Finite Group Databasebeta

← #46822410

Subgroups — #46822410

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×#22575477C211yesmaximal, characteristic, complemented
2#31771254C211yesmaximal, characteristic, complemented
3#31771243C211yesmaximal, characteristic, complemented
4#31770998C211yesmaximal, characteristic, complemented
5#31770860C211yesmaximal, characteristic, complemented
6#31770834C211yesmaximal, characteristic, complemented
7#31770822C211yesmaximal, characteristic, complemented
8#31770808C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770764C211yesmaximal, characteristic, complemented
11#31770754C211yesmaximal, characteristic, complemented
12#31770750C211yesmaximal, characteristic, complemented
13#31770717C211yesmaximal, characteristic, complemented
14#31770676C211yesmaximal, characteristic, complemented
15C2×#135×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×D4×D269C2211yescharacteristic
18#22575477—24no—
19C2×#16864392C2211yescharacteristic, complemented
20C2×#16864391C2211yescharacteristic
21C2×#16864390C2211yescharacteristic
22C2×#16864389C2211yescharacteristic
23C2×#16864387C2211yescharacteristic
24C2×#16864385C2211yescharacteristic
25#22575451C2211yescharacteristic, complemented
26#22575442C2211yescharacteristic
27#22575436—21no—
28#22575436—21no—
29#22575434—21no—
30#22575432C2211yescharacteristic, complemented
31#22575429C2211yescharacteristic
32#22575429C2211yescharacteristic
33#22575427C2211yescharacteristic
34#22575425—21no—
35#22575425—21no—
36#22575421—21no—
37D4×Dic269C2211yescharacteristic
38D4×Dic269C2211yescharacteristic, complemented
39C4×C22⋊D269C2211yescharacteristic
40C2×#16864381—21no—
41C2×#16864379—21no—
42#22575385C2211yescharacteristic, complemented
43#22575382C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575377C2211yescharacteristic, complemented
46#22575377C2211yescharacteristic, complemented
47#22575375C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575374C2211yescharacteristic, complemented
50C22⋊#14524130C2211yescharacteristic, complemented
51#22575365C2211yescharacteristic
52#22575363C2211yescharacteristic, complemented
53C2×#41×C269C2211yescharacteristic
54C2×#41×C269C2211yescharacteristic, complemented
55#135×C269—24no—
56C2×Q16×C269C2211yescharacteristic, complemented
57C2×SD16×C269C2211yescharacteristic
58C2×SD16×C269C2211yescharacteristic, complemented
59C2×D8×C269C2211yescharacteristic
60C22×C8×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864401—22no—
63#16864401—41no—
64C2×Dic538C2311yescharacteristic
65C2×D1076C2311yescharacteristic
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269D411yescharacteristic
70D4×D269—22no—
71D4×D269—41no—
72C22×Dic269—21no—
73C22×Dic269—21no—
74C22×Dic269—21no—
75C2×C4×D269C2311yescharacteristic
76#16864392—41no—
77#16864391—22no—
78#16864390—22no—
79#16864389—41no—
80#16864387—22no—
81#16864385—22no—
82C2×#14524130C2311yescharacteristic
83C2×#14524130C2311yescharacteristic
84C4⋊Dic269C2311yescharacteristic
85C4⋊Dic269C2311yescharacteristic
86#16864381—21no—
87#16864381—21no—
88#16864381—21no—
89#16864381—21no—
90#16864381—21no—
91#16864381D411yescharacteristic
92C22⋊Dic269—21no—
93C22⋊Dic269—21no—
94C22⋊Dic269—21no—
95C22⋊Dic269D411yescharacteristic
96#16864379—21no—
97#16864379—21no—
98#16864379—21no—
99#16864379—21no—
100#16864379—21no—
101#16864379D411yescharacteristic
102C4×Dic269C2311yescharacteristic
103#41×C269—22no—
104#41×C269—22no—
105#41×C269—41no—
106#41×C269—41no—
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269—21no—
110C2×D4×C269—21no—
111C2×D4×C269C2311yescharacteristic
112C2×D4×C269C2311yescharacteristic
113C22×C4×C269—21no—
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—22no—
117SD16×C269—22no—
118SD16×C269—22no—
119D8×C269D414yescomplemented
120C2×C8×C269—22no—
121C2×C8×C269C2311yescharacteristic
122C2×C8×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—42no—
125C22×D269—42no—
126C22×D269—81no—
127C22×D269C2×D411yescharacteristic
128Dic538—21no—
129Dic538—21no—
130D1076—21no—
131D1076—21no—
132C22⋊D269—41no—
133C22⋊D269—42no—
134C22⋊D269—42no—
135C22⋊D269—81no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—21no—
139C2×Dic269—42no—
140C2×Dic269—42no—
141C2×Dic269—81no—
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C2×Dic269C2×D411yescharacteristic
145C4×D269—21no—
146C4×D269—21no—
147#14524130—21no—
148#14524130—21no—
149C23×C269—21no—
150C23×C269—21no—
151C23×C269C2×D411yescharacteristic
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—21no—
155Q8×C269—21no—
156D4×C269—21no—
157D4×C269—21no—
158D4×C269—42no—
159D4×C269—42no—
160D4×C269C2×D412yes—
161D4×C269C2×D412yes—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—42no—
167C2×C4×C269—42no—
168C2×C4×C269C2×D411yescharacteristic
169C2×C4×C269C2411yescharacteristic
170C8×C269—21no—
171C8×C269C2×D412yes—
172C22×C269—21no—
173C22×C269—22no—
174C22×C269—22no—
175C22×C269—21no—
176C22×C269—21no—
177C22×C269—41no—
178C22×C269—41no—
179C22×C269—41no—
180C22×C269—41no—
181C22×C269C22×D411yescharacteristic
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—41no—
191C4×C269—41no—
192C4×C269C22×D411yescharacteristic
193C4×C269H(5,𝔽2)11yescharacteristic
194Dic269—21no—
195Dic269—21no—
196Dic269—21no—
197Dic269—41no—
198Dic269—41no—
199Dic269—41no—
200C2×C269—42no—
201C2×C269—41no—
202C2×C269—42no—
203C2×C269D4211yescharacteristic
204C2×C269#57511yescharacteristic
205C2×C269#57711yescharacteristic
206D269—41no—
207D269—81no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135D26911yescharacteristic, complemented
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#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—24no—
231C2×Q16—2691no—
232C2×Q16C2×D26911yescharacteristic, complemented
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16C2×D26911yescharacteristic
237C2×SD16C2×D26911yescharacteristic, complemented
238C2×D8—2691no—
239C2×D8C2×D26911yescharacteristic
240C2×M16—2691no—
241C22×C8C2×D26911yescharacteristic, complemented
242C4⋊Q8—2691no—
243#124—2691no—
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8—2691no—
249C22⋊Q8—2691no—
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#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—5381no—
285C2×D4—5382no—
286C2×D4—5381no—
287C2×D4—5381no—
288C2×D4—10761no—
289C2×D4C22×D26911yescharacteristic
290C2×D4C22×D26911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2691no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4—5381no—
297C22×C4C22×D26911yescharacteristic
298Q16—22no—
299Q16—10761no—
300SD16—22no—
301SD16—22no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—5382no—
306D8C22⋊D26914yescomplemented
307M16—5382no—
308C2×C8—22no—
309C2×C8—2691no—
310C2×C8—2691no—
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—2691no—
314C4⋊C4—2691no—
315C4⋊C4—2691no—
316C4⋊C4—5381no—
317C4⋊C4—5381no—
318C4⋊C4—5381no—
319C4⋊C4—5381no—
320C4⋊C4—5381no—
321C22⋊C4—2691no—
322C22⋊C4—2691no—
323C22⋊C4—5381no—
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—21no—
334C23—2691no—
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—21no—
348D4—42no—
349D4—42no—
350D4—5381no—
351D4—5381no—
352D4—10761no—
353D4—10762no—
354D4—10762no—
355D4—21521no—
356D4C2×C22⋊D26912yes—
357D4C2×C22⋊D26912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
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—5381no—
372C2×C4—10762no—
373C2×C4—10762no—
374C2×C4—21521no—
375C2×C4D4×D26911yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—21no—
378C8—5381no—
379C8—5381no—
380C8C2×C22⋊D26912yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—5381no—
391C22—5381no—
392C22—5381no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C2×D4×D26911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—5381no—
402C4—5381no—
403C4—5381no—
404C4—10761no—
405C4—10761no—
406C4—10761no—
407C4C22×C22⋊D26911yescharacteristic
408C4#2257549811yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—10761no—
413C2—21521no—
414C2D4×C22⋊D26911yescharacteristic
415C2#3177140211yescharacteristic
416C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#135×C269C2
socleC22×C269C22×D4
hypercenterC2×#135D269
hypocenterC269#3267
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. #31771254C2semidirect
3. #31771243C2semidirect
4. #31770998C2semidirect
5. #31770860C2semidirect
6. #31770834C2semidirect
7. #31770822C2semidirect
8. #31770808C2semidirect
9. #31770767C2semidirect
10. #31770764C2semidirect
11. #31770754C2semidirect
12. #31770750C2semidirect
13. #31770717C2semidirect
14. #31770676C2semidirect
15. C2×#135×C269C2semidirect
19. C2×#16864392C22semidirect
25. #22575451C22semidirect
30. #22575432C22semidirect
38. D4×Dic269C22semidirect
42. #22575385C22semidirect
43. #22575382C22semidirect
44. #22575378C22semidirect
45. #22575377C22semidirect
46. #22575377C22semidirect
49. #22575374C22semidirect
50. C22⋊#14524130C22semidirect
52. #22575363C22semidirect
54. C2×#41×C269C22semidirect
56. C2×Q16×C269C22semidirect
58. C2×SD16×C269C22semidirect
60. C22×C8×C269C22semidirect
119. D8×C269D4semidirect
208. C269#3267semidirect
211. C2×#135D269semidirect
227. C2×#41C2×D269semidirect
232. C2×Q16C2×D269semidirect
237. C2×SD16C2×D269semidirect
241. C22×C8C2×D269semidirect
306. D8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
#317712542
#317712432
#317709982
#317708602
#317708342
#317708222
#317708082
#317707672
#317707642
#317707542
#317707502
#317707172
#317706762
C2×#135×C2692
#3267269