Finite Group Databasebeta

← #24919639

Subgroups — #24919639

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#19128401C211yesmaximal, characteristic
2Q8×D332C211yesmaximal, characteristic, complemented
3#19128070C211yesmaximal, characteristic, complemented
4#19128044C211yesmaximal, characteristic, complemented
5#19128038C211yesmaximal, characteristic, complemented
6#19128008C211yesmaximal, characteristic, complemented
7#19127946C211yesmaximal, characteristic, complemented
8#19127929C211yesmaximal, characteristic, complemented
9#19127906C211yesmaximal, characteristic, complemented
10#19127838C211yesmaximal, characteristic, complemented
11#19127838C211yesmaximal, characteristic, complemented
12#19127774C211yesmaximal, characteristic, complemented
13#19127708C211yesmaximal, characteristic, complemented
14#103×D83C211yesmaximal, characteristic, complemented
15#440×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21C2×C8×D83C2211yescharacteristic
22#14581792—21no—
23#14581792C2211yescharacteristic
24#14581785—21no—
25#14581784C2211yescharacteristic
26#14581775—21no—
27#14581772—21no—
28#14581772—21no—
29#14581772C2211yescharacteristic
30#14581772C2211yescharacteristic
31#14581765—21no—
32#14581763C2211yescharacteristic
33#14581763C2211yescharacteristic
34C4×D332—21no—
35C4×D332C2211yescharacteristic
36C4×C22⋊D83—21no—
37#14581750C2211yescharacteristic
38C4⋊C4×D83C2211yescharacteristic
39#14581725C2211yescharacteristic
40#14581723C2211yescharacteristic
41C4⋊(C83⋊C8)C2211yescharacteristic
42#14581717C2211yescharacteristic
43#14581717C2211yescharacteristic
44#14581717C2211yescharacteristic
45#14581717C2211yescharacteristic
46#14581716C2211yescharacteristic
47#14581716C2211yescharacteristic
48#14581715C2211yescharacteristic
49#14581715C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C2×SD16×C83C2211yescharacteristic
53C4×Q8×C83C2211yescharacteristic
54C4×D4×C83C2211yescharacteristic
55#107×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic
58#97×C83C2211yescharacteristic
59C2×Dic166—21no—
60C2×Dic166—21no—
61C2×Dic166C2311yescharacteristic
62C2×Dic166C2311yescharacteristic
63Q8×D83—22no—
64Q8×D83—22no—
65Q8×D83—41no—
66Q8×D83—41no—
67C2×D332D411yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83C2311yescharacteristic
74C2×C4×D83C2311yescharacteristic
75#13050578—41no—
76#13050578—41no—
77#13050573—41no—
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C8×D83—41no—
81C4⋊Dic83—21no—
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83D411yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87C4⋊Dic83C2311yescharacteristic
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4D411yescharacteristic
97(C2×D83)⋊C4D411yescharacteristic
98C4×Dic83—21no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83C2311yescharacteristic
101C22×C4×C83—21no—
102SD16×C83—22no—
103C2×C8×C83C2311yescharacteristic
104C2×C8×C83C2311yescharacteristic
105C4⋊C4×C83—21no—
106C4⋊C4×C83C2311yescharacteristic
107C4⋊C4×C83C2311yescharacteristic
108C22⋊C4×C83—21no—
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C22×D83C2×D411yescharacteristic
113C22×D83C2×D411yescharacteristic
114Q8×C83—21no—
115Q8×C83#4112yes—
116D4×C83—21no—
117D4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—81no—
122C2×C4×C83C2×D411yescharacteristic
123C2×C4×C83C2×D411yescharacteristic
124C2×C4×C83C2411yescharacteristic
125C22⋊D83—42no—
126C22⋊D83—81no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83C2×D411yescharacteristic
132C2×Dic83C2×D411yescharacteristic
133D332—22no—
134C4×D83—21no—
135C4×D83—21no—
136C4×D83—21no—
137C4×D83—21no—
138C4×D83—42no—
139C4×D83—42no—
140C4×D83—81no—
141Dic166—21no—
142Dic166—21no—
143Dic166—21no—
144Dic166—21no—
145Dic166—42no—
146Dic166—42no—
147C8×C83—21no—
148C8×C83—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83C22×D411yescharacteristic
155C2×D83—21no—
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—21no—
159C2×D83—21no—
160C2×D83#13712yes—
161C4×C83—22no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83C2×#4111yescharacteristic
167C4×C83H(5,𝔽2)11yescharacteristic
168Dic83—21no—
169Dic83—21no—
170Dic83—41no—
171Dic83—41no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—81no—
175C2×C83#54611yescharacteristic
176C2×C83C2×#13711yescharacteristic
177C2×C83#57711yescharacteristic
178D83—22no—
179D83—41no—
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#448—831no—
193#440D8311yescharacteristic, complemented
194#435—831no—
195#419—831no—
196C2×#103—831no—
197C22×Q8—831no—
198C22×Q8—831no—
199C2×SD16—831no—
200C2×SD16—831no—
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—831no—
215#121—1661no—
216C4×Q8C2×D8311yescharacteristic
217C4×D4—831no—
218C4×D4—1661no—
219C4×D4—1661no—
220C4×D4C2×D8311yescharacteristic
221#117—831no—
222C2×C4⋊C4—831no—
223#107—831no—
224#107C2×D8311yescharacteristic
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#103—831no—
235#103C2×D8311yescharacteristic
236#102C2×D8311yescharacteristic
237C22⋊C8—831no—
238C22⋊C8—831no—
239#97C2×D8311yescharacteristic
240C2×Q8—3321no—
241C2×Q8—3321no—
242C2×Q8—831no—
243C2×Q8—831no—
244C2×Q8—1661no—
245C2×Q8—1661no—
246C2×Q8—1662no—
247C2×Q8—1662no—
248C2×Q8C22×D8311yescharacteristic
249C2×D4—831no—
250C2×D4—1661no—
251C2×D4—1661no—
252C2×D4C22×D8311yescharacteristic
253C22×C4—21no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4—1661no—
259SD16—22no—
260SD16—3321no—
261SD16—3321no—
262M16—3321no—
263C2×C8—3321no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8C22×D8311yescharacteristic
267C2×C8C22×D8311yescharacteristic
268C4⋊C4—21no—
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⋊C4C22×D8311yescharacteristic
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—21no—
290C42—1661no—
291C42C22×D8311yescharacteristic
292C23—21no—
293C23—831no—
294C23—831no—
295Q8—21no—
296Q8—3322no—
297Q8—3322no—
298Q8—1661no—
299Q8—1661no—
300Q8—1661no—
301Q8—1661no—
302Q8#1305059012yes—
303D4—21no—
304D4—21no—
305D4—3322no—
306D4—6641no—
307D4—1662no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—3322no—
313C2×C4—3322no—
314C2×C4—831no—
315C2×C4—831no—
316C2×C4—6641no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4D4×D8311yescharacteristic
326C2×C4D4×D8311yescharacteristic
327C2×C4C23×D8311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1661no—
331C8—1661no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—832no—
336C22—1661no—
337C22—1661no—
338C22—1661no—
339C22—1661no—
340C22—1661no—
341C22C2×D4×D8311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3321no—
348C4—3321no—
349C4—3321no—
350C4—3321no—
351C4—1661no—
352C4—1661no—
353C4C2×#1305059011yescharacteristic
354C4#1458183811yescharacteristic
355C2—81no—
356C2—3321no—
357C2—1662no—
358C2#1912834711yescharacteristic
359C2#137×D8311yescharacteristic
360C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#440×C83C2
socleC22×C83C22×D4
prime radical#19128401C2
hypercenter#440D83
hypocenterC83#3320
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. Q8×D332C2semidirect
3. #19128070C2semidirect
4. #19128044C2semidirect
5. #19128038C2semidirect
6. #19128008C2semidirect
7. #19127946C2semidirect
8. #19127929C2semidirect
9. #19127906C2semidirect
10. #19127838C2semidirect
11. #19127838C2semidirect
12. #19127774C2semidirect
13. #19127708C2semidirect
14. #103×D83C2semidirect
15. #440×C83C2semidirect
181. C83#3320semidirect
193. #440D83semidirect

Maximal subgroups

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

subgroupindex
#191284012
Q8×D3322
#191280702
#191280442
#191280382
#191280082
#191279462
#191279292
#191279062
#191278382
#191277742
#191277082
#103×D832
#440×C832
#332083