Finite Group Databasebeta

← #35314798

Subgroups — #35314798

Order 21248.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14581709C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905493C212yesmaximal, complemented
4#24905469C212yesmaximal, complemented
5C83×#2638C211yesmaximal, characteristic
6C2×#14581709—21no—
7C2×C22⋊(C83⋊C8)—21no—
8C4×#13050574—21no—
9#19127470C2212yes—
10#19127427C2212yes—
11#19127425—21no—
12C2×C22⋊Q8×C83C411yescharacteristic
13C2×C4×D4×C83C2211yescharacteristic
14#398×C83C2211yescharacteristic
15#397×C83—21no—
16#397×C83C411yescharacteristic
17#395×C83C2211yescharacteristic
18#394×C83—21no—
19#393×C83C411yescharacteristic
20#387×C83—21no—
21#385×C83—21no—
22#381×C83C411yescharacteristic
23C2×#13050574—22no—
24C22×C83⋊C8—41no—
25#14581709—24no—
26C22⋊(C83⋊C8)—22no—
27C22⋊(C83⋊C8)—41no—
28#14581703—41no—
29C4×C83⋊C8—41no—
30C22×Q8×C83C2×C411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83C2×C411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C22⋊Q8×C83—41no—
35C22⋊Q8×C83—41no—
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×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2311yescharacteristic
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57#13050574—24no—
58#13050574—42no—
59C2×C83⋊C8—42no—
60C2×C83⋊C8—42no—
61C2×C83⋊C8—82no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C22⋊C411yescharacteristic
64C2×Q8×C83—41no—
65C2×Q8×C83—81no—
66C2×D4×C83—22no—
67C2×D4×C83—22no—
68C2×D4×C83—41no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—81no—
78C22×C4×C83—81no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—42no—
89C4⋊C4×C83—81no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—81no—
95C22⋊C4×C83—81no—
96C42×C83—21no—
97C42×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#10412yes—
108C23×C83C2×C22⋊C411yescharacteristic
109Q8×C83—82no—
110D4×C83—44no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—41no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
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—81no—
134C2×C4×C83—81no—
135C2×C4×C83—81no—
136C2×C4×C83#9911yescharacteristic
137C2×C4×C83#9911yescharacteristic
138C2×C4×C83#13512yes—
139C83⋊C8—44no—
140C83⋊C8—82no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83C2×#9911yescharacteristic
154C22×C83#41611yescharacteristic
155C22×C83C2×#10411yescharacteristic
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—81no—
163C4×C83—81no—
164C4×C83—81no—
165C4×C83—81no—
166#11739—831nomaximal, Sylow
167C2×C83—21no—
168C2×C83—21no—
169C2×C83—42no—
170C2×C83—81no—
171C2×C83#150111yescharacteristic
172C2×C83#212111yescharacteristic
173C2×C83#213511yescharacteristic
174#2638D8311yescharacteristic
175C4×#100—831no—
176#1459—831no—
177#1354—832no—
178#1346—832no—
179C83#1173911yescharacteristic, complemented, Sylow
180C2×C22⋊Q8Dic8311yescharacteristic
181C2×C4×D4C2×D8311yescharacteristic
182C2×#100—1661no—
183C2×C22⋊C8—1661no—
184C4×M16—1661no—
185#398C2×D8311yescharacteristic
186#397—21no—
187#397Dic8311yescharacteristic
188#395C2×D8311yescharacteristic
189#394—21no—
190#393Dic8311yescharacteristic
191#387—21no—
192#385—21no—
193#381Dic8311yescharacteristic
194#343—832no—
195#324—832no—
196#323—1661no—
197C22×Q8C2×Dic8311yescharacteristic
198C22×D4C22⋊D8311yescharacteristic
199C23×C4C2×Dic8311yescharacteristic
200C23×C4C22⋊D8311yescharacteristic
201C2×M16—1662no—
202C22×C8—3321no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
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⋊C4C2×Dic8311yescharacteristic
217C2×C22⋊C4C22⋊D8311yescharacteristic
218C2×C42C22⋊D8311yescharacteristic
219#100—1664no—
220C22⋊C8—3321no—
221C22⋊C8—1662no—
222#97—3321no—
223C4×C8—3321no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic8311yescharacteristic
230#95C2×Dic8311yescharacteristic
231C24C22⋊Dic8311yescharacteristic
232C24C22⋊Dic8311yescharacteristic
233C2×Q8—41no—
234C2×Q8—81no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4C22⋊Dic8311yescharacteristic
249C22×C4C22⋊Dic8311yescharacteristic
250C22×C4C22×Dic8311yescharacteristic
251C22×C4C2×C22⋊D8311yescharacteristic
252C22×C4C2×C22⋊D8311yescharacteristic
253M16—3322no—
254M16—1664no—
255C2×C8—3322no—
256C2×C8—3322no—
257C2×C8—6642no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1458171912yes—
282C23C2×C22⋊Dic8311yescharacteristic
283Q8—82no—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
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—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#1458170711yescharacteristic
311C2×C4#1458170711yescharacteristic
312C2×C4#1458181212yes—
313C8—3324no—
314C8—6642no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#1458170711yescharacteristic
328C22#1912769711yescharacteristic
329C22C2×#1458171911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2490595411yescharacteristic
345C2#2490910911yescharacteristic
346C2#2490915811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2638C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C4⋊C4×C83C23
hypercenter#2638D83
hypocenterC83#11739
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #24905493C2semidirect
4. #24905469C2semidirect
179. C83#11739semidirect

Maximal subgroups

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

subgroupindex
C4×#145817092
#249057812
#249054932
#249054692
C83×#26382
#1173983