Finite Group Databasebeta

← #46813930

Subgroups — #46813930

Order 34432.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575425C211yesmaximal, characteristic
2C2×#22575413C211yesmaximal, characteristic, complemented
3#31770499C211yesmaximal, characteristic, complemented
4#31770495C211yesmaximal, characteristic, complemented
5#31770489C211yesmaximal, characteristic, complemented
6#31770486C211yesmaximal, characteristic, complemented
7#31770485C211yesmaximal, characteristic, complemented
8#31770484C211yesmaximal, characteristic, complemented
9#31770484C211yesmaximal, characteristic, complemented
10#31770481C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic269C211yesmaximal, characteristic, complemented
12C4×#16864379C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic269C211yesmaximal, characteristic, complemented
14C2×#22575360C211yesmaximal, characteristic, complemented
15C2×C4×D4×C269C211yesmaximal, characteristic, complemented
16C22×C22⋊D269C2211yescharacteristic
17C22×C22⋊D269C2211yescharacteristic
18C23×Dic269C2211yescharacteristic
19C23×Dic269C2211yescharacteristic
20C22×C4×D269C2211yescharacteristic
21#22575425C418yescomplemented
22#22575413—24no—
23C2×#16864381C2211yescharacteristic
24C2×#16864381C2211yescharacteristic
25C2×C22⋊Dic269C2211yescharacteristic
26C2×C22⋊Dic269C2211yescharacteristic
27C2×C22⋊Dic269C2211yescharacteristic
28C2×C22⋊Dic269C2211yescharacteristic
29C2×#16864379C2211yescharacteristic
30C2×#16864379C2211yescharacteristic
31C2×#16864379C2211yescharacteristic
32C2×#16864379C2211yescharacteristic
33C2×C4×Dic269C2211yescharacteristic
34C2×C4×Dic269C2211yescharacteristic
35C2×C4×Dic269C2211yescharacteristic
36#22575360C2211yescharacteristic
37#22575360C2211yescharacteristic
38#22575360C2211yescharacteristic
39#22575360C2211yescharacteristic
40#22575360C2211yescharacteristic
41#22575360C2211yescharacteristic
42#22575360C2211yescharacteristic
43#22575360C2211yescharacteristic
44#22575360C2211yescharacteristic
45#22575360C2211yescharacteristic
46C22×D4×C269C2211yescharacteristic
47C23×C4×C269C2211yescharacteristic
48C23×C4×C269C2211yescharacteristic
49C4×D4×C269—24no—
50C2×C4⋊C4×C269C2211yescharacteristic
51C2×C22⋊C4×C269C2211yescharacteristic
52C2×C22⋊C4×C269C2211yescharacteristic
53C2×C42×C269C2211yescharacteristic
54C23×D269C2311yescharacteristic
55C2×C22⋊D269—22no—
56C2×C22⋊D269—22no—
57C2×C22⋊D269—41no—
58C2×C22⋊D269—41no—
59C2×C22⋊D269C2×C414yescomplemented
60C2×C22⋊D269C2×C414yescomplemented
61C22×Dic269—22no—
62C22×Dic269—22no—
63C22×Dic269—41no—
64C22×Dic269—41no—
65C22×Dic269—41no—
66C22×Dic269C2×C414yescomplemented
67C22×Dic269C2311yescharacteristic
68C22×Dic269C2311yescharacteristic
69C22×Dic269C2311yescharacteristic
70C22×Dic269C2311yescharacteristic
71C22×Dic269C2311yescharacteristic
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2311yescharacteristic
74C2×C4×D269—22no—
75C2×C4×D269—22no—
76C2×C4×D269—41no—
77#16864381—22no—
78#16864381C2×C414yescomplemented
79C22⋊Dic269—22no—
80C22⋊Dic269—22no—
81C22⋊Dic269—41no—
82C22⋊Dic269C2×C414yescomplemented
83#16864379—22no—
84#16864379—22no—
85#16864379—41no—
86#16864379C2×C414yescomplemented
87C4×Dic269—22no—
88C4×Dic269—41no—
89C4×Dic269—41no—
90C24×C269C2311yescharacteristic
91C24×C269C2311yescharacteristic
92C2×D4×C269—22no—
93C2×D4×C269—22no—
94C2×D4×C269C2×C414yescomplemented
95C22×C4×C269—22no—
96C22×C4×C269—22no—
97C22×C4×C269—22no—
98C22×C4×C269—22no—
99C22×C4×C269—41no—
100C22×C4×C269—41no—
101C22×C4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104C22×C4×C269C2311yescharacteristic
105C22×C4×C269C2311yescharacteristic
106C4⋊C4×C269—22no—
107C22⋊C4×C269—22no—
108C22⋊C4×C269D414yescomplemented
109C42×C269—22no—
110C22×D269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—21no—
116C22×D269C22×C412yes—
117C22⋊D269—28no—
118C22⋊D269—44no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—21no—
125C2×Dic269—21no—
126C2×Dic269—21no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—21no—
130C2×Dic269—21no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—44no—
137C2×Dic269—44no—
138C2×Dic269C22×C412yes—
139C2×Dic269C22×C412yes—
140C2×Dic269C22×C412yes—
141C2×Dic269#4112yes—
142C4×D269—44no—
143C23×C269—21no—
144C23×C269—21no—
145C23×C269—21no—
146C23×C269—21no—
147C23×C269—21no—
148C23×C269—21no—
149C23×C269—21no—
150C23×C269—21no—
151C23×C269—21no—
152C23×C269—21no—
153C23×C269C22×C412yes—
154C23×C269C22×C412yes—
155C23×C269C2×D412yes—
156C23×C269#4112yes—
157C23×C269C2411yescharacteristic
158D4×C269—28no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269—44no—
171C2×C4×C269—44no—
172C2×C4×C269C22×C412yes—
173C2×C4×C269C2×D412yes—
174C2×C4×C269C2×D412yes—
175C2×C4×C269#4112yes—
176C22×C269—22no—
177C22×C269—22no—
178C22×C269—22no—
179C22×C269—22no—
180C22×C269—22no—
181C22×C269—22no—
182C22×C269—22no—
183C22×C269—22no—
184C22×C269—22no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269C4×D414yes—
190C22×C269C23×C411yescharacteristic
191C22×C269C22×D411yescharacteristic
192C22×C269C2×#4111yescharacteristic
193C22×C269C2×#4111yescharacteristic
194C22×C269C2×#4111yescharacteristic
195C22×C269H(5,𝔽2)11yescharacteristic
196C22×C269H(5,𝔽2)11yescharacteristic
197C2×D269—22no—
198C2×D269—22no—
199C2×D269—22no—
200C2×D269—41no—
201C2×D269—41no—
202C2×D269—41no—
203C2×D269—41no—
204C4×C269—22no—
205C4×C269—22no—
206C4×C269—22no—
207C4×C269—22no—
208C4×C269—41no—
209Dic269—22no—
210Dic269—22no—
211Dic269—22no—
212Dic269—22no—
213Dic269—41no—
214Dic269—41no—
215Dic269—41no—
216C2×C269—24no—
217C2×C269—42no—
218C2×C269C2×C4×D411yescharacteristic
219C2×C269C4×#4111yescharacteristic
220C2×C269#51811yescharacteristic
221C2×C269#54611yescharacteristic
222C2×C269#54611yescharacteristic
223C2×C269#55311yescharacteristic
224C2×C269#55311yescharacteristic
225D269—42no—
226C269#236411yescharacteristic, complemented, Sylow
227#2364—2691nomaximal, Sylow
228C2×#121—2691no—
229C2×C4×D4—2691no—
230C2×C4×D4D26911yescharacteristic, complemented
231#388—2691no—
232#388—2691no—
233#387—2691no—
234#387—2691no—
235#386—2691no—
236#386—2691no—
237#386—2691no—
238#386—2691no—
239C4×C22⋊C4—2691no—
240C4×C22⋊C4—2691no—
241C4×C22⋊C4—2691no—
242C2×#95—2691no—
243C22×D4—2691no—
244C22×D4—2691no—
245C22×D4C2×D26911yescharacteristic
246C23×C4—2691no—
247C23×C4—2691no—
248C23×C4—2691no—
249C23×C4C2×D26911yescharacteristic
250C23×C4C2×D26911yescharacteristic
251#121—2698no—
252C4×D4—24no—
253C4×D4—5384no—
254C2×C4⋊C4—2691no—
255C2×C4⋊C4—2691no—
256C2×C4⋊C4C2×D26911yescharacteristic
257C2×C22⋊C4—2691no—
258C2×C22⋊C4—2691no—
259C2×C22⋊C4—2691no—
260C2×C22⋊C4—2691no—
261C2×C22⋊C4—2691no—
262C2×C22⋊C4—2691no—
263C2×C22⋊C4—2691no—
264C2×C22⋊C4—2691no—
265C2×C22⋊C4C2×D26911yescharacteristic
266C2×C22⋊C4C2×D26911yescharacteristic
267C2×C42—2691no—
268C2×C42—2691no—
269C2×C42—2691no—
270C2×C42C2×D26911yescharacteristic
271#95—2691no—
272#95—2691no—
273#95—2691no—
274#95—2691no—
275#95—2691no—
276#95—2691no—
277#95—2691no—
278#95—2691no—
279#95—2691no—
280#95—2691no—
281C24—2691no—
282C24C22×D26911yescharacteristic
283C24C22×D26911yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—2694no—
287C2×D4—2694no—
288C2×D4—5382no—
289C2×D4—5382no—
290C2×D4—10761no—
291C2×D4—10761no—
292C2×D4C4×D26914yescomplemented
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—2691no—
300C22×C4—2691no—
301C22×C4—2691no—
302C22×C4—2691no—
303C22×C4—2694no—
304C22×C4—2691no—
305C22×C4—2691no—
306C22×C4—2691no—
307C22×C4—5382no—
308C22×C4—5382no—
309C22×C4—5382no—
310C22×C4—5382no—
311C22×C4—10761no—
312C22×C4—10761no—
313C22×C4—10761no—
314C22×C4—10761no—
315C22×C4C22×D26911yescharacteristic
316C22×C4C22×D26911yescharacteristic
317C22×C4C22×D26911yescharacteristic
318C22×C4C22×D26911yescharacteristic
319C22×C4C22×D26911yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—2694no—
322C4⋊C4—5382no—
323C22⋊C4—22no—
324C22⋊C4—2694no—
325C22⋊C4—2694no—
326C22⋊C4—5382no—
327C22⋊C4—5382no—
328C22⋊C4—5382no—
329C22⋊C4—5382no—
330C22⋊C4—10761no—
331C22⋊C4—10761no—
332C22⋊C4C22⋊D26914yescomplemented
333C42—22no—
334C42—5382no—
335C42—10761no—
336C42—10761no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2692no—
348C23—5381no—
349C23—5381no—
350C23—5381no—
351C23—5381no—
352C23—5381no—
353C23—5381no—
354C23C2×C4×D26912yes—
355C23C2×C4×D26912yes—
356C23C2×C22⋊D26912yes—
357C23#1686440212yes—
358C23C23×D26911yescharacteristic
359D4—28no—
360D4—5388no—
361D4—10764no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—2692no—
376C2×C4—2692no—
377C2×C4—2692no—
378C2×C4—2692no—
379C2×C4—5381no—
380C2×C4—5381no—
381C2×C4—5381no—
382C2×C4—5381no—
383C2×C4—5381no—
384C2×C4—5381no—
385C2×C4—5381no—
386C2×C4—5381no—
387C2×C4—5381no—
388C2×C4—5381no—
389C2×C4—5381no—
390C2×C4—5381no—
391C2×C4—5381no—
392C2×C4—5381no—
393C2×C4—5381no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—10764no—
397C2×C4—10764no—
398C2×C4—10764no—
399C2×C4C2×C4×D26912yes—
400C2×C4C2×C22⋊D26912yes—
401C2×C4C2×C22⋊D26912yes—
402C2×C4#1686440112yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—5382no—
417C22—5382no—
418C22—5382no—
419C22—10761no—
420C22—10761no—
421C22—10761no—
422C22—10761no—
423C22C4×C22⋊D26914yes—
424C22C22×C4×D26911yescharacteristic
425C22C22×C22⋊D26911yescharacteristic
426C22#41×D26911yescharacteristic
427C22C2×#1686440111yescharacteristic
428C22C2×#1686440211yescharacteristic
429C22#2257549811yescharacteristic
430C22#2257549911yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—5382no—
437C4—5382no—
438C4—5382no—
439C4—5382no—
440C4—10761no—
441C4—10761no—
442C4—10761no—
443C2—24no—
444C2—42no—
445C2—10762no—
446C2C2×C4×C22⋊D26911yescharacteristic
447C2C4×#1686440111yescharacteristic
448C2#3177107711yescharacteristic
449C2#3177124311yescharacteristic
450C2#3177124911yescharacteristic
451C2#3177130111yescharacteristic
452C2#3177130411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
FittingC2×C4×D4×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575425C2
hypercenterC2×C4×D4D269
hypocenterC269#2364
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#22575413C2semidirect
3. #31770499C2semidirect
4. #31770495C2semidirect
5. #31770489C2semidirect
6. #31770486C2semidirect
7. #31770485C2semidirect
8. #31770484C2semidirect
9. #31770484C2semidirect
10. #31770481C2semidirect
11. C22⋊C4×Dic269C2semidirect
12. C4×#16864379C2semidirect
13. C4×C22⋊Dic269C2semidirect
14. C2×#22575360C2semidirect
15. C2×C4×D4×C269C2semidirect
21. #22575425C4semidirect
59. C2×C22⋊D269C2×C4semidirect
60. C2×C22⋊D269C2×C4semidirect
66. C22×Dic269C2×C4semidirect
78. #16864381C2×C4semidirect
82. C22⋊Dic269C2×C4semidirect
86. #16864379C2×C4semidirect
94. C2×D4×C269C2×C4semidirect
108. C22⋊C4×C269D4semidirect
226. C269#2364semidirect
230. C2×C4×D4D269semidirect
292. C2×D4C4×D269semidirect
332. C22⋊C4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754252
C2×#225754132
#317704992
#317704952
#317704892
#317704862
#317704852
#317704842
#317704812
C22⋊C4×Dic2692
C4×#168643792
C4×C22⋊Dic2692
C2×#225753602
C2×C4×D4×C2692
#2364269