12#31770750C211yesmaximal, characteristic, complemented
13#31770715C211yesmaximal, characteristic, complemented
14#31770677C211yesmaximal, characteristic, complemented
15#547×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18#22575478—24no—
19#22575473—24no—
20C2×#16864392C2211yescharacteristic, complemented
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864390C2211yescharacteristic, complemented
23C2×#16864389C2211yescharacteristic
24C2×#16864389C2211yescharacteristic
25C2×#16864389C2211yescharacteristic
26C2×#16864389C2211yescharacteristic
27C2×#16864387C2211yescharacteristic
28C2×#16864387C2211yescharacteristic
29C2×#16864386C2211yescharacteristic
30C22×#14524130C2211yescharacteristic, complemented
31#22575441C2211yescharacteristic
32#22575438—21no—
33#22575432C2211yescharacteristic
34#22575427C2211yescharacteristic
35#22575426—21no—
36#22575421—21no—
37C4×D1076C2211yescharacteristic
38C22⋊C4×D269—21no—
39#22575384C2211yescharacteristic, complemented
40C4⋊#14524130C2211yescharacteristic, complemented
41#22575377C2211yescharacteristic, complemented
42#22575377C2211yescharacteristic, complemented
43#22575376C2211yescharacteristic, complemented
44#22575374C2211yescharacteristic, complemented
45#22575373C2211yescharacteristic
46#22575373C2211yescharacteristic
47C22⋊#14524130C2211yescharacteristic
48C22⋊#14524130C2211yescharacteristic, complemented
49#22575364C2211yescharacteristic, complemented
50C2×#41×C269C2211yescharacteristic
51C2×#41×C269C2211yescharacteristic, complemented
52C4⋊Q8×C269C2211yescharacteristic, complemented
53#123×C269—21no—
54#123×C269—21no—
55C22⋊Q8×C269C2211yescharacteristic
56C22⋊Q8×C269C2211yescharacteristic, complemented
57#121×C269—21no—
58C4×D4×C269C2211yescharacteristic
59C4×D4×C269C2211yescharacteristic, complemented
60C2×C4⋊C4×C269—21no—
61C23×D269—21no—
62#16864402—22no—
63#16864402—41no—
64C2×Dic538C2311yescharacteristic
65C2×D1076—22no—
66C2×D1076—21no—
67C2×D1076—41no—
68C2×D1076C2311yescharacteristic
69C2×D1076C2311yescharacteristic
70C2×C22⋊D269—21no—
71C2×C22⋊D269—21no—
72C2×C4×D269—21no—
73C2×C4×D269—21no—
74#16864392—22no—
75#16864392—41no—
76#16864390—22no—
77#16864389—22no—
78#16864389—22no—
79#16864389—22no—
80#16864389—41no—
81#16864387—22no—
82#16864387D414yescomplemented
83#16864386—22no—
84C2×#14524130—22no—
85C2×#14524130C2311yescharacteristic
86C2×#14524130C2311yescharacteristic
87C2×#14524130C2311yescharacteristic
88C2×#14524130C2311yescharacteristic
89C4⋊Dic269C2311yescharacteristic
90#16864381—21no—
91C22⋊Dic269—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#41×C269—22no—
97#41×C269—22no—
98#41×C269—41no—
99#41×C269—41no—
100C2×Q8×C269C2311yescharacteristic
101C2×Q8×C269C2311yescharacteristic
102C2×D4×C269—21no—
103C2×D4×C269—21no—
104C2×D4×C269D411yescharacteristic
105C2×D4×C269C2311yescharacteristic
106C22×C4×C269—21no—
107C22×C4×C269—21no—
108C22×C4×C269—21no—
109C22×C4×C269C2311yescharacteristic
110C22×C4×C269C2311yescharacteristic
111C4⋊C4×C269—21no—
112C4⋊C4×C269—21no—
113C4⋊C4×C269—21no—
114C4⋊C4×C269—21no—
115C4⋊C4×C269D411yescharacteristic
116C4⋊C4×C269C2311yescharacteristic
117C22⋊C4×C269—21no—
118C22⋊C4×C269—21no—
119C22⋊C4×C269—21no—
120C22⋊C4×C269D411yescharacteristic
121C22⋊C4×C269D411yescharacteristic
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—42no—
126C22×D269—42no—
127C22×D269—81no—
128Dic538—21no—
129Dic538—21no—
130D1076—21no—
131D1076—21no—
132D1076—21no—
133D1076—42no—
134D1076C2×D412yes—
135C22⋊D269—42no—
136C22⋊D269—81no—
137C2×Dic269—21no—
138C2×Dic269—21no—
139C4×D269—42no—
140C4×D269—81no—
141#14524130—21no—
142#14524130—21no—
143#14524130—21no—
144#14524130C2×D412yes—
145C23×C269—21no—
146C23×C269C2×D411yescharacteristic
147C23×C269C2×D411yescharacteristic
148Q8×C269—21no—
149Q8×C269—21no—
150Q8×C269—21no—
151Q8×C269—21no—
152D4×C269—21no—
153D4×C269—41no—
154D4×C269—42no—
155D4×C269—42no—
156D4×C269C2×D412yes—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—42no—
165C2×C4×C269—42no—
166C2×C4×C269—42no—
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2×D411yescharacteristic
169C2×C4×C269C2×D411yescharacteristic
170C2×C4×C269C2×D411yescharacteristic
171C2×C4×C269C2411yescharacteristic
172C22×C269—21no—
173C22×C269—21no—
174C22×C269—22no—
175C22×C269—21no—
176C22×C269—21no—
177C22×C269—21no—
178C22×C269—21no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269C22×D411yescharacteristic
182C2×D269—22no—
183C2×D269—41no—
184C2×D269—41no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—21no—
191C4×C269—21no—
192C4×C269—21no—
193C4×C269—41no—
194C4×C269—41no—
195C4×C269—41no—
196C4×C269C22×D411yescharacteristic
197C4×C269H(5,𝔽2)11yescharacteristic
198Dic269—41no—
199Dic269—41no—
200C2×C269—41no—
201C2×C269—41no—
202C2×C269—42no—
203C2×C269D4211yescharacteristic
204C2×C269#57511yescharacteristic
205C2×C269#57711yescharacteristic
206D269—42no—
207D269—81no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135—2691no—
212#547D26911yescharacteristic, complemented
213#546—2691no—
214#497—2691no—
215#470—2691no—
216#465—2691no—
217#463—2691no—
218#461—2691no—
219#452—2691no—
220#452—2691no—
221#450—2691no—
222#449—2691no—
223#442—2691no—
224#435—2691no—
225C2×#41—2691no—
226C2×#41C2×D26911yescharacteristic
227C2×#41C2×D26911yescharacteristic, complemented
228C22×D4—2691no—
229AGL(1,ℤ/8)—5384no—
230#135—5384no—
231C2×Q16—2691no—
232C2×Q16—2691no—
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16—2691no—
237C2×SD16—2691no—
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16—2691no—
241C22×C8—2691no—
242C4⋊Q8C2×D26911yescharacteristic, complemented
243#124—2691no—
244#123—21no—
245#123—21no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8C2×D26911yescharacteristic
249C22⋊Q8C2×D26911yescharacteristic, complemented
250#121—21no—
251#121—2691no—
252#121—5381no—
253#120—5381no—
254C4×D4—2691no—
255C4×D4C2×D26911yescharacteristic
256C4×D4C2×D26911yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—5381no—
259#106—2691no—
260C4⋊C8—2691no—
261#103—2691no—
262#103—2691no—
263#103—2691no—
264#102—2691no—
265#102—2691no—
266#102—2691no—
267C22⋊C8—2691no—
268C22⋊C8—2691no—
269#97—2691no—
270C24—5381no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—5382no—
276#41—10761no—
277C2×Q8—2691no—
278C2×Q8C22×D26911yescharacteristic
279C2×Q8C22×D26911yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2691no—
283C2×D4—2691no—
284C2×D4—5382no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4—5381no—
288C2×D4—10761no—
289C2×D4C22⋊D26911yescharacteristic
290C2×D4C22×D26911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4C22×D26911yescharacteristic
297C22×C4C22×D26911yescharacteristic
298Q16—5382no—
299Q16—10761no—
300SD16—5382no—
301SD16—5382no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—2694no—
306D8—5382no—
307M16—5382no—
308C2×C8—2691no—
309C2×C8—2691no—
310C2×C8—2691no—
311C2×C8—2691no—
312C2×C8—5382no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—2691no—
318C4⋊C4—5381no—
319C4⋊C4C22⋊D26911yescharacteristic
320C4⋊C4C22×D26911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C22⋊C4—5381no—
327C22⋊C4—5381no—
328C22⋊C4—5381no—
329C22⋊C4C22⋊D26911yescharacteristic
330C22⋊C4C22⋊D26911yescharacteristic
331C42C22×D26911yescharacteristic
332C23—21no—
333C23—5381no—
334C23—5381no—
335C23—10762no—
336C23—10762no—
337C23—21521no—
338C23C2×C22⋊D26911yescharacteristic
339C23C2×C22⋊D26911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5381no—
345Q8—5381no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2692no—
351D4—5381no—
352D4—5381no—
353D4—5381no—
354D4—10762no—
355D4—10762no—
356D4—21521no—
357D4D4×D26912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—5381no—
369C2×C4—5381no—
370C2×C4—10762no—
371C2×C4—21521no—
372C2×C4C2×C22⋊D26911yescharacteristic
373C2×C4C2×C22⋊D26911yescharacteristic
374C2×C4C2×C22⋊D26911yescharacteristic
375C2×C4C2×C22⋊D26911yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—2692no—
378C8—5381no—
379C8—5381no—
380C8—5381no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—5382no—
391C22—10761no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C22×C22⋊D26911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—10761no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549811yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—10762no—
413C2—21521no—
414C2D4×C22⋊D26911yescharacteristic
415C2#3177140311yescharacteristic
416C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#547×C269C2
socleC22×C269C22×D4
hypercenter#547D269
hypocenterC269#3267
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C2×#22575473C2semidirect
3. #31771251C2semidirect
4. #31770997C2semidirect
5. #31770861C2semidirect
6. #31770836C2semidirect
7. #31770820C2semidirect
8. #31770807C2semidirect
9. #31770767C2semidirect
10. #31770765C2semidirect
11. #31770755C2semidirect
12. #31770750C2semidirect
13. #31770715C2semidirect
14. #31770677C2semidirect
15. #547×C269C2semidirect
20. C2×#16864392C22semidirect
21. C2×#16864392C22semidirect
22. C2×#16864390C22semidirect
30. C22×#14524130C22semidirect
39. #22575384C22semidirect
40. C4⋊#14524130C22semidirect
41. #22575377C22semidirect
42. #22575377C22semidirect
43. #22575376C22semidirect
44. #22575374C22semidirect
48. C22⋊#14524130C22semidirect
49. #22575364C22semidirect
51. C2×#41×C269C22semidirect
52. C4⋊Q8×C269C22semidirect
56. C22⋊Q8×C269C22semidirect
59. C4×D4×C269C22semidirect
82. #16864387D4semidirect
208. C269#3267semidirect
212. #547D269semidirect
227. C2×#41C2×D269semidirect
242. C4⋊Q8C2×D269semidirect
249. C22⋊Q8C2×D269semidirect
256. C4×D4C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C2×#225754732
#317712512
#317709972
#317708612
#317708362
#317708202
#317708072
#317707672
#317707652
#317707552
#317707502
#317707152
#317706772
#547×C2692
#3267269