Finite Group Databasebeta

← #35320652

Subgroups — #35320652

Order 21248.

628 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#24910055C211yesmaximal, characteristic
2#24907391C211yesmaximal, characteristic
3#24905580C214yesmaximal, complemented
4C83×#1782C211yesmaximal, characteristic
5C2×#14581751C411yescharacteristic, complemented
6C22×C4⋊Dic83C411yescharacteristic, complemented
7C2×#14581711—21no—
8C2×#14581709C2212yes—
9#19127665—21no—
10#19127608—21no—
11#19127604—21no—
12#19127581C412yescomplemented
13#19127581C2212yes—
14#19127561—21no—
15#19127561—21no—
16C22×#41×C83C2211yescharacteristic
17C2×#99×C83—21no—
18C2×#99×C83C2212yes—
19#381×C83—21no—
20C23×Dic83C2×C412yes—
21C2×#13050574—22no—
22#14581751—42no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—42no—
27C2×C22⋊Dic83—22no—
28C2×C22⋊Dic83—22no—
29C2×C22⋊Dic83C2×C412yes—
30C2×C4×Dic83—21no—
31#14581711—42no—
32#14581709—24no—
33C22⋊(C83⋊C8)—42no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C2×#41×C83—24no—
37C2×#41×C83—42no—
38C2×#41×C83—42no—
39C22×Q8×C83D411yescharacteristic
40C22×D4×C83C2×C411yescharacteristic
41C22×D4×C83D411yescharacteristic
42C22×D4×C83C2311yescharacteristic
43C23×C4×C83C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C23×C4×C83D411yescharacteristic
46C2×C22⋊C4×C83—22no—
47#99×C83—24no—
48#99×C83—42no—
49#95×C83—42no—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—42no—
53C22×Dic83—82no—
54#13050574—42no—
55#13050574—42no—
56C2×C83⋊C8—42no—
57C4⋊Dic83—42no—
58C4⋊Dic83—42no—
59C4⋊Dic83—42no—
60C22⋊Dic83—42no—
61C22⋊Dic83—82no—
62C22⋊Dic83—82no—
63C4×Dic83—42no—
64C24×C83C22⋊C411yescharacteristic
65C24×C83C22×C411yescharacteristic
66C24×C83C2×D411yescharacteristic
67#41×C83—416no—
68C2×Q8×C83—22no—
69C2×Q8×C83—22no—
70C2×Q8×C83—41no—
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C2×D4×C83—22no—
74C2×D4×C83—41no—
75C2×D4×C83—41no—
76C2×D4×C83—41no—
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83#4114yes—
80C22×C4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—22no—
83C22×C4×C83—22no—
84C22×C4×C83—22no—
85C22×C4×C83—22no—
86C22×C4×C83—22no—
87C22×C4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83—42no—
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83C2×C22⋊C411yescharacteristic
110C23×C83#11712yes—
111C23×C83#12312yes—
112Q8×C83—44no—
113D4×C83—44no—
114D4×C83—44no—
115D4×C83—44no—
116C2×C4×C83—28no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—44no—
127C2×C4×C83—44no—
128C2×C4×C83—82no—
129C2×C4×C83—82no—
130C2×C4×C83—82no—
131C2×C4×C83#9912yes—
132C2×C4×C83#9912yes—
133C2×C4×C83#12312yes—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138C2×Dic83—42no—
139C2×Dic83—42no—
140C2×Dic83—84no—
141C83⋊C8—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—81no—
158C22×C83—81no—
159C22×C83#38111yescharacteristic
160C22×C83C2×#9911yescharacteristic
161C22×C83C2×#9911yescharacteristic
162C4×C83—22no—
163C4×C83—22no—
164C4×C83—22no—
165C4×C83—41no—
166C4×C83—162no—
167Dic83—82no—
168Dic83—82no—
169#12855—831nomaximal, Sylow
170C2×C83—21no—
171C2×C83—21no—
172C2×C83—42no—
173C2×C83—42no—
174C2×C83—81no—
175C2×C83#181911yescharacteristic
176C2×C83#212211yescharacteristic
177C2×C83#212311yescharacteristic
178#2311—831no—
179#1785—831no—
180#1782D8311yescharacteristic
181#1390—834no—
182C83#1285511yescharacteristic, complemented, Sylow
183C22×#41C2×D8311yescharacteristic
184C2×#117—831no—
185C22×C4⋊C4—831no—
186C2×#101—1661no—
187C2×#100—832no—
188C2×#99—21no—
189C2×#99C2×D8312yes—
190#407—1661no—
191#391—1661no—
192#390—1661no—
193#386—832no—
194#386—832no—
195#381—21no—
196#380—1661no—
197#380—1661no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8D33211yescharacteristic
202C22×D4C4×D8311yescharacteristic
203C22×D4C22⋊D8311yescharacteristic
204C22×D4C22×D8311yescharacteristic
205C23×C4—832no—
206C23×C4C4×D8311yescharacteristic
207C23×C4D33211yescharacteristic
208C23×C4C22⋊D8311yescharacteristic
209C2×M16—1662no—
210#117—3322no—
211C2×C4⋊C4—3322no—
212C2×C4⋊C4—1661no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—832no—
217C2×C22⋊C4—1662no—
218C2×C22⋊C4—1662no—
219C2×C42—1661no—
220#101—3322no—
221#100—1664no—
222#99—24no—
223#99—42no—
224C22⋊C8—3322no—
225#95—42no—
226#95—1662no—
227#95—1662no—
228C24(C2×D83)⋊C411yescharacteristic
229C24C2×C4×D8311yescharacteristic
230C24C2×C22⋊D8311yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1305058914yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—3322no—
255C22×C4—6642no—
256C22×C4—1662no—
257C22×C4—1662no—
258C22×C4(C2×D83)⋊C411yescharacteristic
259C22×C4(C2×D83)⋊C411yescharacteristic
260C22×C4(C2×D83)⋊C411yescharacteristic
261C22×C4C2×D33211yescharacteristic
262M16—3322no—
263M16—3322no—
264C2×C8—3322no—
265C4⋊C4—3322no—
266C4⋊C4—3322no—
267C4⋊C4—3322no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—3322no—
271C22⋊C4—6642no—
272C22⋊C4—6642no—
273C42—3322no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×(C2×D83)⋊C411yescharacteristic
288C23#1458174812yes—
289C23#1458177312yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—3322no—
310C2×C4—3322no—
311C2×C4—3322no—
312C2×C4—3322no—
313C2×C4—3322no—
314C2×C4—3322no—
315C2×C4—6644no—
316C2×C4#1458170612yes—
317C2×C4#1458170612yes—
318C2×C4#1458177612yes—
319C8—6642no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1912756211yescharacteristic
338C22C2×#1458170611yescharacteristic
339C22C2×#1458170611yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—162no—
345C4—6642no—
346C4—6642no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#2490753511yescharacteristic
353C2#2490911211yescharacteristic
354C2#2490911611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#1782C2
socleC22×C83C2×#99
prime radicalC22×#41×C83C22
cosocleC22×D4×C83C23
hypercenter#1782D83
hypocenterC83#12855
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #24905580C2semidirect
5. C2×#14581751C4semidirect
6. C22×C4⋊Dic83C4semidirect
12. #19127581C4semidirect
182. C83#12855semidirect

Maximal subgroups

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

subgroupindex
#249100552
#249073912
#249055802
C83×#17822
#1285583