Finite Group Databasebeta

← #46820512

Subgroups — #46820512

Order 34432.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575485C211yesmaximal, characteristic, complemented
2C22×#16864385C211yesmaximal, characteristic, complemented
3#31771118C211yesmaximal, characteristic
4C22⋊Q8×D269C211yesmaximal, characteristic, complemented
5#31770862C211yesmaximal, characteristic, complemented
6#31770860C211yesmaximal, characteristic, complemented
7#31770854C211yesmaximal, characteristic, complemented
8#31770852C211yesmaximal, characteristic, complemented
9#31770844C211yesmaximal, characteristic, complemented
10#31770836C211yesmaximal, characteristic, complemented
11#31770834C211yesmaximal, characteristic, complemented
12#31770646C211yesmaximal, characteristic, complemented
13#31770612C211yesmaximal, characteristic, complemented
14#31770605C211yesmaximal, characteristic, complemented
15#467×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C22×C4×D269C2211yescharacteristic
19#22575485—24no—
20C2×#16864392C2211yescharacteristic, complemented
21C2×#16864391C2211yescharacteristic, complemented
22C2×#16864389C2211yescharacteristic, complemented
23C2×#16864385—24no—
24C2×#16864385C2211yescharacteristic
25C2×#16864385C2211yescharacteristic
26C2×#16864385C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C22×#14524130C2211yescharacteristic
29#22575438—21no—
30#22575433C2211yescharacteristic
31#22575432C2211yescharacteristic
32#22575431—21no—
33#22575430—21no—
34#22575429C2211yescharacteristic
35#22575429C2211yescharacteristic
36#22575428C2211yescharacteristic
37#22575427C2211yescharacteristic
38#22575423—21no—
39#22575415—21no—
40#22575413—21no—
41#22575411C2211yescharacteristic
42C4⋊C4×D269—21no—
43C4⋊C4×D269C2211yescharacteristic
44C22⋊C4×D269—21no—
45#22575386C2211yescharacteristic, complemented
46#22575385C2211yescharacteristic, complemented
47#22575384C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575377C2211yescharacteristic, complemented
50#22575377C2211yescharacteristic, complemented
51#22575376C2211yescharacteristic, complemented
52#22575375C2211yescharacteristic, complemented
53#22575373C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C22×C8×C269C2211yescharacteristic
56C22⋊Q8×C269C2211yescharacteristic
57C22⋊Q8×C269C2211yescharacteristic
58#107×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61C23×D269D411yescharacteristic
62#16864403—22no—
63#16864403—22no—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269—22no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72C22×Dic269D411yescharacteristic
73C2×C4×D269—22no—
74C2×C4×D269—22no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269D411yescharacteristic
79C2×C4×D269D411yescharacteristic
80C2×C4×D269C2311yescharacteristic
81C2×C4×D269C2311yescharacteristic
82#16864392—22no—
83#16864391—22no—
84#16864389—22no—
85#16864385—22no—
86#16864385—22no—
87#16864385—22no—
88#16864385D414yescomplemented
89C2×#14524130—22no—
90C2×#14524130C2311yescharacteristic
91C2×#14524130C2311yescharacteristic
92C4⋊Dic269—21no—
93C4⋊Dic269C2311yescharacteristic
94C4⋊Dic269C2311yescharacteristic
95#16864381—21no—
96#16864381—21no—
97#16864381—21no—
98#16864381—21no—
99C22⋊Dic269—21no—
100#16864379—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106C4×Dic269—21no—
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C22×C4×C269C2311yescharacteristic
110Q16×C269—22no—
111C2×C8×C269—22no—
112C2×C8×C269C2311yescharacteristic
113C2×C8×C269C2311yescharacteristic
114C4⋊C4×C269—21no—
115C4⋊C4×C269—21no—
116C4⋊C4×C269C2311yescharacteristic
117C4⋊C4×C269C2311yescharacteristic
118C22⋊C4×C269—21no—
119C22⋊C4×C269—21no—
120C22×D269—22no—
121C22×D269—22no—
122C22×D269—21no—
123C22×D269—41no—
124C22×D269C2×D411yescharacteristic
125C22×D269C2×D411yescharacteristic
126Dic538—21no—
127Dic538—21no—
128Dic538—42no—
129D1076—21no—
130D1076—21no—
131D1076—42no—
132D1076—42no—
133C22⋊D269—42no—
134C2×Dic269—22no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269C2×D411yescharacteristic
139C2×Dic269C2×D411yescharacteristic
140C4×D269—22no—
141C4×D269—21no—
142C4×D269—21no—
143C4×D269—22no—
144C4×D269—42no—
145C4×D269—42no—
146C4×D269—42no—
147C4×D269C2×D412yes—
148C4×D269#4112yes—
149#14524130—21no—
150#14524130C2×D412yes—
151C23×C269C2×D411yescharacteristic
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—21no—
155Q8×C269—21no—
156C2×C4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269C2×D411yescharacteristic
163C2×C4×C269C2411yescharacteristic
164C8×C269—21no—
165C8×C269C2×D412yes—
166C22×C269—21no—
167C22×C269—21no—
168C22×C269#13712yescomplemented
169C22×C269C22×D411yescharacteristic
170C2×D269—21no—
171C2×D269—21no—
172C2×D269—21no—
173C2×D269—21no—
174C2×D269—21no—
175C2×D269—44no—
176C2×D269—41no—
177C2×D269—41no—
178C2×D269—41no—
179C2×D269#12112yes—
180C4×C269—21no—
181C4×C269—41no—
182C4×C269—41no—
183C4×C269—41no—
184C4×C269—41no—
185C4×C269C22×D411yescharacteristic
186C4×C269C2×#4111yescharacteristic
187Dic269—21no—
188Dic269—41no—
189Dic269—41no—
190Dic269—41no—
191Dic269#12112yes—
192C2×C269—22no—
193C2×C269C2×#12111yescharacteristic
194C2×C269C2×#13711yescharacteristic
195C2×C269#57511yescharacteristic
196D269—22no—
197D269—41no—
198D269—81no—
199C269#305611yescharacteristic, complemented, Sylow
200#3056—2691nomaximal, Sylow
201C2×#137—2691no—
202C22×M16—2691no—
203#525—2691no—
204C2×C22⋊Q8—2691no—
205#470—2691no—
206#470—2691no—
207#468—2691no—
208#468—2691no—
209#467—2691no—
210#467D26911yescharacteristic, complemented
211#465—2691no—
212#465—2691no—
213#428—2691no—
214#419—2691no—
215#417—2691no—
216C2×#41—2691no—
217C22×Q8—2691no—
218C23×C4—2691no—
219#137—5384no—
220C2×Q16—2691no—
221C2×Q16C2×D26911yescharacteristic, complemented
222C2×SD16—2691no—
223C2×SD16—2691no—
224C2×M16—2691no—
225C2×M16—2691no—
226C2×M16—2691no—
227C2×M16—2691no—
228C2×M16—5384no—
229C22×C8—2691no—
230C22×C8C2×D26911yescharacteristic
231#123—5381no—
232C22⋊Q8—2691no—
233C22⋊Q8—2691no—
234C22⋊Q8—2691no—
235C22⋊Q8—2691no—
236C22⋊Q8—5381no—
237C22⋊Q8—5381no—
238C22⋊Q8C2×D26911yescharacteristic
239C22⋊Q8C2×D26911yescharacteristic
240#121—2691no—
241#121—2691no—
242#120—5381no—
243C4×D4—5381no—
244C4×D4—5381no—
245#117—2691no—
246C2×C4⋊C4—2691no—
247C2×C4⋊C4—5381no—
248C2×C22⋊C4—5381no—
249#107—2691no—
250#107C2×D26911yescharacteristic, complemented
251#106—2691no—
252#106—2691no—
253#103—2691no—
254#103—2691no—
255#103—2691no—
256#103—2691no—
257#103C2×D26911yescharacteristic, complemented
258#103C2×D26911yescharacteristic, complemented
259#102—2691no—
260#102—2691no—
261C24—2691no—
262#41—5382no—
263#41—5382no—
264C2×Q8—2691no—
265C2×Q8—5381no—
266C2×Q8—5382no—
267C2×Q8—5382no—
268C2×Q8C22×D26911yescharacteristic
269C2×Q8C22×D26911yescharacteristic
270C2×D4—2691no—
271C2×D4—5381no—
272C2×D4—5381no—
273C2×D4—5381no—
274C22×C4—2691no—
275C22×C4—2691no—
276C22×C4—2691no—
277C22×C4—2691no—
278C22×C4—2691no—
279C22×C4—5382no—
280C22×C4—5382no—
281C22×C4—5381no—
282C22×C4—5381no—
283C22×C4—5381no—
284C22×C4C22×D26911yescharacteristic
285Q16—22no—
286Q16—5382no—
287SD16—5382no—
288SD16—5382no—
289M16—2694no—
290M16—5382no—
291M16—5382no—
292M16—5382no—
293C2×C8—22no—
294C2×C8—2691no—
295C2×C8—2691no—
296C2×C8—5382no—
297C2×C8C22×D26911yescharacteristic
298C2×C8C22×D26911yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—2691no—
302C4⋊C4—2691no—
303C4⋊C4—5381no—
304C4⋊C4—5381no—
305C4⋊C4—5381no—
306C4⋊C4—5381no—
307C4⋊C4—5381no—
308C4⋊C4C22×D26911yescharacteristic
309C4⋊C4C22×D26911yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—5381no—
313C22⋊C4—5381no—
314C22⋊C4—5381no—
315C22⋊C4—5381no—
316C22⋊C4—5381no—
317C22⋊C4—5381no—
318C22⋊C4—5381no—
319C42—5381no—
320C23—2691no—
321C23—2691no—
322C23—5382no—
323C23—5382no—
324C23—5381no—
325C23—10761no—
326C23D4×D26911yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—5381no—
332Q8—5381no—
333Q8—10762no—
334D4—5381no—
335D4—5381no—
336D4—10762no—
337D4—10762no—
338D4—10762no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—2691no—
346C2×C4—2692no—
347C2×C4—2692no—
348C2×C4—2691no—
349C2×C4—5382no—
350C2×C4—5382no—
351C2×C4—5382no—
352C2×C4—5381no—
353C2×C4—5381no—
354C2×C4—5381no—
355C2×C4—5381no—
356C2×C4—5381no—
357C2×C4—10762no—
358C2×C4—10762no—
359C2×C4—10762no—
360C2×C4D4×D26911yescharacteristic
361C2×C4C23×D26911yescharacteristic
362C8—21no—
363C8—2692no—
364C8—5381no—
365C8D4×D26912yes—
366C22—21no—
367C22—21no—
368C22—2692no—
369C22—5381no—
370C22—5381no—
371C22—5381no—
372C22—5381no—
373C22—5381no—
374C22—10764no—
375C22—10761no—
376C22—10761no—
377C22—10761no—
378C22#2257548512yescomplemented
379C22C2×D4×D26911yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—2692no—
386C4—5381no—
387C4—10761no—
388C4—10761no—
389C4—10761no—
390C4C2×D4×D26911yescharacteristic
391C4#41×D26911yescharacteristic
392C2—22no—
393C2—5382no—
394C2—10761no—
395C2—21521no—
396C2#121×D26911yescharacteristic
397C2C2×#2257548511yescharacteristic
398C2#3177140211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#467×C269C2
socleC22×C269C22×D4
prime radical#31771118C2
hypercenter#467D269
hypocenterC269#3056
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#22575485C2semidirect
2. C22×#16864385C2semidirect
4. C22⋊Q8×D269C2semidirect
5. #31770862C2semidirect
6. #31770860C2semidirect
7. #31770854C2semidirect
8. #31770852C2semidirect
9. #31770844C2semidirect
10. #31770836C2semidirect
11. #31770834C2semidirect
12. #31770646C2semidirect
13. #31770612C2semidirect
14. #31770605C2semidirect
15. #467×C269C2semidirect
20. C2×#16864392C22semidirect
21. C2×#16864391C22semidirect
22. C2×#16864389C22semidirect
45. #22575386C22semidirect
46. #22575385C22semidirect
47. #22575384C22semidirect
48. #22575378C22semidirect
49. #22575377C22semidirect
50. #22575377C22semidirect
51. #22575376C22semidirect
52. #22575375C22semidirect
53. #22575373C22semidirect
54. C2×Q16×C269C22semidirect
58. #107×C269C22semidirect
59. #103×C269C22semidirect
60. #103×C269C22semidirect
88. #16864385D4semidirect
168. C22×C269#137semidirect
199. C269#3056semidirect
210. #467D269semidirect
221. C2×Q16C2×D269semidirect
250. #107C2×D269semidirect
257. #103C2×D269semidirect
258. #103C2×D269semidirect
378. C22#22575485semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
C22×#168643852
#317711182
C22⋊Q8×D2692
#317708622
#317708602
#317708542
#317708522
#317708442
#317708362
#317708342
#317706462
#317706122
#317706052
#467×C2692
#3056269