Finite Group Databasebeta

← #35313657

Subgroups — #35313657

Order 21248.

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

normal subgroups only

This group has 488 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
1C4×#14581773C211yesmaximal, characteristic
2#24905784C211yesmaximal, characteristic
3#24905412C212yesmaximal, complemented
4#24905392C212yesmaximal, complemented
5C83×#1459C211yesmaximal, characteristic
6C2×#14581773C411yescharacteristic
7C22×C4×Dic83C411yescharacteristic
8#19127754—21no—
9#19127589—21no—
10#19127582—21no—
11#19127581C411yescharacteristic
12#19127573C2211yescharacteristic
13#19127562C2211yescharacteristic
14C4×Dic83⋊C4—21no—
15C4×C22⋊Dic83—21no—
16C4×C22⋊Dic83C411yescharacteristic
17#19127425—21no—
18#19127425C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#324×C83—21no—
22#323×C83C2212yes—
23C23×Dic83C2×C411yescharacteristic
24C23×Dic83C2×C411yescharacteristic
25#14581773—22no—
26#14581773C814yescomplemented
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83C2×C411yescharacteristic
32C2×C22⋊Dic83C2×C411yescharacteristic
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83—21no—
36C2×C4×Dic83—41no—
37C2×C4×Dic83—41no—
38C4⋊(C83⋊C8)—41no—
39C22⋊(C83⋊C8)—22no—
40C4×C83⋊C8—41no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C22×D4×C83C2×C411yescharacteristic
47C23×C4×C83C2×C411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C22×C8×C83—41no—
50C4×D4×C83—41no—
51C4×D4×C83—41no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C22⋊C4×C83C2311yescharacteristic
55C2×C42×C83D411yescharacteristic
56C22⋊C8×C83—22no—
57C22⋊C8×C83—41no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—42no—
64C22×Dic83—41no—
65C22×Dic83—41no—
66C22×Dic83—41no—
67C22×Dic83C2×C812yescomplemented
68C2×C83⋊C8—42no—
69Dic83⋊C4—24no—
70Dic83⋊C4—42no—
71C22⋊Dic83—24no—
72C22⋊Dic83—21no—
73C22⋊Dic83—42no—
74C22⋊Dic83—41no—
75C22⋊Dic83C2×C812yescomplemented
76C4×Dic83—42no—
77C4×Dic83—42no—
78C4×Dic83—41no—
79C4×Dic83—41no—
80C24×C83C22⋊C411yescharacteristic
81C24×C83C22×C411yescharacteristic
82C2×D4×C83—41no—
83C2×D4×C83—41no—
84C2×D4×C83C2×C812yes—
85C2×D4×C83M1612yes—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—42no—
91C22×C4×C83—42no—
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C2×C8×C83—42no—
98C2×C8×C83—82no—
99C4⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C42×C83—21no—
103C42×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83C22⋊C812yes—
113C23×C83C2×C22⋊C411yescharacteristic
114C23×C83C22×C811yescharacteristic
115C23×C83C2×M1611yescharacteristic
116D4×C83—42no—
117D4×C83—42no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—22no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—81no—
131C2×C4×C83C22⋊C812yes—
132C2×C4×C83#9911yescharacteristic
133C2×C4×C83#9911yescharacteristic
134C2×Dic83—22no—
135C2×Dic83—22no—
136C2×Dic83—22no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—42no—
141C2×Dic83—42no—
142C2×Dic83—41no—
143C2×Dic83—41no—
144C2×Dic83—41no—
145C2×Dic83—42no—
146C2×Dic83—42no—
147C2×Dic83—41no—
148C2×Dic83—41no—
149C2×Dic83—41no—
150C8×C83—82no—
151C83⋊C8—82no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—22no—
157C22×C83—42no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—42no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—81no—
166C22×C83C2×C22⋊C811yescharacteristic
167C22×C83C2×#9911yescharacteristic
168C22×C83#41311yescharacteristic
169C4×C83—21no—
170C4×C83—22no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83#35212yes—
175Dic83—42no—
176Dic83—42no—
177Dic83—42no—
178Dic83—81no—
179#11472—831nomaximal, Sylow
180C2×C83—21no—
181C2×C83—21no—
182C2×C83—41no—
183C2×C83—41no—
184C2×C83—42no—
185C2×C83#145911yescharacteristic
186C2×C83C2×#35211yescharacteristic
187C2×C83#212011yescharacteristic
188C4×#123—831no—
189#1460—831no—
190#1459D8311yescharacteristic
191#1326—832no—
192#1317—832no—
193C83#1147211yescharacteristic, complemented, Sylow
194C2×#123—831no—
195C2×C4×D4C2×D8311yescharacteristic
196C22×C42—831no—
197#431—1661no—
198C2×C22⋊C8—21no—
199#388—1661no—
200#387—1661no—
201#386—831no—
202#385—831no—
203#381—831no—
204C4×C4⋊C4—1661no—
205C4×C22⋊C4—831no—
206C4×C22⋊C4—1661no—
207#324—21no—
208#323—832no—
209#323—1661no—
210#323C2×D8312yes—
211C22×D4C4×D8311yescharacteristic
212C23×C4—831no—
213C23×C4—831no—
214C23×C4C4×D8311yescharacteristic
215C23×C4C22⋊D8311yescharacteristic
216C22×C8—41no—
217#123—834no—
218#123—1662no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—1661no—
222C2×C4⋊C4—1661no—
223C2×C4⋊C4C22⋊D8311yescharacteristic
224C2×C22⋊C4—831no—
225C2×C22⋊C4—831no—
226C2×C22⋊C4—1661no—
227C2×C22⋊C4—1661no—
228C2×C22⋊C4D33211yescharacteristic
229C2×C22⋊C4C22×D8311yescharacteristic
230C2×C42—3321no—
231C2×C42—3321no—
232C2×C42—1661no—
233C2×C42—1661no—
234C2×C42—1661no—
235C2×C42D33211yescharacteristic
236C4⋊C8—3321no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—1662no—
240C4×C8—3321no—
241#95—1661no—
242#95—1661no—
243#95—1661no—
244#95—1661no—
245#95—1661no—
246C24(C2×D83)⋊C411yescharacteristic
247C24C2×C4×D8311yescharacteristic
248C2×D4—41no—
249C2×D4—41no—
250C2×D4C8×D8312yes—
251C2×D4#1305057312yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3322no—
262C22×C4—832no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4—1661no—
266C22×C4—1661no—
267C22×C4—1661no—
268C22×C4(C2×D83)⋊C411yescharacteristic
269C22×C4(C2×D83)⋊C411yescharacteristic
270C22×C4(C2×D83)⋊C411yescharacteristic
271C22×C4C2×C22⋊D8311yescharacteristic
272C22×C4C2×D33211yescharacteristic
273C2×C8—42no—
274C2×C8—82no—
275C2×C8—3322no—
276C4⋊C4—41no—
277C4⋊C4—3322no—
278C4⋊C4—1664no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—3322no—
282C22⋊C4—3321no—
283C22⋊C4—832no—
284C22⋊C4—1661no—
285C22⋊C4—1664no—
286C42—21no—
287C42—21no—
288C42—3322no—
289C42—3322no—
290C42—3321no—
291C42—3321no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23(C2×D83)⋊C812yes—
301C23C2×(C2×D83)⋊C411yescharacteristic
302C23C2×C8×D8311yescharacteristic
303C23C2×#1305057311yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—3321no—
322C2×C4—3322no—
323C2×C4—3322no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4—3322no—
328C2×C4—3322no—
329C2×C4—3321no—
330C2×C4—3321no—
331C2×C4—3321no—
332C2×C4—1662no—
333C2×C4—1662no—
334C2×C4—1662no—
335C2×C4(C2×D83)⋊C812yes—
336C2×C4#1458170611yescharacteristic
337C2×C4#1458170611yescharacteristic
338C8—82no—
339C8—6642no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22C2×(C2×D83)⋊C811yescharacteristic
355C22C2×#1458170611yescharacteristic
356C22#1912768311yescharacteristic
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—3322no—
363C4—3322no—
364C4—3322no—
365C4—6641no—
366C4#1912749012yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—42no—
372C2#2490577911yescharacteristic
373C2C2×#1912749011yescharacteristic
374C2#2490910511yescharacteristic
subgroupG/H
centerC2×C4#14581706
commutatorC23×C83C22×C8
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1459C2
socleC22×C83C2×#99
prime radicalC22×D4×C83C2×C4
cosocleC2×C22⋊C4×C83C23
hypercenter#1459D83
hypocenterC83#11472
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #24905412C2semidirect
4. #24905392C2semidirect
26. #14581773C8semidirect
67. C22×Dic83C2×C8semidirect
75. C22⋊Dic83C2×C8semidirect
193. C83#11472semidirect

Maximal subgroups

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

subgroupindex
C4×#145817732
#249057842
#249054122
#249053922
C83×#14592
#1147283