Finite Group Databasebeta

← #24919627

Subgroups — #24919627

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128398C211yesmaximal, characteristic
2Q8×C22⋊D83C211yesmaximal, characteristic, complemented
3#19128068C211yesmaximal, characteristic, complemented
4#19128043C211yesmaximal, characteristic, complemented
5#19128035C211yesmaximal, characteristic, complemented
6#19128006C211yesmaximal, characteristic, complemented
7#19127945C211yesmaximal, characteristic, complemented
8#19127927C211yesmaximal, characteristic, complemented
9#19127904C211yesmaximal, characteristic, complemented
10#19127838C211yesmaximal, characteristic, complemented
11#19127802C211yesmaximal, characteristic, complemented
12#19127775C211yesmaximal, characteristic, complemented
13#19127710C211yesmaximal, characteristic, complemented
14#103×D83C211yesmaximal, characteristic, complemented
15#448×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C2×#13050578C2211yescharacteristic
18C2×#13050574C2211yescharacteristic
19C2×C8×D83C2211yescharacteristic
20#14581790—21no—
21#14581790C2211yescharacteristic
22#14581785—21no—
23#14581783C2211yescharacteristic
24#14581773—21no—
25#14581771—21no—
26#14581771C2211yescharacteristic
27#14581768—21no—
28#14581768C2211yescharacteristic
29#14581763—21no—
30#14581763C2211yescharacteristic
31Q8×Dic83C2211yescharacteristic
32D4×Dic83C2211yescharacteristic
33C4×C22⋊D83—21no—
34C4×C22⋊D83C2211yescharacteristic
35(C22⋊D83)⋊C4—21no—
36#14581751C2211yescharacteristic
37C4⋊C4×D83C2211yescharacteristic
38#14581726C2211yescharacteristic
39#14581725C2211yescharacteristic
40#14581724C2211yescharacteristic
41Dic83⋊C8C2211yescharacteristic
42#14581717C2211yescharacteristic
43#14581717C2211yescharacteristic
44#14581716C2211yescharacteristic
45#14581716C2211yescharacteristic
46#14581716C2211yescharacteristic
47#14581716C2211yescharacteristic
48#14581715C2211yescharacteristic
49#14581713C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581702C2211yescharacteristic
52C22×Q8×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic
57#103×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic
59C2×Dic166—21no—
60C2×Dic166C2311yescharacteristic
61Q8×D83—22no—
62Q8×D83—41no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83D411yescharacteristic
65C22×Dic83—21no—
66C2×C4×D83—21no—
67C2×C4×D83—21no—
68C2×C4×D83C2311yescharacteristic
69#13050578—41no—
70#13050574—41no—
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C8×D83—41no—
74C4⋊Dic83—21no—
75C4⋊Dic83—21no—
76C4⋊Dic83C2311yescharacteristic
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4—21no—
84Dic83⋊C4—21no—
85Dic83⋊C4D411yescharacteristic
86C22⋊Dic83—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83D411yescharacteristic
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4D411yescharacteristic
93C4×Dic83—21no—
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96C2×Q8×C83—21no—
97C2×Q8×C83—22no—
98C2×Q8×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×Q8×C83C2311yescharacteristic
101C2×D4×C83—21no—
102C2×D4×C83C2311yescharacteristic
103C22×C4×C83—21no—
104C22×C4×C83C2311yescharacteristic
105SD16×C83—22no—
106SD16×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C22×D83C2×D411yescharacteristic
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—42no—
118Q8×C83#4112yes—
119D4×C83—21no—
120D4×C83—21no—
121D4×C83—81no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—42no—
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2411yescharacteristic
130C22⋊D83—22no—
131C22⋊D83—42no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—81no—
137C2×Dic83C2×D411yescharacteristic
138C2×Dic83C2×D411yescharacteristic
139C2×Dic83C2×D411yescharacteristic
140C4×D83—21no—
141C4×D83—21no—
142C4×D83—42no—
143C4×D83—81no—
144Dic166—21no—
145Dic166—21no—
146Dic166—42no—
147C8×C83—21no—
148C8×C83—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83C22×D411yescharacteristic
158C2×D83—21no—
159C2×D83—21no—
160C2×D83#13712yes—
161C4×C83—22no—
162C4×C83—21no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83C2×#4111yescharacteristic
166C4×C83H(5,𝔽2)11yescharacteristic
167Dic83—21no—
168Dic83—21no—
169Dic83—21no—
170Dic83—41no—
171Dic83—41no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—41no—
175C2×C83—81no—
176C2×C83#54611yescharacteristic
177C2×C83C2×#13711yescharacteristic
178C2×C83#57711yescharacteristic
179D83—22no—
180#3320—831nomaximal, Sylow
181C83#332011yescharacteristic, complemented, Sylow
182#553—831no—
183D4×Q8—831no—
184#491—831no—
185#485—831no—
186#484—831no—
187#479—831no—
188#468—831no—
189#465—831no—
190#461—831no—
191#448—831no—
192#448D8311yescharacteristic, complemented
193#440—831no—
194#435—831no—
195#419—831no—
196C2×#103—831no—
197C22×Q8—831no—
198C22×Q8C2×D8311yescharacteristic
199C2×SD16—831no—
200C2×SD16C2×D8311yescharacteristic
201C2×SD16C2×D8311yescharacteristic
202C2×M16—831no—
203C22×C8—831no—
204C4⋊Q8—831no—
205C4⋊Q8—1661no—
206#126—1661no—
207#125—831no—
208#123—1661no—
209C22⋊Q8—831no—
210C22⋊Q8—831no—
211C22⋊Q8—1661no—
212C22⋊Q8—1661no—
213#121—831no—
214#121—1661no—
215#121C2×D8311yescharacteristic
216C4×Q8—831no—
217C4×D4—831no—
218C4×D4—831no—
219C4×D4—1661no—
220C4×D4—1661no—
221#117—831no—
222C2×C4⋊C4—831no—
223#107—831no—
224#107—831no—
225#106—831no—
226C4⋊C8—831no—
227#103—831no—
228#103—831no—
229#103—831no—
230#103—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103C2×D8311yescharacteristic
235#103C2×D8311yescharacteristic
236#102—831no—
237C22⋊C8—831no—
238C22⋊C8C2×D8311yescharacteristic
239#97—831no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—3321no—
244C2×Q8—831no—
245C2×Q8—1661no—
246C2×Q8—1662no—
247C2×Q8C22×D8311yescharacteristic
248C2×Q8C22×D8311yescharacteristic
249C2×D4—21no—
250C2×D4—831no—
251C2×D4—1661no—
252C2×D4C22×D8311yescharacteristic
253C22×C4—21no—
254C22×C4—831no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4C22×D8311yescharacteristic
259SD16—22no—
260SD16—41no—
261SD16—3321no—
262M16—3321no—
263C2×C8—3321no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8C22×D8311yescharacteristic
267C2×C8C22×D8311yescharacteristic
268C4⋊C4—831no—
269C4⋊C4—831no—
270C4⋊C4—831no—
271C4⋊C4—831no—
272C4⋊C4—1661no—
273C4⋊C4—1661no—
274C4⋊C4—1661no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4C22×D8311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—831no—
283C22⋊C4—831no—
284C22⋊C4—1661no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C42—831no—
290C42—1661no—
291C42—1661no—
292C23—21no—
293C23—831no—
294C23D4×D8311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—3322no—
300Q8—1661no—
301Q8—1661no—
302Q8#1305058912yes—
303D4—21no—
304D4—21no—
305D4—81no—
306D4—3322no—
307D4—1662no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3322no—
315C2×C4—831no—
316C2×C4—831no—
317C2×C4—831no—
318C2×C4—6641no—
319C2×C4—6641no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4D4×D8311yescharacteristic
327C2×C4C23×D8311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1661no—
331C8—1661no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—832no—
339C22—1661no—
340C22—1661no—
341C22C2×D4×D8311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—3321no—
347C4—3321no—
348C4—3321no—
349C4—3321no—
350C4—1661no—
351C4—1661no—
352C4—1661no—
353C4C2×#1305058911yescharacteristic
354C4#1458183711yescharacteristic
355C2—41no—
356C2—81no—
357C2—1662no—
358C2#1912833811yescharacteristic
359C2#137×D8311yescharacteristic
360C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#448×C83C2
socleC22×C83C22×D4
prime radical#19128398C2
hypercenter#448D83
hypocenterC83#3320
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊D83C2semidirect
3. #19128068C2semidirect
4. #19128043C2semidirect
5. #19128035C2semidirect
6. #19128006C2semidirect
7. #19127945C2semidirect
8. #19127927C2semidirect
9. #19127904C2semidirect
10. #19127838C2semidirect
11. #19127802C2semidirect
12. #19127775C2semidirect
13. #19127710C2semidirect
14. #103×D83C2semidirect
15. #448×C83C2semidirect
181. C83#3320semidirect
192. #448D83semidirect

Maximal subgroups

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

subgroupindex
#191283982
Q8×C22⋊D832
#191280682
#191280432
#191280352
#191280062
#191279452
#191279272
#191279042
#191278382
#191278022
#191277752
#191277102
#103×D832
#448×C832
#332083