Finite Group Databasebeta

← #35314689

Subgroups — #35314689

Order 21248.

592 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#19127468C211yesmaximal, characteristic
2C2×#19127425C211yesmaximal, characteristic
3#24905487C212yesmaximal, complemented
4#24905484C212yesmaximal, complemented
5C83×#2626C211yesmaximal, characteristic
6C22×C22⋊Dic83—21no—
7C2×C22⋊(C83⋊C8)—21no—
8#19127468—21no—
9#19127468C2212yes—
10#19127425—21no—
11#19127425C2212yes—
12C2×C22⋊Q8×C83C411yescharacteristic, complemented
13C22×C22⋊C4×C83C2211yescharacteristic
14#399×C83C411yescharacteristic, complemented
15#399×C83C2211yescharacteristic
16#396×C83—21no—
17#396×C83—21no—
18#393×C83C411yescharacteristic, complemented
19#393×C83C2211yescharacteristic
20#385×C83—21no—
21#380×C83C411yescharacteristic, complemented
22#379×C83—21no—
23C23×Dic83—41no—
24C22×C83⋊C8—41no—
25C2×C22⋊Dic83—24no—
26C2×C22⋊Dic83—22no—
27C2×C22⋊Dic83—41no—
28C2×C22⋊Dic83—42no—
29C22⋊(C83⋊C8)—22no—
30C22⋊(C83⋊C8)—41no—
31C25×C83D411yescharacteristic
32C22×Q8×C83C2×C411yescharacteristic, complemented
33C23×C4×C83C2×C411yescharacteristic
34C23×C4×C83D411yescharacteristic
35C22⋊Q8×C83—41no—
36C22⋊Q8×C83—41no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83C2×C411yescharacteristic, complemented
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83—41no—
49C2×C22⋊C4×C83—41no—
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C22⋊C4×C83C2311yescharacteristic
54#95×C83—21no—
55#95×C83—21no—
56#95×C83C2×C411yescharacteristic, complemented
57#95×C83C2×C411yescharacteristic, complemented
58C22×Dic83—44no—
59C22×Dic83—42no—
60C22×Dic83—82no—
61C22×Dic83—82no—
62C2×C83⋊C8—42no—
63C2×C83⋊C8—82no—
64C22⋊Dic83—24no—
65C22⋊Dic83—44no—
66C22⋊Dic83—42no—
67C24×C83—21no—
68C24×C83—21no—
69C24×C83—22no—
70C24×C83—81no—
71C24×C83—81no—
72C24×C83C22⋊C411yescharacteristic
73C24×C83C22⋊C411yescharacteristic
74C24×C83SD1612yes—
75C24×C83Q1612yes—
76C24×C83C2×D411yescharacteristic
77C2×Q8×C83—41no—
78C2×Q8×C83—81no—
79C22×C4×C83—21no—
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—81no—
88C22×C4×C83—81no—
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C4⋊C4×C83—41no—
94C4⋊C4×C83—42no—
95C4⋊C4×C83—81no—
96C4⋊C4×C83—81no—
97C4⋊C4×C83—81no—
98C22⋊C4×C83—41no—
99C22⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—41no—
104C22⋊C4×C83—42no—
105C22⋊C4×C83—81no—
106C22⋊C4×C83—81no—
107C22⋊C4×C83—81no—
108C22⋊C4×C83—81no—
109C23×C83—22no—
110C23×C83—22no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—42no—
120C23×C83—42no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—81no—
126C23×C83—81no—
127C23×C83—82no—
128C23×C83—82no—
129C23×C83—84no—
130C23×C83#9911yescharacteristic
131C23×C83#9911yescharacteristic
132C23×C83#10312yes—
133C23×C83#10312yes—
134C23×C83#10412yes—
135C23×C83C2×C22⋊C411yescharacteristic
136C23×C83C2×SD1611yescharacteristic
137C23×C83C2×Q1611yescharacteristic
138Q8×C83—82no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
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—41no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—41no—
159C2×C4×C83—41no—
160C2×C4×C83—81no—
161C2×C4×C83—81no—
162C2×C4×C83—81no—
163C2×Dic83—44no—
164C2×Dic83—88no—
165C2×Dic83—82no—
166C2×Dic83—82no—
167C83⋊C8—82no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—22no—
176C22×C83—21no—
177C22×C83—42no—
178C22×C83—42no—
179C22×C83—42no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—44no—
183C22×C83—44no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—41no—
188C22×C83—81no—
189C22×C83—81no—
190C22×C83—82no—
191C22×C83—82no—
192C22×C83#32812yes—
193C22×C83#32812yes—
194C22×C83C2×#9911yescharacteristic
195C22×C83C2×#10311yescharacteristic
196C22×C83C2×#10411yescharacteristic
197C4×C83—41no—
198C4×C83—41no—
199C4×C83—41no—
200C4×C83—41no—
201C4×C83—81no—
202C4×C83—81no—
203C4×C83—81no—
204C4×C83—81no—
205Dic83—84no—
206#11716—831nomaximal, Sylow
207C2×C83—21no—
208C2×C83—22no—
209C2×C83—21no—
210C2×C83—22no—
211C2×C83—41no—
212C2×C83—42no—
213C2×C83—41no—
214C2×C83C2×#32811yescharacteristic
215C2×C83#212111yescharacteristic
216C2×C83#212711yescharacteristic
217#2626D8311yescharacteristic
218C2×#342—831no—
219C2×#323—831no—
220#1352—832no—
221#1351—832no—
222C83#1171611yescharacteristic, complemented, Sylow
223C2×C22⋊Q8Dic8311yescharacteristic, complemented
224C22×C22⋊C4—1661no—
225C22×C22⋊C4C2×D8311yescharacteristic
226C2×C22⋊C8—1661no—
227#399Dic8311yescharacteristic, complemented
228#399C2×D8311yescharacteristic
229#396—21no—
230#396—21no—
231#393Dic8311yescharacteristic, complemented
232#393C2×D8311yescharacteristic
233#385—21no—
234#380Dic8311yescharacteristic, complemented
235#379—21no—
236#342—832no—
237#342—1661no—
238#323—832no—
239#323—1661no—
240C25C22⋊D8311yescharacteristic
241C22×Q8C2×Dic8311yescharacteristic, complemented
242C23×C4—3321no—
243C23×C4C2×Dic8311yescharacteristic
244C23×C4C22⋊D8311yescharacteristic
245C22×C8—3321no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×Dic8311yescharacteristic, complemented
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—3321no—
262C2×C22⋊C4—3322no—
263C2×C22⋊C4—1664no—
264C2×C22⋊C4—1662no—
265C2×C22⋊C4C2×Dic8311yescharacteristic
266C2×C22⋊C4C22⋊D8311yescharacteristic
267C2×C22⋊C4C22⋊D8311yescharacteristic
268C2×C22⋊C4C22×D8311yescharacteristic
269C22⋊C8—3321no—
270C22⋊C8—1662no—
271#95—21no—
272#95—21no—
273#95C2×Dic8311yescharacteristic, complemented
274#95C2×Dic8311yescharacteristic, complemented
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24C22⋊Dic8311yescharacteristic
281C24C22⋊Dic8311yescharacteristic
282C24#1305057712yes—
283C24#1305058012yes—
284C24C2×C22⋊D8311yescharacteristic
285C2×Q8—41no—
286C2×Q8—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—3322no—
298C22×C4—3324no—
299C22×C4—6642no—
300C22×C4—6642no—
301C22×C4C22⋊Dic8311yescharacteristic
302C22×C4C22⋊Dic8311yescharacteristic
303C22×C4C22×Dic8311yescharacteristic
304C22×C4C2×C22⋊D8311yescharacteristic
305C2×C8—3322no—
306C2×C8—6642no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—81no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—3324no—
324C22⋊C4—3322no—
325C22⋊C4—1664no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1458170711yescharacteristic
348C23#1458170711yescharacteristic
349C23#1458171612yes—
350C23#1458171612yes—
351C23#1458171912yes—
352C23C2×C22⋊Dic8311yescharacteristic
353C23C2×#1305057711yescharacteristic
354C23C2×#1305058011yescharacteristic
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
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—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—3324no—
381C2×C4—6648no—
382C2×C4—6642no—
383C2×C4—6642no—
384C8—6642no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1912743812yes—
410C22#1912743812yes—
411C22C2×#1458170711yescharacteristic
412C22C2×#1458171611yescharacteristic
413C22C2×#1458171911yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—6644no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—41no—
429C2—42no—
430C2C2×#1912743811yescharacteristic
431C2#2490910911yescharacteristic
432C2#2490913211yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2626C2
socleC22×C83C2×#99
prime radicalC25×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#2626D83
hypocenterC83#11716
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #24905487C2semidirect
4. #24905484C2semidirect
12. C2×C22⋊Q8×C83C4semidirect
14. #399×C83C4semidirect
18. #393×C83C4semidirect
21. #380×C83C4semidirect
32. C22×Q8×C83C2×C4semidirect
41. C2×C4⋊C4×C83C2×C4semidirect
56. #95×C83C2×C4semidirect
57. #95×C83C2×C4semidirect
222. C83#11716semidirect
223. C2×C22⋊Q8Dic83semidirect
227. #399Dic83semidirect
231. #393Dic83semidirect
234. #380Dic83semidirect
241. C22×Q8C2×Dic83semidirect
252. C2×C4⋊C4C2×Dic83semidirect
273. #95C2×Dic83semidirect
274. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191274682
C2×#191274252
#249054872
#249054842
C83×#26262
#1171683