Finite Group Databasebeta

← #35322688

Subgroups — #35322688

Order 21248.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19127491C214yesmaximal
2#24907451C211yesmaximal, characteristic
3C4×#14581709C211yesmaximal, characteristic
4C83×#2486C211yesmaximal, characteristic
5C2×#14581709C2212yes—
6C2×#14581707C2212yes—
7C4×#13050574—21no—
8#19127561—21no—
9#19127491—28no—
10#19127470—21no—
11#19127468—21no—
12C2×#123×C83C2212yes—
13C2×C4×D4×C83C2211yescharacteristic
14C22×C4⋊C4×C83C411yescharacteristic, complemented
15#398×C83—21no—
16#397×C83—21no—
17#389×C83—21no—
18#381×C83C412yescomplemented
19#380×C83—21no—
20#380×C83C411yescharacteristic, complemented
21C2×#13050574—22no—
22C2×C4⋊Dic83—41no—
23C2×C22⋊Dic83—22no—
24#14581709—22no—
25#14581709D414yes—
26#14581707—24no—
27#14581703—41no—
28C4×C83⋊C8—41no—
29C83⋊#95—41no—
30C22×D4×C83C2311yescharacteristic
31C23×C4×C83C2×C412yes—
32C23×C4×C83C2×C411yescharacteristic
33C23×C4×C83D411yescharacteristic
34#123×C83—24no—
35#123×C83—24no—
36C4×D4×C83—41no—
37C4×D4×C83—41no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83—41no—
43C2×C4⋊C4×C83—41no—
44C2×C4⋊C4×C83D411yescharacteristic
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83C2×C412yes—
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50#95×C83—22no—
51#95×C83—21no—
52#95×C83—22no—
53C22×Dic83—42no—
54#13050574—24no—
55#13050574—42no—
56C2×C83⋊C8—42no—
57C4⋊Dic83—82no—
58C22⋊Dic83—42no—
59C22⋊Dic83—42no—
60C24×C83C22×C411yescharacteristic
61C24×C83C2×D411yescharacteristic
62C2×D4×C83—41no—
63C2×D4×C83—41no—
64C2×D4×C83C2×D412yes—
65C2×D4×C83#4112yes—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—22no—
71C22×C4×C83—22no—
72C22×C4×C83—24no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C4⋊C4×C83—41no—
83C4⋊C4×C83—42no—
84C4⋊C4×C83—42no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—44no—
88C22⋊C4×C83—24no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—21no—
94C42×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83C2×C22⋊C411yescharacteristic
104C23×C83C4×D412yes—
105C23×C83#12411yescharacteristic
106C23×C83#12711yescharacteristic
107D4×C83—42no—
108D4×C83—42no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—22no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—81no—
128C2×C4×C83—84no—
129C2×C4×C83#9911yescharacteristic
130C2×C4×C83#9911yescharacteristic
131C2×C4×C83#12112yes—
132C2×Dic83—82no—
133C2×Dic83—82no—
134C2×Dic83—82no—
135C83⋊C8—44no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—22no—
140C22×C83—21no—
141C22×C83—42no—
142C22×C83—41no—
143C22×C83—41no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—81no—
150C22×C83#39011yescharacteristic
151C22×C83C2×#9911yescharacteristic
152C22×C83#41011yescharacteristic
153C4×C83—21no—
154C4×C83—22no—
155C4×C83—41no—
156C4×C83—41no—
157C4×C83—41no—
158C4×C83—82no—
159C4×C83—82no—
160C4×C83#35212yescomplemented
161Dic83—162no—
162#13217—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83—41no—
166C2×C83—42no—
167C2×C83—41no—
168C2×C83#196411yescharacteristic
169C2×C83C2×#35211yescharacteristic
170C2×C83#212311yescharacteristic
171#2486D8311yescharacteristic
172C2×#352—834no—
173#1799—831no—
174C4×#100—831no—
175C83#1321711yescharacteristic, complemented, Sylow
176C2×#123C2×D8312yes—
177C2×C4×D4C2×D8311yescharacteristic
178C22×C4⋊C4Dic8311yescharacteristic, complemented
179C2×#100—832no—
180C2×#99—832no—
181C4×M16—1661no—
182#398—21no—
183#397—21no—
184#389—21no—
185#381Dic8312yescomplemented
186#380—21no—
187#380—1661no—
188#380Dic8311yescharacteristic, complemented
189#352—1668no—
190#343—1661no—
191#342—1661no—
192C22×D4C22×D8311yescharacteristic
193C23×C4C2×Dic8312yes—
194C23×C4C2×Dic8311yescharacteristic
195C23×C4C22⋊D8311yescharacteristic
196C2×M16—1662no—
197#123—24no—
198#123—24no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—3321no—
208C2×C4⋊C4C22⋊D8311yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1662no—
211C2×C22⋊C4C2×Dic8312yes—
212C2×C22⋊C4C2×Dic8311yescharacteristic
213C2×C22⋊C4C22⋊D8311yescharacteristic
214C2×C42C22⋊D8311yescharacteristic
215#100—834no—
216#100—1662no—
217#99—1664no—
218#97—3321no—
219C4×C8—3321no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—3321no—
224C24C22×Dic8311yescharacteristic
225C24C2×C22⋊D8311yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D8312yes—
229C2×D4#1305058912yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—3322no—
242C22×C4C22⋊Dic8311yescharacteristic
243C22×C4C22⋊Dic8311yescharacteristic
244C22×C4C22⋊Dic8311yescharacteristic
245C22×C4C22⋊Dic8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247M16—3322no—
248M16—1664no—
249C2×C8—3322no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—6642no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—3322no—
263C22⋊C4—3322no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊Dic8311yescharacteristic
275C23D4×Dic8312yes—
276C23#1458177911yescharacteristic
277C23#1458178811yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—84no—
300C2×C4—6642no—
301C2×C4—6642no—
302C2×C4—6642no—
303C2×C4#1458170711yescharacteristic
304C2×C4#1458170711yescharacteristic
305C2×C4#1458176312yes—
306C8—3324no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1912760411yescharacteristic
322C22C2×#1458170711yescharacteristic
323C22#1912767511yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—13282no—
332C4#1912749112yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2490830011yescharacteristic
339C2C2×#1912749111yescharacteristic
340C2#2490911711yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#2486C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenter#2486D83
hypocenterC83#13217
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C83C4semidirect
18. #381×C83C4semidirect
20. #380×C83C4semidirect
160. C4×C83#352semidirect
175. C83#13217semidirect
178. C22×C4⋊C4Dic83semidirect
185. #381Dic83semidirect
188. #380Dic83semidirect
332. C4#19127491semidirect

Maximal subgroups

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

subgroupindex
C2×#191274912
#249074512
C4×#145817092
C83×#24862
#1321783