Finite Group Databasebeta

← #46822694

Subgroups — #46822694

Order 34432.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771256C211yesmaximal, characteristic, complemented
2D4×C22⋊D269C211yesmaximal, characteristic, complemented
3#31770950C211yesmaximal, characteristic, complemented
4#31770918C211yesmaximal, characteristic, complemented
5#31770917C211yesmaximal, characteristic, complemented
6#31770883C211yesmaximal, characteristic, complemented
7#31770841C211yesmaximal, characteristic, complemented
8#31770840C211yesmaximal, characteristic, complemented
9#31770799C211yesmaximal, characteristic, complemented
10#31770740C211yesmaximal, characteristic, complemented
11#31770739C211yesmaximal, characteristic, complemented
12#31770685C211yesmaximal, characteristic, complemented
13C8×C22⋊D269C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15C2×#102×C269C211yesmaximal, characteristic, complemented
16C2×#16864402—21no—
17C22×C22⋊D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×D2152C2211yescharacteristic, complemented
20C2×#16864387C2211yescharacteristic
21C2×#16864387C2211yescharacteristic, complemented
22C2×C8×D269C2211yescharacteristic, complemented
23#22575452C2211yescharacteristic, complemented
24#22575449C2211yescharacteristic
25#22575438—21no—
26#22575436—21no—
27#22575431—21no—
28#22575428C2211yescharacteristic, complemented
29#22575427C2211yescharacteristic
30#22575427C2211yescharacteristic, complemented
31#22575425—21no—
32#22575424C2211yescharacteristic
33#22575422—21no—
34#22575421—21no—
35D4×Dic269C2211yescharacteristic
36#22575415C2211yescharacteristic, complemented
37C4×C22⋊D269C2211yescharacteristic
38C4⋊C4×D269C2211yescharacteristic, complemented
39#22575387C2211yescharacteristic, complemented
40#22575386C2211yescharacteristic, complemented
41#22575386C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575375C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic, complemented
45#22575374C2211yescharacteristic
46#22575374C2211yescharacteristic, complemented
47#22575373C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic, complemented
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C22×D4×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54C2×C4⋊C4×C269C2211yescharacteristic, complemented
55#102×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60#16864402—44no—
61C2×Dic538—21no—
62C2×D1076—21no—
63C2×D1076C2311yescharacteristic
64C2×D1076C2311yescharacteristic
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269—44no—
72C2×C22⋊D269D411yescharacteristic
73D4×D269—22no—
74D4×D269—41no—
75C22×Dic269—21no—
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269C2311yescharacteristic
79D2152—41no—
80#16864387—22no—
81#16864387—41no—
82C2×#14524130C2311yescharacteristic
83C2×#14524130C2311yescharacteristic
84C8×D269—41no—
85C4⋊Dic269C2311yescharacteristic
86C4⋊Dic269C2311yescharacteristic
87#16864381—21no—
88#16864381—21no—
89#16864381D411yescharacteristic
90C22⋊Dic269—21no—
91C22⋊Dic269—21no—
92C22⋊Dic269D411yescharacteristic
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379D411yescharacteristic
98C4×Dic269C2311yescharacteristic
99C24×C269—21no—
100C2×D4×C269—22no—
101C2×D4×C269—21no—
102C2×D4×C269—41no—
103C2×D4×C269C2311yescharacteristic
104C2×D4×C269C2311yescharacteristic
105C22×C4×C269—21no—
106C22×C4×C269C2311yescharacteristic
107C2×C8×C269—41no—
108C2×C8×C269C2311yescharacteristic
109C2×C8×C269C2311yescharacteristic
110C4⋊C4×C269—21no—
111C4⋊C4×C269C2311yescharacteristic
112C4⋊C4×C269C2311yescharacteristic
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—81no—
118C22×D269C2×D411yescharacteristic
119Dic538—42no—
120D1076—21no—
121D1076—21no—
122D1076—21no—
123D1076—21no—
124D1076—42no—
125C22⋊D269—42no—
126C22⋊D269—42no—
127C22⋊D269—42no—
128C22⋊D269—42no—
129C22⋊D269—42no—
130C22⋊D269—81no—
131C22⋊D269D814yescomplemented
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—42no—
135C2×Dic269C2×D411yescharacteristic
136C2×Dic269C2×D411yescharacteristic
137C2×Dic269C2×D411yescharacteristic
138C4×D269—21no—
139C4×D269—21no—
140C4×D269—42no—
141C4×D269—42no—
142#14524130—21no—
143#14524130—21no—
144C23×C269—21no—
145C23×C269—21no—
146C23×C269—42no—
147C23×C269—42no—
148C23×C269—81no—
149C23×C269C2×D411yescharacteristic
150D4×C269—21no—
151D4×C269—21no—
152D4×C269—21no—
153D4×C269—42no—
154D4×C269#4112yescomplemented
155C2×C4×C269—21no—
156C2×C4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—42no—
160C2×C4×C269C2×D411yescharacteristic
161C2×C4×C269C2411yescharacteristic
162C8×C269—21no—
163C8×C269—21no—
164C22×C269—21no—
165C22×C269—21no—
166C22×C269—22no—
167C22×C269—41no—
168C22×C269—41no—
169C22×C269—41no—
170C22×C269—41no—
171C22×C269—41no—
172C22×C269—84no—
173C22×C269C2×D812yes—
174C22×C269C22×D411yescharacteristic
175C2×D269—21no—
176C2×D269—21no—
177C2×D269—41no—
178C2×D269—41no—
179C2×D269—41no—
180C2×D269—41no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—84no—
184C2×D269C2×D812yes—
185C4×C269—21no—
186C4×C269—41no—
187C4×C269—41no—
188C4×C269C2×#4111yescharacteristic
189C4×C269H(5,𝔽2)11yescharacteristic
190Dic269—21no—
191Dic269—21no—
192Dic269—41no—
193Dic269—41no—
194Dic269C2×D812yes—
195C2×C269—22no—
196C2×C269—42no—
197C2×C269—81no—
198C2×C269#54611yescharacteristic
199C2×C269C22×D811yescharacteristic
200C2×C269#57711yescharacteristic
201D269—22no—
202D269—81no—
203D269—81no—
204C269#329411yescharacteristic, complemented, Sylow
205#3294—2691nomaximal, Sylow
206#547—2691no—
207D42—2691no—
208#486—2691no—
209#480—2691no—
210#480—2691no—
211#474—2691no—
212#466—2691no—
213#466—2691no—
214#459—2691no—
215#447—2691no—
216#447—2691no—
217C4×D8—2691no—
218C8×D4—2691no—
219C2×#102—2691no—
220C2×#102D26911yescharacteristic, complemented
221C2×#41—5381no—
222C22×D4—2691no—
223C22×D4—5381no—
224C22×D4C2×D26911yescharacteristic
225C2×D8—2691no—
226C2×D8—2691no—
227C2×D8—2691no—
228C22×C8—2691no—
229C22×C8C2×D26911yescharacteristic, complemented
230C4⋊Q8—2691no—
231#127—2691no—
232#123—5381no—
233#123—5381no—
234C22⋊Q8—5381no—
235#121—2691no—
236#121—2691no—
237#121—2691no—
238#121—2691no—
239#121—5381no—
240#120—5381no—
241#120—5381no—
242C4×D4—2691no—
243C4×D4—2691no—
244C4×D4—2691no—
245C2×C4⋊C4—2691no—
246C2×C4⋊C4C2×D26911yescharacteristic, complemented
247#107—2691no—
248#107—2691no—
249#107—2691no—
250C4⋊C8—2691no—
251#102—2691no—
252#102—2691no—
253#102—2691no—
254#102—2691no—
255#102—2691no—
256#102—2691no—
257#102C2×D26911yescharacteristic
258#102C2×D26911yescharacteristic
259#102C2×D26911yescharacteristic, complemented
260#102C2×D26911yescharacteristic, complemented
261C22⋊C8—2691no—
262C22⋊C8—2691no—
263C4×C8—2691no—
264C24—21no—
265C24—5381no—
266#41—10764no—
267C2×Q8—5381no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—2691no—
272C2×D4—2691no—
273C2×D4—2691no—
274C2×D4—5381no—
275C2×D4—5381no—
276C2×D4—5381no—
277C2×D4—5381no—
278C2×D4—5382no—
279C2×D4—5381no—
280C2×D4—5381no—
281C2×D4—5381no—
282C2×D4—10761no—
283C2×D4—10764no—
284C2×D4C22×D26911yescharacteristic
285C2×D4C22×D26911yescharacteristic
286C22×C4—21no—
287C22×C4—2691no—
288C22×C4—5381no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4C22×D26911yescharacteristic
292D8—5382no—
293D8—10761no—
294D8—10761no—
295C2×C8—41no—
296C2×C8—2691no—
297C2×C8—2691no—
298C2×C8—10761no—
299C2×C8C22×D26911yescharacteristic
300C2×C8C22×D26911yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—2691no—
303C4⋊C4—2691no—
304C4⋊C4—2691no—
305C4⋊C4—5381no—
306C4⋊C4—5381no—
307C4⋊C4C22×D26911yescharacteristic
308C4⋊C4C22×D26911yescharacteristic
309C22⋊C4—2691no—
310C22⋊C4—2691no—
311C22⋊C4—5381no—
312C22⋊C4—5381no—
313C22⋊C4—5381no—
314C22⋊C4—5381no—
315C22⋊C4—5381no—
316C22⋊C4—5381no—
317C42—2691no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—2691no—
324C23—5381no—
325C23—5381no—
326C23—10762no—
327C23—10762no—
328C23—21521no—
329C23D4×D26911yescharacteristic
330Q8—10762no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—2694no—
336D4—5381no—
337D4—5381no—
338D4—5381no—
339D4—5381no—
340D4—10762no—
341D4—10762no—
342D4—10762no—
343D4—10762no—
344D4—10762no—
345D4—10762no—
346D4—21521no—
347D4#1686440212yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—2691no—
354C2×C4—2691no—
355C2×C4—2691no—
356C2×C4—5381no—
357C2×C4—5381no—
358C2×C4—5381no—
359C2×C4—5381no—
360C2×C4—10762no—
361C2×C4—10762no—
362C2×C4—10762no—
363C2×C4D4×D26911yescharacteristic
364C2×C4C23×D26911yescharacteristic
365C8—21no—
366C8—21no—
367C8—5381no—
368C8—5381no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—2692no—
379C22—5381no—
380C22—5381no—
381C22—10761no—
382C22—10761no—
383C22—10761no—
384C22—10761no—
385C22—10761no—
386C22—10761no—
387C22—21524no—
388C22D8×D26912yes—
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—2692no—
394C4—5381no—
395C4—5381no—
396C4—10761no—
397C4—10761no—
398C4C2×#1686440211yescharacteristic
399C4#2257549811yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—5382no—
404C2—21521no—
405C2—21521no—
406C2#3177124511yescharacteristic
407C2C2×D8×D26911yescharacteristic
408C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#102×C269C2
socleC22×C269C22×D4
hypercenterC2×#102D269
hypocenterC269#3294
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #31771256C2semidirect
2. D4×C22⋊D269C2semidirect
3. #31770950C2semidirect
4. #31770918C2semidirect
5. #31770917C2semidirect
6. #31770883C2semidirect
7. #31770841C2semidirect
8. #31770840C2semidirect
9. #31770799C2semidirect
10. #31770740C2semidirect
11. #31770739C2semidirect
12. #31770685C2semidirect
13. C8×C22⋊D269C2semidirect
14. #102×D269C2semidirect
15. C2×#102×C269C2semidirect
19. C2×D2152C22semidirect
21. C2×#16864387C22semidirect
22. C2×C8×D269C22semidirect
23. #22575452C22semidirect
28. #22575428C22semidirect
30. #22575427C22semidirect
36. #22575415C22semidirect
38. C4⋊C4×D269C22semidirect
39. #22575387C22semidirect
40. #22575386C22semidirect
41. #22575386C22semidirect
42. #22575382C22semidirect
43. #22575375C22semidirect
44. #22575375C22semidirect
46. #22575374C22semidirect
47. #22575373C22semidirect
48. #22575373C22semidirect
49. C22⋊#14524130C22semidirect
50. #22575365C22semidirect
51. C8×Dic269C22semidirect
53. C22×C8×C269C22semidirect
54. C2×C4⋊C4×C269C22semidirect
57. #102×C269C22semidirect
58. #102×C269C22semidirect
131. C22⋊D269D8semidirect
154. D4×C269#41semidirect
204. C269#3294semidirect
220. C2×#102D269semidirect
229. C22×C8C2×D269semidirect
246. C2×C4⋊C4C2×D269semidirect
259. #102C2×D269semidirect
260. #102C2×D269semidirect
347. D4#16864402semidirect

Maximal subgroups

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

subgroupindex
#317712562
D4×C22⋊D2692
#317709502
#317709182
#317709172
#317708832
#317708412
#317708402
#317707992
#317707402
#317707392
#317706852
C8×C22⋊D2692
#102×D2692
C2×#102×C2692
#3294269