Finite Group Databasebeta

← #24918972

Subgroups — #24918972

Order 10624.

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×#14581824C211yesmaximal, characteristic, complemented
2C2×SD16×D83C211yesmaximal, characteristic, complemented
3#19128366C211yesmaximal, characteristic, complemented
4#19128344C211yesmaximal, characteristic
5#19127944C211yesmaximal, characteristic, complemented
6#19127939C211yesmaximal, characteristic, complemented
7#19127920C211yesmaximal, characteristic, complemented
8#19127903C211yesmaximal, characteristic, complemented
9#19127862C211yesmaximal, characteristic, complemented
10#19127855C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127838C211yesmaximal, characteristic, complemented
13#19127803C211yesmaximal, characteristic, complemented
14#19127774C211yesmaximal, characteristic, complemented
15#497×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic
18C2×Q8×D83C2211yescharacteristic, complemented
19C2×D4×D83C2211yescharacteristic
20#14581824—24no—
21C2×#13050580C2211yescharacteristic, complemented
22C2×#13050579C2211yescharacteristic
23C2×#13050578C2211yescharacteristic
24C2×#13050577C2211yescharacteristic
25C2×#13050577C2211yescharacteristic
26SD16×D83—24no—
27C2×#13050573C2211yescharacteristic, complemented
28C2×C8×D83C2211yescharacteristic
29#14581791—21no—
30#14581774—21no—
31#14581770—21no—
32#14581770—21no—
33#14581768C2211yescharacteristic
34#14581768C2211yescharacteristic
35#14581768C2211yescharacteristic, complemented
36#14581765—21no—
37#14581763C2211yescharacteristic
38#14581762—21no—
39C4×Dic166C2211yescharacteristic, complemented
40C4×D332C2211yescharacteristic
41C4×D332C2211yescharacteristic
42#14581754—21no—
43C22⋊C4×D83—21no—
44#14581726C2211yescharacteristic, complemented
45C4⋊(C83⋊C8)C2211yescharacteristic, complemented
46#14581717C2211yescharacteristic, complemented
47#14581716C2211yescharacteristic, complemented
48#14581716C2211yescharacteristic, complemented
49#14581716C2211yescharacteristic, complemented
50#14581714C2211yescharacteristic
51#14581713C2211yescharacteristic
52(C2×D83)⋊C8C2211yescharacteristic
53(C2×D83)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C83C2211yescharacteristic, complemented
55C2×Q16×C83C2211yescharacteristic, complemented
56C2×SD16×C83C2211yescharacteristic
57C2×SD16×C83C2211yescharacteristic
58C4⋊Q8×C83C2211yescharacteristic, complemented
59#124×C83C2211yescharacteristic
60#97×C83C2211yescharacteristic, complemented
61C23×D83—21no—
62#13050591—22no—
63#13050591—41no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166C2311yescharacteristic
67Q8×D83—22no—
68Q8×D83—22no—
69Q8×D83—41no—
70Q8×D83—41no—
71C2×D332—21no—
72C2×D332D411yescharacteristic
73C2×D332C2311yescharacteristic
74C2×C22⋊D83—21no—
75C2×C22⋊D83—21no—
76D4×D83—22no—
77D4×D83—41no—
78C2×C4×D83—21no—
79C2×C4×D83—21no—
80C2×C4×D83—21no—
81C2×C4×D83—21no—
82C2×C4×D83C2311yescharacteristic
83C2×C4×D83C2311yescharacteristic
84#13050580—22no—
85#13050579D414yescomplemented
86#13050578—22no—
87#13050577—22no—
88#13050577—22no—
89#13050573—22no—
90C2×C83⋊C8C2311yescharacteristic
91C2×C83⋊C8C2311yescharacteristic
92C8×D83—22no—
93C4⋊Dic83D411yescharacteristic
94C4⋊Dic83C2311yescharacteristic
95C4⋊Dic83C2311yescharacteristic
96Dic83⋊C4—21no—
97Dic83⋊C4—21no—
98Dic83⋊C4—21no—
99Dic83⋊C4—21no—
100C22⋊Dic83—21no—
101(C2×D83)⋊C4—21no—
102(C2×D83)⋊C4—21no—
103(C2×D83)⋊C4—21no—
104(C2×D83)⋊C4—21no—
105(C2×D83)⋊C4D411yescharacteristic
106(C2×D83)⋊C4D411yescharacteristic
107C4×Dic83—21no—
108C2×Q8×C83C2311yescharacteristic
109C2×Q8×C83C2311yescharacteristic
110C2×Q8×C83C2311yescharacteristic
111C2×D4×C83C2311yescharacteristic
112Q16×C83—22no—
113Q16×C83—41no—
114SD16×C83—22no—
115SD16×C83—41no—
116C2×C8×C83C2311yescharacteristic
117C2×C8×C83C2311yescharacteristic
118C4⋊C4×C83—21no—
119C4⋊C4×C83—21no—
120C22⋊C4×C83—21no—
121C22⋊C4×C83—21no—
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C22×D83—21no—
125C22×D83—42no—
126C22×D83—42no—
127C22×D83—81no—
128C22×D83C2×D411yescharacteristic
129C22×D83C2×D411yescharacteristic
130Q8×C83—21no—
131Q8×C83—21no—
132Q8×C83—21no—
133Q8×C83—21no—
134Q8×C83—21no—
135Q8×C83—21no—
136D4×C83—21no—
137D4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83C2×D411yescharacteristic
142C2×C4×C83C2×D411yescharacteristic
143C2×C4×C83C2411yescharacteristic
144C22⋊D83—42no—
145C22⋊D83—81no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83C2×D411yescharacteristic
150C2×Dic83C2×D411yescharacteristic
151D332—22no—
152D332—21no—
153D332—42no—
154D332C2×D412yes—
155C4×D83—21no—
156C4×D83—21no—
157C4×D83—21no—
158C4×D83—21no—
159C4×D83—42no—
160C4×D83—42no—
161C4×D83—42no—
162C4×D83—81no—
163Dic166—21no—
164Dic166—42no—
165Dic166—42no—
166Dic166C2×D412yes—
167C8×C83—21no—
168C8×C83C2×D412yes—
169C83⋊C8—21no—
170C83⋊C8—21no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83C22×D411yescharacteristic
175C2×D83—21no—
176C2×D83—21no—
177C2×D83—21no—
178C2×D83—22no—
179C2×D83—21no—
180C2×D83—21no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—84no—
184C2×D83#13712yescomplemented
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83—41no—
190C4×C83C22×D411yescharacteristic
191C4×C83H(5,𝔽2)11yescharacteristic
192Dic83—22no—
193Dic83—21no—
194Dic83—21no—
195Dic83—41no—
196Dic83—41no—
197C2×C83—81no—
198C2×C83D4211yescharacteristic
199C2×C83C2×#13711yescharacteristic
200C2×C83#57711yescharacteristic
201D83—22no—
202D83—41no—
203D83—42no—
204#3266—831nomaximal, Sylow
205C83#326611yescharacteristic, complemented, Sylow
206C2×#137—831no—
207C22×SD16—831no—
208D4×Q8—831no—
209#546—831no—
210#497D8311yescharacteristic, complemented
211#468—831no—
212#467—831no—
213#464—831no—
214#461—831no—
215#452—831no—
216#451—831no—
217#450—831no—
218#448—831no—
219#440—831no—
220#435—831no—
221C2×#41—831no—
222C22×Q8—831no—
223C22×Q8—831no—
224C22×D4—831no—
225#137—1664no—
226C2×Q16—831no—
227C2×Q16C2×D8311yescharacteristic, complemented
228C2×Q16C2×D8311yescharacteristic, complemented
229C2×SD16—831no—
230C2×SD16—831no—
231C2×SD16—831no—
232C2×SD16—831no—
233C2×SD16—1664no—
234C2×SD16C2×D8311yescharacteristic
235C2×SD16C2×D8311yescharacteristic
236C2×M16—831no—
237C22×C8—831no—
238C4⋊Q8—1661no—
239C4⋊Q8C2×D8311yescharacteristic, complemented
240#124C2×D8311yescharacteristic
241#123—1661no—
242C22⋊Q8—831no—
243C22⋊Q8—831no—
244C22⋊Q8—831no—
245C22⋊Q8—1661no—
246C22⋊Q8—1661no—
247#121—831no—
248#121—1661no—
249#120—1661no—
250C4×Q8—831no—
251C4×D4—831no—
252C4×D4—831no—
253C4×D4—1661no—
254C2×C22⋊C4—1661no—
255#107—831no—
256C4⋊C8—831no—
257#103—831no—
258#103—831no—
259#103—831no—
260#103—831no—
261#102—831no—
262#102—831no—
263C22⋊C8—831no—
264C22⋊C8—831no—
265#97C2×D8311yescharacteristic, complemented
266C24—1661no—
267#41—3321no—
268#41—1662no—
269C2×Q8—3321no—
270C2×Q8—3321no—
271C2×Q8—831no—
272C2×Q8—1661no—
273C2×Q8—1661no—
274C2×Q8—1662no—
275C2×Q8—1662no—
276C2×Q8C22×D8311yescharacteristic
277C2×Q8C22×D8311yescharacteristic
278C2×Q8C22×D8311yescharacteristic
279C2×D4—3321no—
280C2×D4—831no—
281C2×D4—831no—
282C2×D4—1661no—
283C2×D4—1661no—
284C2×D4—1662no—
285C2×D4—1661no—
286C2×D4C22×D8311yescharacteristic
287C22×C4—831no—
288C22×C4—831no—
289C22×C4—1661no—
290C22×C4—1661no—
291C22×C4—1661no—
292C22×C4—1661no—
293Q16—22no—
294Q16—41no—
295Q16—1662no—
296SD16—22no—
297SD16—41no—
298SD16—834no—
299SD16—1662no—
300SD16—1662no—
301SD16—1662no—
302M16—1662no—
303C2×C8—831no—
304C2×C8—831no—
305C2×C8—1662no—
306C2×C8C22×D8311yescharacteristic
307C2×C8C22×D8311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—831no—
311C4⋊C4—831no—
312C4⋊C4—831no—
313C4⋊C4—1661no—
314C4⋊C4—1661no—
315C4⋊C4—1661no—
316C4⋊C4—1661no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—831no—
320C22⋊C4—831no—
321C22⋊C4—1661no—
322C22⋊C4—1661no—
323C22⋊C4—1661no—
324C22⋊C4—1661no—
325C22⋊C4—1661no—
326C42—1661no—
327C42C22×D8311yescharacteristic
328C23—21no—
329C23—3322no—
330C23—3322no—
331C23—831no—
332C23—831no—
333C23—6641no—
334C23—1661no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—3322no—
342Q8—3322no—
343Q8—832no—
344Q8—1661no—
345D4—21no—
346D4—21no—
347D4—3322no—
348D4—3322no—
349D4—832no—
350D4—6641no—
351D4—1662no—
352D4—1661no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3322no—
357C2×C4—3322no—
358C2×C4—3322no—
359C2×C4—831no—
360C2×C4—831no—
361C2×C4—6641no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4D4×D8311yescharacteristic
370C2×C4D4×D8311yescharacteristic
371C2×C4C23×D8311yescharacteristic
372C8—21no—
373C8—1661no—
374C8—1661no—
375C8D4×D8312yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3321no—
380C22—3321no—
381C22—832no—
382C22—6644no—
383C22—1661no—
384C22—1662no—
385C22—1661no—
386C22—1661no—
387C22—1661no—
388C22—1661no—
389C22C2×D4×D8311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3321no—
396C4—3321no—
397C4—1661no—
398C4—1662no—
399C4—1661no—
400C4C2×D4×D8311yescharacteristic
401C4#1458183711yescharacteristic
402C2—81no—
403C2—3322no—
404C2—3321no—
405C2—1662no—
406C2#1912833611yescharacteristic
407C2#137×D8311yescharacteristic
408C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
prime radical#19128344C2
hypercenter#497D83
hypocenterC83#3266
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14581824C2semidirect
2. C2×SD16×D83C2semidirect
3. #19128366C2semidirect
5. #19127944C2semidirect
6. #19127939C2semidirect
7. #19127920C2semidirect
8. #19127903C2semidirect
9. #19127862C2semidirect
10. #19127855C2semidirect
11. #19127850C2semidirect
12. #19127838C2semidirect
13. #19127803C2semidirect
14. #19127774C2semidirect
15. #497×C83C2semidirect
18. C2×Q8×D83C22semidirect
21. C2×#13050580C22semidirect
27. C2×#13050573C22semidirect
35. #14581768C22semidirect
39. C4×Dic166C22semidirect
44. #14581726C22semidirect
45. C4⋊(C83⋊C8)C22semidirect
46. #14581717C22semidirect
47. #14581716C22semidirect
48. #14581716C22semidirect
49. #14581716C22semidirect
53. (C2×D83)⋊C8C22semidirect
54. C2×Q16×C83C22semidirect
55. C2×Q16×C83C22semidirect
58. C4⋊Q8×C83C22semidirect
60. #97×C83C22semidirect
85. #13050579D4semidirect
184. C2×D83#137semidirect
205. C83#3266semidirect
210. #497D83semidirect
227. C2×Q16C2×D83semidirect
228. C2×Q16C2×D83semidirect
239. C4⋊Q8C2×D83semidirect
265. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818242
C2×SD16×D832
#191283662
#191283442
#191279442
#191279392
#191279202
#191279032
#191278622
#191278552
#191278502
#191278382
#191278032
#191277742
#497×C832
#326683