Finite Group Databasebeta

← #35315433

Subgroups — #35315433

Order 21248.

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

normal subgroups only

This group has 522 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#24907451C211yesmaximal, characteristic, complemented
2#24905784C211yesmaximal, characteristic, complemented
3#24905463C212yesmaximal, complemented
4#24905460C212yesmaximal, complemented
5C83×#2543C211yesmaximal, characteristic
6#19127754—21no—
7C2×#14581707—21no—
8#19127561—21no—
9#19127468C2212yes—
10#19127425—21no—
11#19127425C2212yes—
12C2×#121×C83—21no—
13C2×#121×C83C411yescharacteristic, complemented
14C2×C4×D4×C83C2211yescharacteristic
15#398×C83C411yescharacteristic, complemented
16#397×C83C2211yescharacteristic
17#394×C83C411yescharacteristic, complemented
18#392×C83C2211yescharacteristic
19#388×C83—21no—
20#386×C83—21no—
21#386×C83—21no—
22C2×#95×C83C411yescharacteristic, complemented
23C2×C4⋊Dic83—41no—
24C2×C22⋊Dic83—22no—
25C4⋊(C83⋊C8)—41no—
26#14581707—42no—
27C22⋊(C83⋊C8)—22no—
28C4×C83⋊C8—41no—
29C83⋊#95—41no—
30C22×D4×C83—21no—
31C22×D4×C83—21no—
32C22×D4×C83C2×C411yescharacteristic
33C22×D4×C83D411yescharacteristic
34C23×C4×C83—21no—
35C23×C4×C83C2×C411yescharacteristic
36C23×C4×C83D411yescharacteristic
37#121×C83—42no—
38#121×C83—42no—
39#121×C83—41no—
40#121×C83—41no—
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83—21no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C22⋊C4×C83C2311yescharacteristic
53C2×C42×C83D411yescharacteristic
54#95×C83—21no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83C2×C411yescharacteristic
58#95×C83C2×C411yescharacteristic
59C22×Dic83—42no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—82no—
62C22⋊Dic83—42no—
63C22⋊Dic83—42no—
64C24×C83—21no—
65C24×C83C22⋊C411yescharacteristic
66C24×C83C2×D411yescharacteristic
67C2×D4×C83—41no—
68C2×D4×C83—41no—
69C2×D4×C83—41no—
70C2×D4×C83—42no—
71C2×D4×C83—42no—
72C2×D4×C83—42no—
73C2×D4×C83—42no—
74C2×D4×C83—41no—
75C2×D4×C83—41no—
76C2×D4×C83—81no—
77C2×D4×C83—81no—
78C2×D4×C83D812yes—
79C2×D4×C83SD1612yes—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—42no—
88C22×C4×C83—42no—
89C22×C4×C83—81no—
90C22×C4×C83—81no—
91C22×C4×C83—81no—
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83C22×C411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—42no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—42no—
103C22⋊C4×C83—41no—
104C22⋊C4×C83—41no—
105C22⋊C4×C83—41no—
106C22⋊C4×C83—41no—
107C22⋊C4×C83—42no—
108C22⋊C4×C83—81no—
109C42×C83—21no—
110C42×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—41no—
116C23×C83—41no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—41no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—41no—
126C23×C83#10212yes—
127C23×C83C2×C22⋊C411yescharacteristic
128C23×C83C2×D811yescharacteristic
129C23×C83C2×SD1611yescharacteristic
130D4×C83—42no—
131D4×C83—42no—
132D4×C83—82no—
133D4×C83—82no—
134D4×C83—82no—
135D4×C83—84no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—22no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—41no—
143C2×C4×C83—42no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—81no—
156C2×C4×C83—81no—
157C2×C4×C83—81no—
158C2×C4×C83—84no—
159C2×C4×C83#9911yescharacteristic
160C2×C4×C83#9911yescharacteristic
161C2×C4×C83#10212yes—
162C2×Dic83—82no—
163C2×Dic83—82no—
164C2×Dic83—82no—
165C83⋊C8—82no—
166C22×C83—21no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—22no—
170C22×C83—21no—
171C22×C83—42no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83—42no—
175C22×C83—41no—
176C22×C83—41no—
177C22×C83—41no—
178C22×C83—41no—
179C22×C83—81no—
180C22×C83—81no—
181C22×C83—81no—
182C22×C83—81no—
183C22×C83—81no—
184C22×C83—81no—
185C22×C83—81no—
186C22×C83—81no—
187C22×C83C2×#9911yescharacteristic
188C22×C83C2×#10211yescharacteristic
189C22×C83#42111yescharacteristic
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—22no—
193C4×C83—42no—
194C4×C83—41no—
195C4×C83—41no—
196C4×C83—41no—
197C4×C83—81no—
198C4×C83—81no—
199Dic83—162no—
200#11876—831nomaximal, Sylow
201C2×C83—21no—
202C2×C83—21no—
203C2×C83—41no—
204C2×C83—42no—
205C2×C83—41no—
206C2×C83—161no—
207C2×C83#151911yescharacteristic
208C2×C83#212111yescharacteristic
209C2×C83#212611yescharacteristic
210#2543D8311yescharacteristic
211#1799—831no—
212#1460—831no—
213#1344—832no—
214#1343—832no—
215C83#1187611yescharacteristic, complemented, Sylow
216C2×#121—21no—
217C2×#121Dic8311yescharacteristic, complemented
218C2×C4×D4C2×D8311yescharacteristic
219#431—1661no—
220C2×#99—1661no—
221#398Dic8311yescharacteristic, complemented
222#397C2×D8311yescharacteristic
223#394Dic8311yescharacteristic, complemented
224#392C2×D8311yescharacteristic
225#388—21no—
226#386—21no—
227#386—21no—
228#380—1661no—
229C2×#95Dic8311yescharacteristic, complemented
230#342—832no—
231#323—832no—
232#323—1661no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2×Dic8311yescharacteristic
236C22×D4C22⋊D8311yescharacteristic
237C23×C4—21no—
238C23×C4C2×Dic8311yescharacteristic
239C23×C4C22⋊D8311yescharacteristic
240#121—42no—
241#121—42no—
242#121—41no—
243#121—41no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—3321no—
248C2×C4⋊C4C2×Dic8311yescharacteristic
249C2×C4⋊C4C2×Dic8311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1662no—
256C2×C22⋊C4C22⋊D8311yescharacteristic
257C2×C22⋊C4C22×D8311yescharacteristic
258C2×C42C22⋊D8311yescharacteristic
259C4⋊C8—3321no—
260#99—3322no—
261C22⋊C8—1662no—
262C4×C8—3321no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—3321no—
267#95C2×Dic8311yescharacteristic
268#95C2×Dic8311yescharacteristic
269C24—21no—
270C24C22⋊Dic8311yescharacteristic
271C24C2×C22⋊D8311yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4#1305057512yes—
284C2×D4#1305057812yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—3322no—
298C22×C4C22⋊Dic8311yescharacteristic
299C22×C4C22⋊Dic8311yescharacteristic
300C22×C4C22⋊Dic8311yescharacteristic
301C22×C4C22×Dic8311yescharacteristic
302C22×C4C2×C22⋊D8311yescharacteristic
303C2×C8—3322no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—6642no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—81no—
317C22⋊C4—3322no—
318C22⋊C4—3322no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1458171312yes—
337C23C2×C22⋊Dic8311yescharacteristic
338C23C2×#1305057511yescharacteristic
339C23C2×#1305057811yescharacteristic
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—6642no—
370C2×C4—6642no—
371C2×C4—6642no—
372C2×C4#1458170711yescharacteristic
373C2×C4#1458170711yescharacteristic
374C2×C4#1458171312yes—
375C8—6642no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×#1458170711yescharacteristic
398C22C2×#1458171311yescharacteristic
399C22#1912771811yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4—13282no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—161no—
416C2#2490603111yescharacteristic
417C2#2490910911yescharacteristic
418C2#2490912811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
Fitting, prime radicalC83×#2543C2
socleC22×C83C2×#99
cosocleC2×C22⋊C4×C83C23
hypercenter#2543D83
hypocenterC83#11876
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #24907451C2semidirect
2. #24905784C2semidirect
3. #24905463C2semidirect
4. #24905460C2semidirect
13. C2×#121×C83C4semidirect
15. #398×C83C4semidirect
17. #394×C83C4semidirect
22. C2×#95×C83C4semidirect
215. C83#11876semidirect
217. C2×#121Dic83semidirect
221. #398Dic83semidirect
223. #394Dic83semidirect
229. C2×#95Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074512
#249057842
#249054632
#249054602
C83×#25432
#1187683