Finite Group Databasebeta

← #35293344

Subgroups — #35293344

Order 21248.

472 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 91 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#24908709C211yesmaximal, characteristic, complemented
2#24908604C211yesmaximal, characteristic, complemented
3#24908599C211yesmaximal, characteristic
4#24905289C211yesmaximal, characteristic, complemented
5#24905286C211yesmaximal, characteristic, complemented
6#24905284C211yesmaximal, characteristic
7C83×#2732C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9C2×#14581716—21no—
10C2×#14581715—21no—
11C2×#14581715—21no—
12C2×#14581713—21no—
13#19127463—21no—
14#19127463—21no—
15#19127456C2211yescharacteristic
16#19127456C2211yescharacteristic
17#19127456C2211yescharacteristic
18C2×C22⋊Q8×C83—21no—
19#399×C83—21no—
20#398×C83—21no—
21#396×C83—21no—
22#391×C83C2211yescharacteristic
23#390×C83C2211yescharacteristic
24#389×C83C2211yescharacteristic
25#386×C83—21no—
26#385×C83—21no—
27C22×Dic166—21no—
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30C22×C83⋊C8—21no—
31#14581792—42no—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83—21no—
34#14581716—42no—
35#14581715—42no—
36#14581715—42no—
37#14581713—42no—
38C22×Q8×C83D411yescharacteristic
39C22×D4×C83D411yescharacteristic
40C23×C4×C83—21no—
41C22⋊Q8×C83—42no—
42C22⋊Q8×C83—42no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83—21no—
47C2×C4⋊C4×C83D411yescharacteristic
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—21no—
56C2×C42×C83C2311yescharacteristic
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—21no—
61C2×Dic166—22no—
62C2×Dic166—41no—
63C2×Dic166—41no—
64C2×Dic166—41no—
65C2×Dic166—41no—
66C2×Dic166—41no—
67C22×Dic83—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8—41no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8—41no—
77C4⋊Dic83—22no—
78C4⋊Dic83—22no—
79C4⋊Dic83—41no—
80C4⋊Dic83—41no—
81C24×C83—21no—
82C2×Q8×C83—21no—
83C2×Q8×C83—41no—
84C2×Q8×C83—41no—
85C2×Q8×C83SD1612yes—
86C2×D4×C83—21no—
87C2×D4×C83—41no—
88C2×D4×C83—41no—
89C2×D4×C83Q1612yes—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83—42no—
96C22×C4×C83—81no—
97C22×C4×C83—81no—
98C22×C4×C83C2×D411yescharacteristic
99C22×C4×C83C2×D411yescharacteristic
100C22×C4×C83C2×D411yescharacteristic
101C4⋊C4×C83—21no—
102C4⋊C4×C83—21no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—41no—
106C4⋊C4×C83—42no—
107C4⋊C4×C83—42no—
108C4⋊C4×C83—41no—
109C4⋊C4×C83—41no—
110C4⋊C4×C83—81no—
111C22⋊C4×C83—42no—
112C22⋊C4×C83—41no—
113C22⋊C4×C83—41no—
114C22⋊C4×C83—41no—
115C22⋊C4×C83—41no—
116C22⋊C4×C83—81no—
117C22⋊C4×C83—81no—
118C42×C83—41no—
119C23×C83—22no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—41no—
126C23×C83#12011yescharacteristic
127Q8×C83—42no—
128Q8×C83—81no—
129D4×C83—42no—
130D4×C83—81no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—22no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
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—84no—
153C2×C4×C83C2×SD1611yescharacteristic
154C2×C4×C83C2×Q1611yescharacteristic
155C2×C4×C83#13511yescharacteristic
156C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C83#13711yescharacteristic
159C2×Dic83—42no—
160C2×Dic83—81no—
161C2×Dic83—81no—
162Dic166—42no—
163Dic166—42no—
164Dic166—81no—
165Dic166—81no—
166C83⋊C8—81no—
167C83⋊C8—81no—
168C83⋊C8—81no—
169C22×C83—42no—
170C22×C83—42no—
171C22×C83—42no—
172C22×C83—81no—
173C22×C83—81no—
174C22×C83—81no—
175C22×C83—81no—
176C22×C83#44911yescharacteristic
177C22×C83#45011yescharacteristic
178C22×C83#45111yescharacteristic
179C22×C83#45411yescharacteristic
180C22×C83#45511yescharacteristic
181C22×C83#45511yescharacteristic
182C22×C83B(4,𝔽2)11yescharacteristic
183C4×C83—21no—
184C4×C83—21no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—42no—
190C4×C83—81no—
191C4×C83—81no—
192C4×C83—81no—
193C4×C83—81no—
194Dic83—82no—
195#7774—831nomaximal, Sylow
196C2×C83—82no—
197C2×C83#159711yescharacteristic
198C2×C83#160111yescharacteristic
199C2×C83#160211yescharacteristic
200C2×C83#163711yescharacteristic
201C2×C83#164211yescharacteristic
202C2×C83#164411yescharacteristic
203C2×C83#166211yescharacteristic
204#2732D8311yescharacteristic
205#2043—831no—
206#2024—831no—
207#2023—831no—
208#1272—831no—
209#1271—831no—
210#1270—831no—
211C83#777411yescharacteristic, complemented, Sylow
212C2×C4⋊Q8—831no—
213C2×C22⋊Q8—21no—
214C2×#103—1661no—
215C2×#103—1661no—
216C2×#103—1661no—
217C2×#102—1661no—
218#399—21no—
219#398—21no—
220#396—21no—
221#391C2×D8311yescharacteristic
222#390C2×D8311yescharacteristic
223#389C2×D8311yescharacteristic
224#386—21no—
225#385—21no—
226#340—1661no—
227#340—1661no—
228#336—831no—
229#336—831no—
230#336—831no—
231C22×Q8—1661no—
232C22×Q8C22⋊D8311yescharacteristic
233C22×D4C22⋊D8311yescharacteristic
234C23×C4—21no—
235C22×C8—1661no—
236C22×C8—1661no—
237C22×C8—1661no—
238C4⋊Q8—3322no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—1661no—
246C2×C4⋊C4—1661no—
247C2×C4⋊C4C22⋊D8311yescharacteristic
248C2×C4⋊C4C22⋊D8311yescharacteristic
249C2×C4⋊C4C22⋊D8311yescharacteristic
250C2×C4⋊C4C22⋊D8311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42C22×D8311yescharacteristic
257#103—3322no—
258#103—3322no—
259#103—3322no—
260#102—3322no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—3321no—
270C2×Q8—3321no—
271C2×Q8—3321no—
272C2×Q8—3321no—
273C2×Q8—3321no—
274C2×Q8—1662no—
275C2×Q8#1305057812yes—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4#1305058012yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—3321no—
289C22×C4C2×C22⋊D8311yescharacteristic
290C22×C4C2×C22⋊D8311yescharacteristic
291C22×C4C2×C22⋊D8311yescharacteristic
292C2×C8—3321no—
293C2×C8—3321no—
294C2×C8—3321no—
295C2×C8—3321no—
296C2×C8—3321no—
297C2×C8—3321no—
298C2×C8—3321no—
299C2×C8—3321no—
300C2×C8—3321no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—81no—
311C4⋊C4—3321no—
312C4⋊C4—3321no—
313C4⋊C4—1662no—
314C4⋊C4—1662no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—41no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#1458176111yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—3322no—
334Q8—3322no—
335Q8—6641no—
336Q8—6641no—
337D4—42no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—3322no—
362C2×C4—6641no—
363C2×C4—6641no—
364C2×C4C2×#1305057811yescharacteristic
365C2×C4C2×#1305058011yescharacteristic
366C2×C4#1458181211yescharacteristic
367C2×C4#1458181711yescharacteristic
368C2×C4#1458181811yescharacteristic
369C2×C4#1458182311yescharacteristic
370C8—6641no—
371C8—6641no—
372C8—6641no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22#1912784611yescharacteristic
381C22#1912785211yescharacteristic
382C22#1912785711yescharacteristic
383C22#1912787411yescharacteristic
384C22#1912787911yescharacteristic
385C22#1912787911yescharacteristic
386C22#1912788811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—6642no—
399C2—82no—
400C2#2490639211yescharacteristic
401C2#2490641511yescharacteristic
402C2#2490642011yescharacteristic
403C2#2490662811yescharacteristic
404C2#2490666311yescharacteristic
405C2#2490667711yescharacteristic
406C2#2490678611yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2732C2
socleC23×C83#120
prime radicalC22×D4×C83D4
hypercenter#2732D83
hypocenterC83#7774
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
1. #24908709C2semidirect
2. #24908604C2semidirect
4. #24905289C2semidirect
5. #24905286C2semidirect
211. C83#7774semidirect

Maximal subgroups

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

subgroupindex
#249087092
#249086042
#249085992
#249052892
#249052862
#249052842
C83×#27322
#777483