Finite Group Databasebeta

← #35320646

Subgroups — #35320646

Order 21248.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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#24907391C211yesmaximal, characteristic
2#24907381C211yesmaximal, characteristic
3#24905581C214yesmaximal, complemented
4C83×#2297C211yesmaximal, characteristic
5C2×#14581711—21no—
6C2×#14581709C2212yes—
7C2×#14581707—21no—
8C2×#14581707C2212yes—
9#19127665—21no—
10#19127564—21no—
11C22×#41×C83C2211yescharacteristic
12C2×#117×C83C411yescharacteristic, complemented
13C22×C42×C83C411yescharacteristic, complemented
14#391×C83—21no—
15#390×C83—21no—
16#386×C83C412yescomplemented
17#386×C83C2212yes—
18C4×C22⋊C4×C83—21no—
19C4×C22⋊C4×C83—21no—
20C2×#13050574—22no—
21C2×C22⋊Dic83—22no—
22#14581711—42no—
23#14581709—24no—
24#14581707—24no—
25#14581707—42no—
26C22⋊(C83⋊C8)—42no—
27C83⋊#95—42no—
28C2×#41×C83—24no—
29C2×#41×C83—42no—
30C2×#41×C83—42no—
31C22×Q8×C83D411yescharacteristic
32C22×D4×C83C2×C411yescharacteristic
33C22×D4×C83D411yescharacteristic
34C22×D4×C83C2311yescharacteristic
35C23×C4×C83C2×C411yescharacteristic
36C23×C4×C83C2×C412yes—
37C23×C4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39#117×C83—42no—
40C2×C4⋊C4×C83—21no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83C2×C412yes—
44C2×C42×C83—21no—
45C2×C42×C83—21no—
46C2×C42×C83—21no—
47C2×C42×C83—42no—
48#95×C83—22no—
49#95×C83—22no—
50C22×Dic83—42no—
51#13050574—42no—
52#13050574—42no—
53C2×C83⋊C8—42no—
54C22⋊Dic83—42no—
55C22⋊Dic83—42no—
56C24×C83C22⋊C411yescharacteristic
57C24×C83C22×C411yescharacteristic
58C24×C83C2×D411yescharacteristic
59#41×C83—416no—
60C2×Q8×C83—22no—
61C2×Q8×C83—22no—
62C2×Q8×C83—41no—
63C2×D4×C83—22no—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—41no—
67C2×D4×C83—41no—
68C2×D4×C83—41no—
69C2×D4×C83—41no—
70C2×D4×C83—41no—
71C2×D4×C83#4114yes—
72C22×C4×C83—22no—
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83—22no—
77C22×C4×C83—22no—
78C22×C4×C83—22no—
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83—41no—
82C22×C4×C83—41no—
83C22×C4×C83—44no—
84C22×C4×C83—42no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C4⋊C4×C83—42no—
90C22⋊C4×C83—44no—
91C22⋊C4×C83—44no—
92C22⋊C4×C83—42no—
93C42×C83—42no—
94C42×C83—42no—
95C42×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—42no—
104C23×C83—42no—
105C23×C83—42no—
106C23×C83C2×C22⋊C411yescharacteristic
107C23×C83#11712yes—
108C23×C83#12312yes—
109Q8×C83—44no—
110D4×C83—44no—
111D4×C83—44no—
112D4×C83—44no—
113C2×C4×C83—28no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—24no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—44no—
125C2×C4×C83—44no—
126C2×C4×C83—42no—
127C2×C4×C83—44no—
128C2×C4×C83—42no—
129C2×C4×C83—44no—
130C2×C4×C83—42no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83#9912yes—
134C2×C4×C83#9912yes—
135C2×C4×C83#12312yes—
136C2×Dic83—82no—
137C2×Dic83—84no—
138C83⋊C8—82no—
139C22×C83—22no—
140C22×C83—22no—
141C22×C83—22no—
142C22×C83—42no—
143C22×C83—42no—
144C22×C83—42no—
145C22×C83—42no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83—41no—
149C22×C83—42no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83#38111yescharacteristic
153C22×C83C2×#9911yescharacteristic
154C22×C83C2×#9911yescharacteristic
155C4×C83—22no—
156C4×C83—22no—
157C4×C83—22no—
158C4×C83—41no—
159C4×C83—44no—
160C4×C83—82no—
161Dic83—162no—
162#12853—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83—42no—
166C2×C83—42no—
167C2×C83—81no—
168C2×C83#181911yescharacteristic
169C2×C83#212012yes—
170#2297D8311yescharacteristic
171#1785—831no—
172#1782—831no—
173#1390—834no—
174C83#1285311yescharacteristic, complemented, Sylow
175C22×#41C2×D8311yescharacteristic
176C2×#117Dic8311yescharacteristic, complemented
177C22×C42Dic8311yescharacteristic, complemented
178C2×#101—1661no—
179C2×#100—832no—
180C2×#99—832no—
181C2×#99—1661no—
182#407—1661no—
183#391—21no—
184#390—21no—
185#386Dic8312yescomplemented
186#386C2×D8312yes—
187#381—1661no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8C22⋊D8311yescharacteristic
194C22×D4C2×Dic8311yescharacteristic
195C22×D4C22⋊D8311yescharacteristic
196C22×D4C22×D8311yescharacteristic
197C23×C4C2×Dic8311yescharacteristic
198C23×C4C2×Dic8312yes—
199C23×C4C22⋊D8311yescharacteristic
200C23×C4C22⋊D8311yescharacteristic
201C2×M16—1662no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1662no—
207C2×C22⋊C4C2×Dic8312yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—42no—
212#101—3322no—
213#100—1664no—
214#99—3322no—
215#99—1664no—
216C22⋊C8—3322no—
217#95—22no—
218#95—22no—
219#95—3322no—
220C24C22⋊Dic8311yescharacteristic
221C24C22×Dic8311yescharacteristic
222C24C2×C22⋊D8311yescharacteristic
223#41—416no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4#1305059014yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—3322no—
250C22×C4C22⋊Dic8311yescharacteristic
251C22×C4C22⋊Dic8311yescharacteristic
252C22×C4C22⋊Dic8311yescharacteristic
253C22×C4C2×C22⋊D8311yescharacteristic
254M16—3322no—
255M16—3322no—
256C2×C8—3322no—
257C4⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—3322no—
262C22⋊C4—3322no—
263C42—42no—
264C42—42no—
265C42—42no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C2×C22⋊Dic8311yescharacteristic
277C23#1458175112yes—
278C23#1458177512yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—6642no—
304C2×C4—6644no—
305C2×C4#1458170712yes—
306C2×C4#1458170712yes—
307C2×C4#1458177512yes—
308C8—6642no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22#1912756411yescharacteristic
323C22C2×#1458170711yescharacteristic
324C22C2×#1458170711yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—44no—
330C4—82no—
331C4—13282no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2#2490753711yescharacteristic
338C2#2490910412yes—
subgroupG/H
centerC22C2×#14581707
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#2297C2
socleC22×C83C2×#99
prime radicalC22×#41×C83C22
cosocleC22×D4×C83C23
hypercenter#2297D83
hypocenterC83#12853
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #24905581C2semidirect
12. C2×#117×C83C4semidirect
13. C22×C42×C83C4semidirect
16. #386×C83C4semidirect
174. C83#12853semidirect
176. C2×#117Dic83semidirect
177. C22×C42Dic83semidirect
185. #386Dic83semidirect

Maximal subgroups

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

subgroupindex
#249073912
#249073812
#249055812
C83×#22972
#1285383