Finite Group Databasebeta

← #35315428

Subgroups — #35315428

Order 21248.

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

normal subgroups only

This group has 536 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#24905781C211yesmaximal, characteristic, complemented
3#24905463C212yesmaximal, complemented
4#24905460C212yesmaximal, complemented
5C83×#2535C211yesmaximal, characteristic
6C2×#14581707—21no—
7C2×C22⋊(C83⋊C8)—21no—
8#19127561—21no—
9#19127468C2212yes—
10#19127427—21no—
11#19127425C2212yes—
12C2×#121×C83—21no—
13C2×C4×D4×C83C411yescharacteristic, complemented
14C2×C4×D4×C83C2211yescharacteristic
15#397×C83C411yescharacteristic, complemented
16#397×C83C2211yescharacteristic
17#392×C83C411yescharacteristic, complemented
18#392×C83C2211yescharacteristic
19#386×C83—21no—
20#386×C83—21no—
21#382×C83C411yescharacteristic, complemented
22#380×C83—21no—
23C22×C83⋊C8—41no—
24C2×C4⋊Dic83—41no—
25C2×C22⋊Dic83—22no—
26#14581707—42no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(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×C83D411yescharacteristic
36C23×C4×C83D411yescharacteristic
37#121×C83—42no—
38#121×C83—42no—
39C4×D4×C83—41no—
40C4×D4×C83—41no—
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
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×C83C2×C411yescharacteristic
52C2×C22⋊C4×C83C2311yescharacteristic
53C2×C42×C83C2×C411yescharacteristic
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—
61C2×C83⋊C8—82no—
62C4⋊Dic83—82no—
63C22⋊Dic83—42no—
64C22⋊Dic83—42no—
65C24×C83—21no—
66C24×C83C22⋊C411yescharacteristic
67C24×C83C2×D411yescharacteristic
68C2×D4×C83—41no—
69C2×D4×C83—41no—
70C2×D4×C83—41no—
71C2×D4×C83—42no—
72C2×D4×C83—42no—
73C2×D4×C83—42no—
74C2×D4×C83—42no—
75C2×D4×C83—41no—
76C2×D4×C83—41no—
77C2×D4×C83—81no—
78C2×D4×C83—81no—
79C2×D4×C83D812yes—
80C2×D4×C83SD1612yes—
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—21no—
88C22×C4×C83—42no—
89C22×C4×C83—42no—
90C22×C4×C83—42no—
91C22×C4×C83—81no—
92C22×C4×C83—81no—
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83C22⋊C411yescharacteristic
96C22×C4×C83C22×C411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—42no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—42no—
104C22⋊C4×C83—41no—
105C22⋊C4×C83—41no—
106C22⋊C4×C83—41no—
107C22⋊C4×C83—41no—
108C22⋊C4×C83—81no—
109C22⋊C4×C83—81no—
110C42×C83—21no—
111C42×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—22no—
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—41no—
127C23×C83#10212yes—
128C23×C83C2×C22⋊C411yescharacteristic
129C23×C83C2×D811yescharacteristic
130C23×C83C2×SD1611yescharacteristic
131D4×C83—42no—
132D4×C83—42no—
133D4×C83—42no—
134D4×C83—42no—
135D4×C83—84no—
136D4×C83—82no—
137D4×C83—82no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—41no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
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—41no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—81no—
159C2×C4×C83—81no—
160C2×C4×C83—84no—
161C2×C4×C83#9911yescharacteristic
162C2×C4×C83#9911yescharacteristic
163C2×C4×C83#10212yes—
164C2×Dic83—82no—
165C2×Dic83—82no—
166C2×Dic83—82no—
167C83⋊C8—82no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—22no—
172C22×C83—21no—
173C22×C83—42no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83—42no—
177C22×C83—42no—
178C22×C83—41no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—42no—
183C22×C83—81no—
184C22×C83—81no—
185C22×C83—81no—
186C22×C83—81no—
187C22×C83—81no—
188C22×C83—81no—
189C22×C83C2×#9911yescharacteristic
190C22×C83C2×#10211yescharacteristic
191C22×C83#42111yescharacteristic
192C4×C83—21no—
193C4×C83—21no—
194C4×C83—22no—
195C4×C83—41no—
196C4×C83—41no—
197C4×C83—41no—
198C4×C83—81no—
199C4×C83—81no—
200C4×C83—81no—
201Dic83—162no—
202#11875—831nomaximal, Sylow
203C2×C83—21no—
204C2×C83—21no—
205C2×C83—41no—
206C2×C83—42no—
207C2×C83—41no—
208C2×C83—82no—
209C2×C83#151911yescharacteristic
210C2×C83#212011yescharacteristic
211C2×C83#212711yescharacteristic
212#2535D8311yescharacteristic
213#1799—831no—
214#1459—831no—
215#1344—832no—
216#1343—832no—
217C83#1187511yescharacteristic, complemented, Sylow
218C2×#121—21no—
219C2×C4×D4Dic8311yescharacteristic, complemented
220C2×C4×D4C2×D8311yescharacteristic
221C2×#99—1661no—
222C2×C22⋊C8—1661no—
223#397Dic8311yescharacteristic, complemented
224#397C2×D8311yescharacteristic
225#392Dic8311yescharacteristic, complemented
226#392C2×D8311yescharacteristic
227#386—21no—
228#386—21no—
229#382Dic8311yescharacteristic, complemented
230#380—21no—
231#380—1661no—
232#342—832no—
233#324—1661no—
234#323—832no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic8311yescharacteristic
238C22×D4C22⋊D8311yescharacteristic
239C23×C4—21no—
240C23×C4C22⋊D8311yescharacteristic
241C23×C4C22⋊D8311yescharacteristic
242C22×C8—3321no—
243#121—42no—
244#121—42no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—3321no—
251C2×C4⋊C4C2×Dic8311yescharacteristic
252C2×C4⋊C4C22⋊D8311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1662no—
259C2×C22⋊C4C2×Dic8311yescharacteristic
260C2×C22⋊C4C22×D8311yescharacteristic
261C2×C42C2×Dic8311yescharacteristic
262#99—3322no—
263C22⋊C8—3321no—
264C22⋊C8—1662no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—3321no—
269#95C2×Dic8311yescharacteristic
270#95C2×Dic8311yescharacteristic
271C24—21no—
272C24C22⋊Dic8311yescharacteristic
273C24C2×C22⋊D8311yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4#1305057512yes—
286C2×D4#1305057812yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—3322no—
300C22×C4C22⋊Dic8311yescharacteristic
301C22×C4C22⋊Dic8311yescharacteristic
302C22×C4C22⋊Dic8311yescharacteristic
303C22×C4C22×Dic8311yescharacteristic
304C22×C4C2×C22⋊D8311yescharacteristic
305C2×C8—3322no—
306C2×C8—6642no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—6642no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C22⋊C4—3322no—
321C22⋊C4—3322no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#1458171312yes—
340C23C2×C22⋊Dic8311yescharacteristic
341C23C2×#1305057511yescharacteristic
342C23C2×#1305057811yescharacteristic
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—82no—
348D4—82no—
349D4—84no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—6642no—
374C2×C4—6642no—
375C2×C4—6642no—
376C2×C4#1458170711yescharacteristic
377C2×C4#1458170711yescharacteristic
378C2×C4#1458171312yes—
379C8—6642no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×#1458170711yescharacteristic
402C22C2×#1458171311yescharacteristic
403C22#1912771811yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—13282no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—42no—
419C2—82no—
420C2#2490603111yescharacteristic
421C2#2490910411yescharacteristic
422C2#2490913211yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
Fitting, prime radicalC83×#2535C2
socleC22×C83C2×#99
cosocleC2×C22⋊C4×C83C23
hypercenter#2535D83
hypocenterC83#11875
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #24907451C2semidirect
2. #24905781C2semidirect
3. #24905463C2semidirect
4. #24905460C2semidirect
13. C2×C4×D4×C83C4semidirect
15. #397×C83C4semidirect
17. #392×C83C4semidirect
21. #382×C83C4semidirect
217. C83#11875semidirect
219. C2×C4×D4Dic83semidirect
223. #397Dic83semidirect
225. #392Dic83semidirect
229. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074512
#249057812
#249054632
#249054602
C83×#25352
#1187583