Finite Group Databasebeta

← #35323073

Subgroups — #35323073

Order 21248.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#24912416C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905587C214yesmaximal
4C83×#1799C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581771—21no—
7C2×#14581771—21no—
8C2×#14581707C2212yes—
9#19127637—21no—
10#19127637—21no—
11#19127604C2212yes—
12#19127570C412yescomplemented
13C4×C4⋊Dic83C411yescharacteristic, complemented
14C4×C22⋊Dic83—21no—
15#19127468—21no—
16C2×C4×D4×C83C2211yescharacteristic
17C2×#99×C83C2212yes—
18#380×C83—21no—
19#342×C83—21no—
20C22×Dic166—21no—
21#14581792—41no—
22#14581792—41no—
23#14581771—44no—
24#14581771—44no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2×C412yes—
28C2×Dic83⋊C4—22no—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83—22no—
33C2×C4×Dic83—41no—
34C2×C4×Dic83C2×C412yes—
35#14581707—22no—
36#14581707Q814yescomplemented
37C83⋊#95—21no—
38C83⋊#95—41no—
39C22×D4×C83C2311yescharacteristic
40C23×C4×C83D411yescharacteristic
41C23×C4×C83D411yescharacteristic
42C4×D4×C83—41no—
43C4×D4×C83—41no—
44C2×C4⋊C4×C83—41no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83D411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83C2×C411yescharacteristic
50#99×C83—24no—
51#95×C83—41no—
52C2×Dic166—42no—
53C2×Dic166—42no—
54C2×Dic166—41no—
55C2×Dic166—41no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58C22×Dic83—42no—
59C4⋊Dic83—24no—
60C4⋊Dic83—42no—
61C4⋊Dic83—42no—
62Dic83⋊C4—44no—
63Dic83⋊C4—44no—
64C22⋊Dic83—24no—
65C22⋊Dic83—44no—
66C22⋊Dic83—44no—
67C22⋊Dic83—42no—
68C4×Dic83—42no—
69C4×Dic83—82no—
70C24×C83C2×D411yescharacteristic
71C24×C83C2×D411yescharacteristic
72C2×D4×C83—41no—
73C2×D4×C83—41no—
74C2×D4×C83C2×Q812yes—
75C2×D4×C83#4112yes—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22×C411yescharacteristic
88C4⋊C4×C83—41no—
89C4⋊C4×C83—82no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C42×C83—21no—
95C42×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83C2×C22⋊C411yescharacteristic
105C23×C83C22⋊Q812yes—
106C23×C83#12411yescharacteristic
107C23×C83C4⋊Q811yescharacteristic
108D4×C83—42no—
109D4×C83—42no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—22no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—81no—
122C2×C4×C83—81no—
123C2×C4×C83—82no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83C4×Q812yes—
129C2×Dic83—24no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—44no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—82no—
137C2×Dic83—82no—
138Dic166—82no—
139Dic166—82no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—22no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83#39111yescharacteristic
155C22×C83C2×#9911yescharacteristic
156C22×C83#41011yescharacteristic
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—162no—
163C4×C83#35312yes—
164Dic83—44no—
165Dic83—84no—
166Dic83—82no—
167#13298—831nomaximal, Sylow
168C2×C83—21no—
169C2×C83—21no—
170C2×C83—42no—
171C2×C83—41no—
172C2×C83—41no—
173C2×C83#199611yescharacteristic
174C2×C83C2×#35311yescharacteristic
175C2×C83#212911yescharacteristic
176#2542—831no—
177#1799D8311yescharacteristic
178C4×#99—831no—
179#1393—834no—
180C83#1329811yescharacteristic, complemented, Sylow
181C2×C4⋊Q8—831no—
182C2×C22⋊Q8—1661no—
183C2×C22⋊Q8—1661no—
184C2×C4×D4C2×D8311yescharacteristic
185C2×#99—832no—
186C2×#99C2×D8312yes—
187#399—1661no—
188#399—1661no—
189#390—832no—
190#384—832no—
191#380—21no—
192C4×C4⋊C4—831no—
193C4×C22⋊C4—1661no—
194#342—21no—
195#342—1661no—
196C22×Q8—1661no—
197C22×D4C22×D8311yescharacteristic
198C23×C4D33211yescharacteristic
199C23×C4C22⋊D8311yescharacteristic
200C4⋊Q8—3321no—
201C4⋊Q8—3321no—
202C22⋊Q8—3324no—
203C22⋊Q8—3324no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—832no—
208C2×C4⋊C4—1662no—
209C2×C4⋊C4—1662no—
210C2×C4⋊C4—1661no—
211C2×C4⋊C4—1661no—
212C2×C4⋊C4C4×D8311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1662no—
215C2×C22⋊C4—1662no—
216C2×C22⋊C4—1662no—
217C2×C22⋊C4D33211yescharacteristic
218C2×C22⋊C4C22⋊D8311yescharacteristic
219C2×C42—3321no—
220C2×C42—832no—
221C2×C42C4×D8311yescharacteristic
222#99—24no—
223#99—834no—
224#99—1662no—
225#95—41no—
226#95—3321no—
227#95—1661no—
228C24C2×C22⋊D8311yescharacteristic
229C24C2×D33211yescharacteristic
230C2×Q8—3322no—
231C2×Q8—3322no—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4Q8×D8312yes—
237C2×D4#1305058912yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—3322no—
246C22×C4—1662no—
247C22×C4—1662no—
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4(C2×D83)⋊C411yescharacteristic
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4C2×C4×D8311yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—82no—
255C4⋊C4—3322no—
256C4⋊C4—3322no—
257C4⋊C4—3324no—
258C4⋊C4—3324no—
259C4⋊C4—1664no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—3322no—
265C22⋊C4—3324no—
266C22⋊C4—3324no—
267C22⋊C4—1664no—
268C42—21no—
269C42—21no—
270C42—3322no—
271C42—6642no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D83)⋊C411yescharacteristic
281C23#1458177212yes—
282C23#1458177911yescharacteristic
283C23#1458179011yescharacteristic
284Q8—6642no—
285Q8—6642no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—3322no—
305C2×C4—3322no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3324no—
309C2×C4—3322no—
310C2×C4—6642no—
311C2×C4—6642no—
312C2×C4—1664no—
313C2×C4#1458170611yescharacteristic
314C2×C4#1458170611yescharacteristic
315C2×C4#1458175712yes—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1912760611yescharacteristic
331C22C2×#1458170611yescharacteristic
332C22#1912767411yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—162no—
339C4—3324no—
340C4—6644no—
341C4—6642no—
342C4#1912749312yes—
343C2—21no—
344C2—21no—
345C2—41no—
346C2—42no—
347C2—41no—
348C2#2490844711yescharacteristic
349C2C2×#1912749311yescharacteristic
350C2#2490913711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#1799C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenter#1799D83
hypocenterC83#13298
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C2×#14581792C4semidirect
12. #19127570C4semidirect
13. C4×C4⋊Dic83C4semidirect
36. #14581707Q8semidirect
180. C83#13298semidirect

Maximal subgroups

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

subgroupindex
#249124162
C4×#145817072
#249055872
C83×#17992
#1329883