Finite Group Databasebeta

← #35299489

Subgroups — #35299489

Order 21248.

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

normal subgroups only

This group has 516 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
1#24916548C211yesmaximal, characteristic, complemented
2C4×#14581711C211yesmaximal, characteristic, complemented
3#24905906C212yesmaximal
4#24905356C212yesmaximal, complemented
5C83×#2398C211yesmaximal, characteristic
6#19127832—24no—
7#19127758—21no—
8#19127758C2212yes—
9#19127754—21no—
10C2×#14581711—21no—
11C4×#13050574C2212yes—
12#19127470—21no—
13D4⋊(C83⋊C8)—24no—
14C2×C4×Q8×C83C2211yescharacteristic
15C2×C4×D4×C83C2211yescharacteristic
16#400×C83C411yescharacteristic
17#400×C83C411yescharacteristic
18#397×C83—21no—
19#394×C83C411yescharacteristic
20#394×C83C411yescharacteristic
21#393×C83—21no—
22C4×C22⋊C4×C83—21no—
23C4×C22⋊C4×C83—21no—
24#376×C83C2211yescharacteristic
25C2×#13050574—22no—
26C4⋊(C83⋊C8)—22no—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)D414yes—
29#14581711—24no—
30C22⋊(C83⋊C8)—22no—
31#14581703—22no—
32#14581703—22no—
33#14581703—22no—
34C4×C83⋊C8—22no—
35C4×C83⋊C8—22no—
36C22×Q8×C83D411yescharacteristic
37C22×D4×C83C2×C411yescharacteristic
38C23×C4×C83—21no—
39C4×Q8×C83—22no—
40C4×Q8×C83—22no—
41C4×D4×C83—22no—
42C4×D4×C83—22no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83—21no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C42×C83—21no—
53C2×C42×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C2×C42×C83C2311yescharacteristic
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—21no—
61#95×C83C2×C411yescharacteristic
62#95×C83C2×C411yescharacteristic
63#13050574—24no—
64#13050574—42no—
65C2×C83⋊C8—24no—
66C2×C83⋊C8—42no—
67C24×C83—21no—
68C2×Q8×C83—22no—
69C2×Q8×C83—22no—
70C2×Q8×C83—41no—
71C2×D4×C83—22no—
72C2×D4×C83—42no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87C4⋊C4×C83—22no—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—22no—
90C4⋊C4×C83—22no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—42no—
101C42×C83C2×D412yes—
102C42×C83#4112yes—
103C23×C83—22no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83C2×C22⋊C411yescharacteristic
111Q8×C83—44no—
112D4×C83—24no—
113D4×C83—42no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—82no—
137C2×C4×C83—82no—
138C2×C4×C83#10412yes—
139C2×C4×C83C4×D412yes—
140C2×C4×C83#12011yescharacteristic
141C2×C4×C83#12112yes—
142C2×C4×C83#12311yescharacteristic
143C83⋊C8—44no—
144C83⋊C8—44no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83#38611yescharacteristic
155C22×C83C2×#10411yescharacteristic
156C22×C83#42111yescharacteristic
157C4×C83—22no—
158C4×C83—22no—
159C4×C83—21no—
160C4×C83—22no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—81no—
165C4×C83—81no—
166C4×C83—81no—
167C4×C83#45412yes—
168#9032—831nomaximal, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—82no—
172C2×C83#187111yescharacteristic
173C2×C83#188811yescharacteristic
174C2×C83#189111yescharacteristic
175#2983—831no—
176#2398D8311yescharacteristic
177C4×#101—831no—
178#1490—832no—
179#1300—832no—
180C83#903211yescharacteristic, complemented, Sylow
181C2×C4×Q8C2×D8311yescharacteristic
182C2×C4×D4C2×D8311yescharacteristic
183#446—1664no—
184#432—832no—
185#432—1661no—
186#431—1661no—
187C2×#101—1661no—
188C4×M16—832no—
189#400Dic8311yescharacteristic
190#400Dic8311yescharacteristic
191#397—21no—
192#394Dic8311yescharacteristic
193#394Dic8311yescharacteristic
194#393—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197#376C2×D8311yescharacteristic
198#343—1661no—
199D4⋊C8—1664no—
200C22×Q8C22⋊D8311yescharacteristic
201C22×D4C2×Dic8311yescharacteristic
202C23×C4—21no—
203C2×M16—1662no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×D4—22no—
207C4×D4—22no—
208C2×C4⋊C4C2×Dic8311yescharacteristic
209C2×C4⋊C4C2×Dic8311yescharacteristic
210C2×C4⋊C4C2×Dic8311yescharacteristic
211C2×C4⋊C4C22⋊D8311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42C22⋊D8311yescharacteristic
219C2×C42C22⋊D8311yescharacteristic
220C2×C42C22×D8311yescharacteristic
221C4⋊C8—834no—
222C4⋊C8—1662no—
223C4⋊C8—1662no—
224#101—1664no—
225C22⋊C8—1662no—
226#97—1662no—
227#97—1662no—
228#97—1662no—
229C4×C8—1662no—
230C4×C8—1662no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C2×Dic8311yescharacteristic
237#95C2×Dic8311yescharacteristic
238C24—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C22⋊Dic8311yescharacteristic
252C22×C4C22⋊Dic8311yescharacteristic
253C22×C4C22⋊Dic8311yescharacteristic
254C22×C4C22⋊Dic8311yescharacteristic
255C22×C4C22×Dic8311yescharacteristic
256C22×C4C2×C22⋊D8311yescharacteristic
257C22×C4C2×C22⋊D8311yescharacteristic
258M16—3322no—
259M16—1664no—
260C2×C8—3322no—
261C2×C8—1664no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42D4×D8312yes—
277C42#1305058912yes—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊Dic8311yescharacteristic
286Q8—44no—
287D4—24no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#1458171912yes—
314C2×C4D4×Dic8312yes—
315C2×C4#1458176011yescharacteristic
316C2×C4#1458176412yes—
317C2×C4#1458177311yescharacteristic
318C8—3324no—
319C8—3324no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#1912758111yescharacteristic
330C22C2×#1458171911yescharacteristic
331C22#1912771811yescharacteristic
332C4—21no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#1912787312yes—
343C2—21no—
344C2—21no—
345C2—82no—
346C2#2490776611yescharacteristic
347C2#2490786711yescharacteristic
348C2#2490788211yescharacteristic
subgroupG/H
centerC2×C4#14581760
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2398C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#2398D83
hypocenterC83#9032
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #24916548C2semidirect
2. C4×#14581711C2semidirect
4. #24905356C2semidirect
180. C83#9032semidirect

Maximal subgroups

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

subgroupindex
#249165482
C4×#145817112
#249059062
#249053562
C83×#23982
#903283