Finite Group Databasebeta

← #46822394

Subgroups — #46822394

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
1C2×#22575485C211yesmaximal, characteristic, complemented
2C2×SD16×D269C211yesmaximal, characteristic, complemented
3#31771270C211yesmaximal, characteristic, complemented
4#31771248C211yesmaximal, characteristic
5#31770848C211yesmaximal, characteristic, complemented
6#31770843C211yesmaximal, characteristic, complemented
7#31770824C211yesmaximal, characteristic, complemented
8#31770807C211yesmaximal, characteristic, complemented
9#31770766C211yesmaximal, characteristic, complemented
10#31770759C211yesmaximal, characteristic, complemented
11#31770754C211yesmaximal, characteristic, complemented
12#31770742C211yesmaximal, characteristic, complemented
13#31770707C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#497×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575485—24no—
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864391C2211yescharacteristic
23C2×#16864390C2211yescharacteristic
24C2×#16864389C2211yescharacteristic
25C2×#16864389C2211yescharacteristic
26SD16×D269—24no—
27C2×#16864385C2211yescharacteristic, complemented
28C2×C8×D269C2211yescharacteristic
29#22575452—21no—
30#22575435—21no—
31#22575431—21no—
32#22575431—21no—
33#22575429C2211yescharacteristic
34#22575429C2211yescharacteristic
35#22575429C2211yescharacteristic, complemented
36#22575426—21no—
37#22575424C2211yescharacteristic
38#22575423—21no—
39C4×Dic538C2211yescharacteristic, complemented
40C4×D1076C2211yescharacteristic
41C4×D1076C2211yescharacteristic
42#22575415—21no—
43C22⋊C4×D269—21no—
44#22575387C2211yescharacteristic, complemented
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575378C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575377C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic
51#22575374C2211yescharacteristic
52#22575365C2211yescharacteristic
53#22575365C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×Q16×C269C2211yescharacteristic, complemented
56C2×SD16×C269C2211yescharacteristic
57C2×SD16×C269C2211yescharacteristic
58C4⋊Q8×C269C2211yescharacteristic, complemented
59#124×C269C2211yescharacteristic
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864403—22no—
63#16864403—41no—
64C2×Dic538—21no—
65C2×Dic538—21no—
66C2×Dic538C2311yescharacteristic
67Q8×D269—22no—
68Q8×D269—22no—
69Q8×D269—41no—
70Q8×D269—41no—
71C2×D1076—21no—
72C2×D1076D411yescharacteristic
73C2×D1076C2311yescharacteristic
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—22no—
77D4×D269—41no—
78C2×C4×D269—21no—
79C2×C4×D269—21no—
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269C2311yescharacteristic
83C2×C4×D269C2311yescharacteristic
84#16864392—22no—
85#16864391D414yescomplemented
86#16864390—22no—
87#16864389—22no—
88#16864389—22no—
89#16864385—22no—
90C2×#14524130C2311yescharacteristic
91C2×#14524130C2311yescharacteristic
92C8×D269—22no—
93C4⋊Dic269D411yescharacteristic
94C4⋊Dic269C2311yescharacteristic
95C4⋊Dic269C2311yescharacteristic
96#16864381—21no—
97#16864381—21no—
98#16864381—21no—
99#16864381—21no—
100C22⋊Dic269—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379D411yescharacteristic
106#16864379D411yescharacteristic
107C4×Dic269—21no—
108C2×Q8×C269C2311yescharacteristic
109C2×Q8×C269C2311yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×D4×C269C2311yescharacteristic
112Q16×C269—22no—
113Q16×C269—41no—
114SD16×C269—22no—
115SD16×C269—41no—
116C2×C8×C269C2311yescharacteristic
117C2×C8×C269C2311yescharacteristic
118C4⋊C4×C269—21no—
119C4⋊C4×C269—21no—
120C22⋊C4×C269—21no—
121C22⋊C4×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269—42no—
125C22×D269—42no—
126C22×D269—81no—
127C22×D269C2×D411yescharacteristic
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—42no—
131Dic538—42no—
132Dic538C2×D412yes—
133D1076—22no—
134D1076—21no—
135D1076—42no—
136D1076C2×D412yes—
137C22⋊D269—42no—
138C22⋊D269—81no—
139C2×Dic269—21no—
140C2×Dic269—21no—
141C2×Dic269—21no—
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C4×D269—21no—
145C4×D269—21no—
146C4×D269—21no—
147C4×D269—21no—
148C4×D269—42no—
149C4×D269—42no—
150C4×D269—42no—
151C4×D269—81no—
152#14524130—21no—
153#14524130—21no—
154C23×C269—21no—
155Q8×C269—21no—
156Q8×C269—21no—
157Q8×C269—21no—
158Q8×C269—21no—
159Q8×C269—21no—
160Q8×C269—21no—
161D4×C269—21no—
162D4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269C2×D411yescharacteristic
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2411yescharacteristic
169C8×C269—21no—
170C8×C269C2×D412yes—
171C22×C269—41no—
172C22×C269—41no—
173C22×C269—41no—
174C22×C269C22×D411yescharacteristic
175C2×D269—21no—
176C2×D269—21no—
177C2×D269—21no—
178C2×D269—22no—
179C2×D269—21no—
180C2×D269—21no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—84no—
184C2×D269#13712yescomplemented
185C4×C269—21no—
186C4×C269—21no—
187C4×C269—41no—
188C4×C269—41no—
189C4×C269—41no—
190C4×C269C22×D411yescharacteristic
191C4×C269H(5,𝔽2)11yescharacteristic
192Dic269—21no—
193Dic269—21no—
194Dic269—22no—
195Dic269—41no—
196Dic269—41no—
197C2×C269—81no—
198C2×C269D4211yescharacteristic
199C2×C269C2×#13711yescharacteristic
200C2×C269#57711yescharacteristic
201D269—22no—
202D269—41no—
203D269—42no—
204C269#326611yescharacteristic, complemented, Sylow
205#3266—2691nomaximal, Sylow
206C2×#137—2691no—
207C22×SD16—2691no—
208D4×Q8—2691no—
209#546—2691no—
210#497D26911yescharacteristic, complemented
211#468—2691no—
212#467—2691no—
213#464—2691no—
214#461—2691no—
215#452—2691no—
216#451—2691no—
217#450—2691no—
218#448—2691no—
219#440—2691no—
220#435—2691no—
221C2×#41—2691no—
222C22×Q8—2691no—
223C22×Q8—2691no—
224C22×D4—2691no—
225#137—5384no—
226C2×Q16—2691no—
227C2×Q16C2×D26911yescharacteristic, complemented
228C2×Q16C2×D26911yescharacteristic, complemented
229C2×SD16—2691no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×SD16—2691no—
233C2×SD16—5384no—
234C2×SD16C2×D26911yescharacteristic
235C2×SD16C2×D26911yescharacteristic
236C2×M16—2691no—
237C22×C8—2691no—
238C4⋊Q8—5381no—
239C4⋊Q8C2×D26911yescharacteristic, complemented
240#124C2×D26911yescharacteristic
241#123—5381no—
242C22⋊Q8—2691no—
243C22⋊Q8—2691no—
244C22⋊Q8—2691no—
245C22⋊Q8—5381no—
246C22⋊Q8—5381no—
247#121—2691no—
248#121—5381no—
249#120—5381no—
250C4×Q8—2691no—
251C4×D4—2691no—
252C4×D4—2691no—
253C4×D4—5381no—
254C2×C22⋊C4—5381no—
255#107—2691no—
256C4⋊C8—2691no—
257#103—2691no—
258#103—2691no—
259#103—2691no—
260#103—2691no—
261#102—2691no—
262#102—2691no—
263C22⋊C8—2691no—
264C22⋊C8—2691no—
265#97C2×D26911yescharacteristic, complemented
266C24—5381no—
267#41—5382no—
268#41—10761no—
269C2×Q8—2691no—
270C2×Q8—5381no—
271C2×Q8—5381no—
272C2×Q8—5382no—
273C2×Q8—5382no—
274C2×Q8—10761no—
275C2×Q8—10761no—
276C2×Q8C22×D26911yescharacteristic
277C2×Q8C22×D26911yescharacteristic
278C2×Q8C22×D26911yescharacteristic
279C2×D4—2691no—
280C2×D4—2691no—
281C2×D4—5381no—
282C2×D4—5381no—
283C2×D4—5382no—
284C2×D4—5381no—
285C2×D4—10761no—
286C2×D4C22×D26911yescharacteristic
287C22×C4—2691no—
288C22×C4—2691no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4—5381no—
292C22×C4—5381no—
293Q16—22no—
294Q16—41no—
295Q16—5382no—
296SD16—22no—
297SD16—41no—
298SD16—2694no—
299SD16—5382no—
300SD16—5382no—
301SD16—5382no—
302M16—5382no—
303C2×C8—2691no—
304C2×C8—2691no—
305C2×C8—5382no—
306C2×C8C22×D26911yescharacteristic
307C2×C8C22×D26911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—2691no—
311C4⋊C4—2691no—
312C4⋊C4—2691no—
313C4⋊C4—5381no—
314C4⋊C4—5381no—
315C4⋊C4—5381no—
316C4⋊C4—5381no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2691no—
320C22⋊C4—2691no—
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C42—5381no—
327C42C22×D26911yescharacteristic
328C23—21no—
329C23—2691no—
330C23—2691no—
331C23—5381no—
332C23—10762no—
333C23—10762no—
334C23—21521no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—2692no—
342Q8—5381no—
343Q8—10762no—
344Q8—10762no—
345D4—21no—
346D4—21no—
347D4—2692no—
348D4—5382no—
349D4—5381no—
350D4—10762no—
351D4—10762no—
352D4—21521no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—2691no—
357C2×C4—2691no—
358C2×C4—5381no—
359C2×C4—5381no—
360C2×C4—5381no—
361C2×C4—5381no—
362C2×C4—5381no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—10762no—
366C2×C4—10762no—
367C2×C4—10762no—
368C2×C4—21521no—
369C2×C4D4×D26911yescharacteristic
370C2×C4D4×D26911yescharacteristic
371C2×C4C23×D26911yescharacteristic
372C8—21no—
373C8—5381no—
374C8—5381no—
375C8D4×D26912yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2692no—
380C22—5381no—
381C22—5381no—
382C22—5381no—
383C22—5381no—
384C22—5381no—
385C22—5382no—
386C22—10761no—
387C22—10761no—
388C22—21524no—
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—5381no—
396C4—5381no—
397C4—5382no—
398C4—10761no—
399C4—10761no—
400C4C2×D4×D26911yescharacteristic
401C4#2257549811yescharacteristic
402C2—81no—
403C2—5382no—
404C2—10761no—
405C2—10762no—
406C2#3177124011yescharacteristic
407C2#137×D26911yescharacteristic
408C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#497×C269C2
socleC22×C269C22×D4
prime radical#31771248C2
hypercenter#497D269
hypocenterC269#3266
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22575485C2semidirect
2. C2×SD16×D269C2semidirect
3. #31771270C2semidirect
5. #31770848C2semidirect
6. #31770843C2semidirect
7. #31770824C2semidirect
8. #31770807C2semidirect
9. #31770766C2semidirect
10. #31770759C2semidirect
11. #31770754C2semidirect
12. #31770742C2semidirect
13. #31770707C2semidirect
14. #31770678C2semidirect
15. #497×C269C2semidirect
18. C2×Q8×D269C22semidirect
21. C2×#16864392C22semidirect
27. C2×#16864385C22semidirect
35. #22575429C22semidirect
39. C4×Dic538C22semidirect
44. #22575387C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575378C22semidirect
47. #22575377C22semidirect
48. #22575377C22semidirect
49. #22575377C22semidirect
53. #22575365C22semidirect
54. C2×Q16×C269C22semidirect
55. C2×Q16×C269C22semidirect
58. C4⋊Q8×C269C22semidirect
60. #97×C269C22semidirect
85. #16864391D4semidirect
184. C2×D269#137semidirect
204. C269#3266semidirect
210. #497D269semidirect
227. C2×Q16C2×D269semidirect
228. C2×Q16C2×D269semidirect
239. C4⋊Q8C2×D269semidirect
265. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
C2×SD16×D2692
#317712702
#317712482
#317708482
#317708432
#317708242
#317708072
#317707662
#317707592
#317707542
#317707422
#317707072
#317706782
#497×C2692
#3266269