Finite Group Databasebeta

← #35422095

Subgroups — #35422095

Order 21248.

766 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127739C211yesmaximal, characteristic
2#24910358C211yesmaximal, characteristic
3#24910350C211yesmaximal, characteristic
4#24908204C212yesmaximal, complemented
5#24908054C212yesmaximal, complemented
6#24908033C212yesmaximal, complemented
7#24907555C211yesmaximal, characteristic
8#24907555C211yesmaximal, characteristic
9#24907362C212yesmaximal, complemented
10#24907239C211yesmaximal, characteristic
11C83×#1808C211yesmaximal, characteristic
12C2×#14581751C2211yescharacteristic
13C2×#14581751C2211yescharacteristic
14#19127739C418yescomplemented
15C2×#14581726C2211yescharacteristic
16C2×#14581726C2211yescharacteristic
17C2×#14581724C2211yescharacteristic
18C2×#14581724C2211yescharacteristic
19C2×C22⋊(C83⋊C8)C2211yescharacteristic
20C2×#14581702C2212yes—
21#19127597C2212yes—
22#19127597C2212yes—
23#19127595—22no—
24#19127589—22no—
25#19127568—21no—
26#19127565—21no—
27C4⋊C4×Dic83C2212yes—
28C4⋊C4×Dic83C2212yes—
29#19127552—21no—
30C4×C22⋊Dic83C2211yescharacteristic
31#19127546—21no—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127463C2212yes—
36#19127455C2212yes—
37C23×C8×C83C2211yescharacteristic
38C2×#107×C83C2212yes—
39#380×C83C2211yescharacteristic
40#380×C83C2211yescharacteristic
41#340×C83C2212yes—
42C22×C83⋊C8C2312yes—
43#14581751—22no—
44#14581751—22no—
45#14581751C2×C414yes—
46#14581751C2×C414yes—
47C2×C4⋊Dic83C2311yescharacteristic
48C2×C4⋊Dic83C2311yescharacteristic
49C2×C4⋊Dic83C2311yescharacteristic
50C2×C4⋊Dic83C2311yescharacteristic
51C2×Dic83⋊C4—22no—
52C2×Dic83⋊C4—22no—
53C2×C22⋊Dic83—21no—
54C2×C22⋊Dic83—21no—
55C2×C22⋊Dic83D412yes—
56C2×C4×Dic83—22no—
57C2×C4×Dic83—22no—
58C2×C4×Dic83—21no—
59C2×C4×Dic83—21no—
60C2×C4×Dic83C2312yes—
61#14581726C2×C414yes—
62#14581726C2×C414yes—
63#14581724C2×C414yes—
64#14581724C2×C414yes—
65C22⋊(C83⋊C8)—22no—
66#14581702—24no—
67C83⋊#95—22no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—22no—
71C83⋊#95D411yescharacteristic
72C83⋊#95D411yescharacteristic
73C23×C4×C83C2311yescharacteristic
74C22×C8×C83—24no—
75C22×C8×C83C2×C414yes—
76C22×C8×C83C2311yescharacteristic
77C22×C8×C83C2311yescharacteristic
78C2×C4⋊C4×C83C2312yes—
79C2×C4⋊C4×C83C2312yes—
80C2×C22⋊C4×C83—21no—
81C2×C22⋊C4×C83—21no—
82#107×C83—24no—
83#95×C83—21no—
84#95×C83—21no—
85C22×Dic83—21no—
86C22×Dic83—21no—
87C22×Dic83—21no—
88C22×Dic83—21no—
89C22×Dic83C2×D412yes—
90C22×Dic83C2×D412yes—
91C2×C83⋊C8—22no—
92C2×C83⋊C8—22no—
93C2×C83⋊C8—22no—
94C4⋊Dic83—21no—
95C4⋊Dic83—21no—
96C4⋊Dic83—21no—
97C4⋊Dic83—21no—
98C4⋊Dic83C22×C412yes—
99C4⋊Dic83C22×C412yes—
100C4⋊Dic83C22×C412yes—
101C4⋊Dic83C22×C412yes—
102Dic83⋊C4—44no—
103Dic83⋊C4—82no—
104C22⋊Dic83—24no—
105C22⋊Dic83—24no—
106C22⋊Dic83—42no—
107C4×Dic83—24no—
108C4×Dic83—24no—
109C4×Dic83—22no—
110C4×Dic83—22no—
111C4×Dic83—44no—
112C4×Dic83—82no—
113C24×C83C2×D411yescharacteristic
114C22×C4×C83—22no—
115C22×C4×C83—22no—
116C22×C4×C83—22no—
117C22×C4×C83—22no—
118C22×C4×C83C22×C412yes—
119C22×C4×C83C2×D411yescharacteristic
120C22×C4×C83#4112yes—
121C22×C4×C83C2411yescharacteristic
122C2×C8×C83—22no—
123C2×C8×C83—28no—
124C2×C8×C83C22×C412yes—
125C2×C8×C83C22×C412yes—
126C2×C8×C83#4114yes—
127C4⋊C4×C83—22no—
128C4⋊C4×C83—22no—
129C4⋊C4×C83—22no—
130C4⋊C4×C83—22no—
131C22⋊C4×C83—81no—
132C22⋊C4×C83—81no—
133C23×C83—22no—
134C23×C83—22no—
135C23×C83—21no—
136C23×C83—21no—
137C23×C83C4×D412yes—
138C23×C83C22×D411yescharacteristic
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—22no—
142C2×C4×C83—24no—
143C2×C4×C83—42no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83—42no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83C4×D412yes—
150C2×C4×C83C23×C411yescharacteristic
151C2×C4×C83C2×#4112yes—
152C2×C4×C83C2×#4111yescharacteristic
153C2×C4×C832−1+412yes—
154C2×Dic83—22no—
155C2×Dic83—22no—
156C2×Dic83—22no—
157C2×Dic83—22no—
158C2×Dic83—22no—
159C2×Dic83—22no—
160C2×Dic83—22no—
161C2×Dic83—22no—
162C2×Dic83—22no—
163C2×Dic83—22no—
164C2×Dic83—42no—
165C2×Dic83—42no—
166C2×Dic83—41no—
167C2×Dic83—41no—
168C2×Dic83—41no—
169C2×Dic83—41no—
170C8×C83—22no—
171C8×C83#11714yes—
172C83⋊C8—42no—
173C22×C83—22no—
174C22×C83—22no—
175C22×C83—22no—
176C22×C83—42no—
177C22×C83—42no—
178C22×C83C2×C4×D411yescharacteristic
179C22×C83#54712yes—
180C22×C83#57512yes—
181C22×C83#57611yescharacteristic
182C22×C83#57811yescharacteristic
183C4×C83—22no—
184C4×C83—82no—
185C4×C83—82no—
186C4×C83C2×#11711yescharacteristic
187C4×C83#51911yescharacteristic
188C4×C83#55212yes—
189Dic83—42no—
190Dic83—42no—
191Dic83—42no—
192Dic83—42no—
193Dic83—42no—
194Dic83—42no—
195#22616—831nomaximal, Sylow
196C2×C83—42no—
197C2×C83#234411yescharacteristic
198C2×C83#294311yescharacteristic
199C2×C83#295111yescharacteristic
200C2×C83#334712yes—
201C2×C83#334912yes—
202C2×#427—831no—
203#2347—831no—
204#2346—831no—
205#1944—832no—
206#1921—832no—
207#1918—832no—
208#1825—831no—
209#1825—831no—
210#1808D8311yescharacteristic
211#1776—832no—
212#1752—831no—
213C83#2261611yescharacteristic, complemented, Sylow
214C23×C8C2×D8311yescharacteristic
215C2×#117—831no—
216C2×#117—831no—
217#427—838no—
218C2×#107—831no—
219C2×#107—831no—
220C2×#107C2×D8312yes—
221C2×#106—831no—
222C2×#106—831no—
223C2×C22⋊C8—831no—
224C2×#97—832no—
225#389—832no—
226#389—832no—
227#388—1662no—
228#388—1662no—
229#383—1661no—
230#382—1661no—
231#380C2×D8311yescharacteristic
232#380C2×D8311yescharacteristic
233C4×C4⋊C4—832no—
234C4×C4⋊C4—832no—
235C4×C4⋊C4—1661no—
236C4×C22⋊C4—831no—
237#376—1661no—
238#340—832no—
239#340—832no—
240#340—832no—
241#340—832no—
242#340C2×D8312yes—
243#336—832no—
244C23×C4C22×D8311yescharacteristic
245C22×C8—24no—
246C22×C8—832no—
247C22×C8C4×D8314yes—
248C22×C8C22×D8311yescharacteristic
249C22×C8C22×D8311yescharacteristic
250#117—834no—
251#117—834no—
252#117—1662no—
253#117—1662no—
254C2×C4⋊C4—831no—
255C2×C4⋊C4—831no—
256C2×C4⋊C4—831no—
257C2×C4⋊C4—831no—
258C2×C4⋊C4—1662no—
259C2×C4⋊C4—1662no—
260C2×C4⋊C4C22×D8312yes—
261C2×C4⋊C4C22×D8312yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—832no—
265C2×C22⋊C4—1661no—
266C2×C22⋊C4—1661no—
267C2×C42—832no—
268C2×C42—1662no—
269C2×C42—1662no—
270C2×C42—1661no—
271C2×C42—1661no—
272#107—24no—
273#107—834no—
274#107—834no—
275#106—834no—
276#106—834no—
277C22⋊C8—1662no—
278#97—1664no—
279#95—21no—
280#95—21no—
281#95—831no—
282#95—831no—
283#95—1662no—
284#95—1662no—
285#95—1662no—
286#95—1662no—
287C24D4×D8311yescharacteristic
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—832no—
293C22×C4—832no—
294C22×C4—1661no—
295C22×C4—1661no—
296C22×C4—1661no—
297C22×C4—1661no—
298C22×C4C2×C4×D8312yes—
299C22×C4D4×D8311yescharacteristic
300C22×C4#1305059112yes—
301C22×C4C23×D8311yescharacteristic
302C2×C8—22no—
303C2×C8—28no—
304C2×C8—1662no—
305C2×C8—1662no—
306C2×C8—1662no—
307C2×C8C2×C4×D8312yes—
308C2×C8C2×C4×D8312yes—
309C2×C8#1305058914yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—3324no—
315C4⋊C4—832no—
316C4⋊C4—832no—
317C4⋊C4—832no—
318C4⋊C4—832no—
319C4⋊C4—6642no—
320C4⋊C4—1661no—
321C4⋊C4—1661no—
322C4⋊C4—1661no—
323C4⋊C4—1661no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—3322no—
327C22⋊C4—1664no—
328C22⋊C4—1664no—
329C42—3324no—
330C42—6642no—
331C42—1662no—
332C42—1662no—
333C42—1664no—
334C42—1664no—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23#1458175412yes—
340C23C2×D4×D8311yescharacteristic
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—3322no—
352C2×C4—3322no—
353C2×C4—3321no—
354C2×C4—3321no—
355C2×C4—3321no—
356C2×C4—3321no—
357C2×C4—1662no—
358C2×C4—1662no—
359C2×C4—1662no—
360C2×C4—1662no—
361C2×C4—1662no—
362C2×C4—1662no—
363C2×C4—1662no—
364C2×C4—1662no—
365C2×C4—1662no—
366C2×C4—1662no—
367C2×C4#1458175412yes—
368C2×C4C22×C4×D8311yescharacteristic
369C2×C4C2×#1305058912yes—
370C2×C4C2×#1305059111yescharacteristic
371C2×C4#1458183912yes—
372C8—22no—
373C8—3322no—
374C8#1458174814yes—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22C2×#1458175411yescharacteristic
381C22#1912835112yes—
382C22#1912849812yes—
383C22#1912850211yescharacteristic
384C22#1912851411yescharacteristic
385C4—22no—
386C4—82no—
387C4—82no—
388C4—3322no—
389C4—3322no—
390C4—3322no—
391C4—3322no—
392C4—3322no—
393C4—3322no—
394C4C2×#1458174811yescharacteristic
395C4#1912817811yescharacteristic
396C4#1912839312yes—
397C2—42no—
398C2#2491033411yescharacteristic
399C2#2491626311yescharacteristic
400C2#2491633711yescharacteristic
401C2#2491996912yes—
402C2#2491999912yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1808C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1808D83
hypocenterC83#22616
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
4. #24908204C2semidirect
5. #24908054C2semidirect
6. #24908033C2semidirect
9. #24907362C2semidirect
14. #19127739C4semidirect
213. C83#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#191277392
#249103582
#249103502
#249082042
#249080542
#249080332
#249075552
#249073622
#249072392
C83×#18082
#2261683