Finite Group Databasebeta

← #24912393

Subgroups — #24912393

Order 10624.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581775C211yesmaximal, characteristic, complemented
2C2×#14581764C211yesmaximal, characteristic, complemented
3C2×#14581762C211yesmaximal, characteristic, complemented
4C2×#14581760C211yesmaximal, characteristic, complemented
5C22×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
6#19127636C212yesmaximal, complemented
7#19127610C212yesmaximal, complemented
8#19127602C212yesmaximal, complemented
9#19127601C212yesmaximal, complemented
10C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
11C2×#123×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×D332C2212yes—
14C22×C22⋊D83C2211yescharacteristic
15C22×C22⋊D83C2212yes—
16C22×C22⋊D83C2211yescharacteristic, complemented
17C23×Dic83C2211yescharacteristic
18#14581775—24no—
19#14581764—24no—
20#14581762—24no—
21#14581760—24no—
22C2×Dic83⋊C4C2212yescomplemented
23C2×Dic83⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic83C2211yescharacteristic
25C2×C22⋊Dic83C2211yescharacteristic, complemented
26C2×(C2×D83)⋊C4—24no—
27C2×(C2×D83)⋊C4C2212yes—
28C2×(C2×D83)⋊C4C2212yes—
29C2×(C2×D83)⋊C4C2212yes—
30C2×(C2×D83)⋊C4C2212yes—
31C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D83)⋊C4C2212yescomplemented
33C2×C4×Dic83C2212yescomplemented
34C83⋊#95C2212yescomplemented
35C22×D4×C83C2211yescharacteristic
36C23×C4×C83C2211yescharacteristic
37#123×C83—24no—
38C2×C4⋊C4×C83C2212yescomplemented
39C2×C22⋊C4×C83C2211yescharacteristic
40C2×C22⋊C4×C83C2212yescomplemented
41C23×D83—22no—
42C23×D83—24no—
43C23×D83—22no—
44C23×D83—22no—
45C23×D83—42no—
46C23×D83C2311yescharacteristic
47C23×D83C2312yes—
48C2×D332—24no—
49C2×D332—24no—
50C2×D332—42no—
51C2×C22⋊D83—22no—
52C2×C22⋊D83—22no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—24no—
55C2×C22⋊D83—24no—
56C2×C22⋊D83—22no—
57C2×C22⋊D83—42no—
58C2×C22⋊D83—42no—
59C22×Dic83—42no—
60C22×Dic83D414yescomplemented
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2312yes—
65Dic83⋊C4—22no—
66Dic83⋊C4—24no—
67C22⋊Dic83—22no—
68C22⋊Dic83D414yescomplemented
69(C2×D83)⋊C4—24no—
70(C2×D83)⋊C4—24no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4—24no—
73(C2×D83)⋊C4—24no—
74(C2×D83)⋊C4—24no—
75C4×Dic83—42no—
76C24×C83C2311yescharacteristic
77C24×C83C2311yescharacteristic
78C2×D4×C83—22no—
79C2×D4×C83—22no—
80C2×D4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—22no—
83C22×C4×C83—22no—
84C22×C4×C83C2312yes—
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2312yes—
87C4⋊C4×C83—24no—
88C22⋊C4×C83—24no—
89C22⋊C4×C83D414yescomplemented
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—22no—
96C23×C83—22no—
97C23×C83C2×D412yes—
98C23×C83C2×D412yes—
99C23×C83C2×D412yes—
100C23×C83C2411yescharacteristic
101C22×D83—28no—
102C22×D83—22no—
103C22×D83—22no—
104C22×D83—22no—
105C22×D83—22no—
106C22×D83—22no—
107C22×D83—22no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—22no—
112C22×D83—22no—
113C22×D83—48no—
114C22×D83—48no—
115C22×D83—44no—
116C22×D83—44no—
117C22×D83#4114yes—
118D4×C83—44no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—21no—
123C2×C4×C83—22no—
124C2×C4×C83—22no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—44no—
128C2×C4×C83C2×D414yes—
129C22⋊D83—48no—
130C22⋊D83—44no—
131C22⋊D83—44no—
132C2×Dic83—22no—
133C2×Dic83—22no—
134C2×Dic83—22no—
135C2×Dic83—22no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—44no—
140C2×Dic83C2×D412yes—
141C2×Dic83C2×D414yes—
142C2×Dic83C2×D412yes—
143D332—48no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—24no—
147C22×C83—24no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83#12714yes—
154C22×C83C22×D411yescharacteristic
155C22×C83C22×D411yescharacteristic
156C22×C83C22×D411yescharacteristic
157C22×C83C2×#4112yes—
158C22×C83H(5,𝔽2)12yes—
159C2×D83—24no—
160C2×D83—24no—
161C2×D83—24no—
162C2×D83—416no—
163C2×D83—42no—
164C2×D83—42no—
165C2×D83—42no—
166C2×D83—42no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—42no—
171C2×D83—42no—
172C4×C83—24no—
173C4×C83—42no—
174C4×C83—41no—
175Dic83—22no—
176Dic83—22no—
177Dic83—24no—
178Dic83—41no—
179C2×C83—24no—
180C2×C83—81no—
181C2×C83#52511yescharacteristic
182C2×C83C2×#12711yescharacteristic
183C2×C83#53411yescharacteristic
184C2×C83#54612yes—
185C2×C83#54612yes—
186D83—44no—
187D83—81no—
188#2539—831nomaximal, Sylow
189C83#253911yescharacteristic, complemented, Sylow
190C2×#123—831no—
191C2×#123D8311yescharacteristic, complemented
192C2×#121—831no—
193C2×#120—831no—
194C2×#120—831no—
195C22×C22⋊C4—831no—
196#399—832no—
197#392—832no—
198#390—832no—
199#390—832no—
200C4×C22⋊C4—831no—
201C25—831no—
202C22×D4—831no—
203C22×D4—832no—
204C22×D4—831no—
205C22×D4—832no—
206C22×D4C2×D8311yescharacteristic
207C23×C4—831no—
208C23×C4C2×D8311yescharacteristic
209#123—24no—
210#123—1664no—
211#121—1664no—
212#120—1664no—
213#120—1664no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—832no—
216C2×C4⋊C4C2×D8312yescomplemented
217C2×C22⋊C4—831no—
218C2×C22⋊C4—831no—
219C2×C22⋊C4—832no—
220C2×C22⋊C4—832no—
221C2×C22⋊C4—832no—
222C2×C22⋊C4—832no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4—832no—
225C2×C22⋊C4—1664no—
226C2×C22⋊C4C2×D8311yescharacteristic
227C2×C22⋊C4C2×D8312yescomplemented
228C2×C42—832no—
229#95—832no—
230C24—3322no—
231C24—832no—
232C24—831no—
233C24—1664no—
234C24—1662no—
235C24—1662no—
236C24—1662no—
237C24C22×D8311yescharacteristic
238C24C22×D8311yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—3322no—
243C2×D4—3322no—
244C2×D4—3322no—
245C2×D4—1664no—
246C2×D4—1664no—
247C2×D4—1662no—
248C2×D4—1664no—
249C2×D4—1662no—
250C2×D4—1662no—
251C2×D4—1662no—
252C2×D4—1664no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—3322no—
257C22×C4—834no—
258C22×C4—831no—
259C22×C4—832no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4C22×D8312yes—
263C22×C4C22×D8311yescharacteristic
264C22×C4C22×D8312yes—
265C4⋊C4—24no—
266C4⋊C4—1662no—
267C4⋊C4—1664no—
268C22⋊C4—24no—
269C22⋊C4—834no—
270C22⋊C4—1664no—
271C22⋊C4—1664no—
272C22⋊C4—1664no—
273C22⋊C4—1662no—
274C22⋊C4—1664no—
275C22⋊C4—1662no—
276C22⋊C4—1664no—
277C22⋊C4C22⋊D8314yescomplemented
278C42—3322no—
279C23—22no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—3324no—
287C23—3328no—
288C23—3328no—
289C23—3324no—
290C23—834no—
291C23—1668no—
292C23—1661no—
293C23—1661no—
294C23—1661no—
295C23—1662no—
296C23—1662no—
297C23—1662no—
298C23—1662no—
299C23—1662no—
300C23—1662no—
301C23—1662no—
302C23—1662no—
303C23D4×D8312yes—
304C23D4×D8312yes—
305C23C2×C22⋊D8312yes—
306C23C23×D8311yescharacteristic
307D4—44no—
308D4—3328no—
309D4—3324no—
310D4—3324no—
311D4—3328no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—3324no—
322C2×C4—834no—
323C2×C4—832no—
324C2×C4—832no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1662no—
329C2×C4—1662no—
330C2×C4—1662no—
331C2×C4—1662no—
332C2×C4C2×C22⋊D8314yes—
333C22—24no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—33216no—
343C22—3322no—
344C22—3322no—
345C22—3322no—
346C22—3322no—
347C22—3322no—
348C22—3322no—
349C22—3321no—
350C22—3321no—
351C22—3321no—
352C22—1664no—
353C22—1664no—
354C22—1664no—
355C22#1458178814yes—
356C22C2×D4×D8311yescharacteristic
357C22C2×D4×D8311yescharacteristic
358C22C22×C22⋊D8311yescharacteristic
359C22#41×D8312yes—
360C22#1458183812yes—
361C4—24no—
362C4—42no—
363C4—41no—
364C4—3321no—
365C4—1664no—
366C4—1662no—
367C4—1662no—
368C2—24no—
369C2—81no—
370C2—3324no—
371C2—6641no—
372C2#1912820911yescharacteristic
373C2C2×#1458178811yescharacteristic
374C2#1912826211yescharacteristic
375C2#1912833912yes—
376C2#1912834312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
hypercenterC2×#123D83
hypocenterC83#2539
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14581775C2semidirect
2. C2×#14581764C2semidirect
3. C2×#14581762C2semidirect
4. C2×#14581760C2semidirect
5. C22×(C2×D83)⋊C4C2semidirect
6. #19127636C2semidirect
7. #19127610C2semidirect
8. #19127602C2semidirect
9. #19127601C2semidirect
10. C22⋊C4×Dic83C2semidirect
11. C2×#123×C83C2semidirect
16. C22×C22⋊D83C22semidirect
22. C2×Dic83⋊C4C22semidirect
23. C2×Dic83⋊C4C22semidirect
25. C2×C22⋊Dic83C22semidirect
31. C2×(C2×D83)⋊C4C22semidirect
32. C2×(C2×D83)⋊C4C22semidirect
33. C2×C4×Dic83C22semidirect
34. C83⋊#95C22semidirect
38. C2×C4⋊C4×C83C22semidirect
40. C2×C22⋊C4×C83C22semidirect
60. C22×Dic83D4semidirect
68. C22⋊Dic83D4semidirect
89. C22⋊C4×C83D4semidirect
189. C83#2539semidirect
191. C2×#123D83semidirect
216. C2×C4⋊C4C2×D83semidirect
227. C2×C22⋊C4C2×D83semidirect
277. C22⋊C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817642
C2×#145817622
C2×#145817602
C22×(C2×D83)⋊C42
#191276362
#191276102
#191276022
#191276012
C22⋊C4×Dic832
C2×#123×C832
#253983