Finite Group Databasebeta

← #35298404

Subgroups — #35298404

Order 21248.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910081C211yesmaximal, characteristic
2#24907658C212yesmaximal
3#24905716C211yesmaximal, characteristic
4#24905343C212yesmaximal
5C2×#324×C83C211yesmaximal, characteristic
6C2×#14581751—21no—
7C22×C4⋊Dic83C2211yescharacteristic
8C2×#14581724—22no—
9#19127600—22no—
10#19127600—22no—
11#19127561—21no—
12#19127561C2212yes—
13C2×C83⋊#95—21no—
14C2×C83⋊#95—21no—
15C2×C83⋊#95C2211yescharacteristic
16#19127464—22no—
17C22×C4⋊C4×C83C2211yescharacteristic
18C2×C22⋊C8×C83C2212yes—
19#324×C83C414yescomplemented
20C23×Dic83—21no—
21C23×Dic83—21no—
22#14581751—42no—
23#14581751—42no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—22no—
29C2×C4⋊Dic83—42no—
30C2×C4⋊Dic83D412yes—
31C2×C4⋊Dic83D412yes—
32C2×C22⋊Dic83—22no—
33C2×C4×Dic83—22no—
34#14581724—44no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C83⋊#95—42no—
39C83⋊#95—42no—
40C23×C4×C83D411yescharacteristic
41C23×C4×C83D411yescharacteristic
42C23×C4×C83C2311yescharacteristic
43C22×C8×C83—22no—
44C2×C4⋊C4×C83—22no—
45C2×C4⋊C4×C83—22no—
46C2×C4⋊C4×C83C2×C412yes—
47C2×C4⋊C4×C83Q812yes—
48C22⋊C8×C83—22no—
49C22⋊C8×C83C2×C414yes—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—41no—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83—41no—
59C22×Dic83—41no—
60C22×Dic83—41no—
61C22×Dic83—41no—
62C22×Dic83—41no—
63C22×Dic83—41no—
64C22×Dic83—41no—
65C22×Dic83—41no—
66C4⋊Dic83—24no—
67C4⋊Dic83—42no—
68C4⋊Dic83—42no—
69C4⋊Dic83—42no—
70C4⋊Dic83—42no—
71C4⋊Dic83—42no—
72C4⋊Dic83—81no—
73C4⋊Dic83—81no—
74C22⋊Dic83—42no—
75C22⋊Dic83—42no—
76C4×Dic83—44no—
77C24×C83C2×D411yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—41no—
85C22×C4×C83—41no—
86C22×C4×C83—41no—
87C22×C4×C83—41no—
88C22×C4×C83C4⋊C412yes—
89C22×C4×C83C4⋊C412yes—
90C22×C4×C83SD1612yes—
91C22×C4×C83SD1612yes—
92C22×C4×C83C22×C411yescharacteristic
93C22×C4×C83C2×D412yes—
94C22×C4×C83C2×Q811yescharacteristic
95C22×C4×C83#4112yes—
96C2×C8×C83—24no—
97C2×C8×C83—42no—
98C2×C8×C83—42no—
99C4⋊C4×C83—24no—
100C4⋊C4×C83—24no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83C2×C4⋊C411yescharacteristic
106C23×C83#12011yescharacteristic
107C23×C83#12311yescharacteristic
108C23×C83C2×SD1611yescharacteristic
109C23×C83C2×SD1611yescharacteristic
110C23×C83AGL(1,ℤ/8)11yescharacteristic
111C23×C83#13711yescharacteristic
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—42no—
127C2×C4×C83—81no—
128C2×C4×C83#10614yescomplemented
129C2×C4×C83C4×D412yes—
130C2×C4×C83C22⋊Q812yes—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138C2×Dic83—42no—
139C2×Dic83—42no—
140C2×Dic83—42no—
141C2×Dic83—42no—
142C2×Dic83—42no—
143C2×Dic83—81no—
144C2×Dic83—81no—
145C2×Dic83—81no—
146C2×Dic83—81no—
147C2×Dic83—81no—
148C2×Dic83—81no—
149C2×Dic83—81no—
150C2×Dic83—81no—
151C8×C83—44no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83#38511yescharacteristic
165C22×C83C2×#10611yescharacteristic
166C22×C83#42811yescharacteristic
167C22×C83#44811yescharacteristic
168C22×C83#45011yescharacteristic
169C22×C83#45411yescharacteristic
170C22×C83#45611yescharacteristic
171C22×C83B(4,𝔽2)11yescharacteristic
172C22×C83#45811yescharacteristic
173C22×C83#48111yescharacteristic
174C22×C83#48311yescharacteristic
175C4×C83—24no—
176C4×C83—22no—
177C4×C83—22no—
178C4×C83—42no—
179Dic83—82no—
180Dic83—82no—
181Dic83—82no—
182Dic83—82no—
183#8857—831nomaximal, Sylow
184C2×C83—21no—
185C2×C83—21no—
186C2×C83—21no—
187C2×C83—21no—
188C2×C83#159911yescharacteristic
189C2×C83#160411yescharacteristic
190C2×C83#161111yescharacteristic
191C2×C83#161611yescharacteristic
192C2×C83#185311yescharacteristic
193C2×C83#185711yescharacteristic
194C2×C83#186711yescharacteristic
195#2316—831no—
196#1853—832no—
197C2×#324D8311yescharacteristic
198#1441—831no—
199#1294—832no—
200C83#885711yescharacteristic, complemented, Sylow
201C2×#117—1661no—
202C22×C4⋊C4—831no—
203C22×C4⋊C4C2×D8311yescharacteristic
204C2×#106—1662no—
205C2×C22⋊C8C2×D8312yes—
206#389—1662no—
207#389—1662no—
208#380—832no—
209#380—1661no—
210C2×#95—831no—
211C2×#95—1661no—
212C2×#95—1661no—
213#340—1662no—
214#324Dic8314yescomplemented
215C23×C4—1661no—
216C23×C4—1661no—
217C23×C4D33211yescharacteristic
218C23×C4D33211yescharacteristic
219C23×C4C22×D8311yescharacteristic
220C22×C8—22no—
221#117—3322no—
222#117—3322no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—3322no—
226C2×C4⋊C4—832no—
227C2×C4⋊C4—832no—
228C2×C4⋊C4—1661no—
229C2×C4⋊C4—1661no—
230C2×C4⋊C4—1662no—
231C2×C4⋊C4—1661no—
232C2×C4⋊C4—1661no—
233C2×C4⋊C4Dic16612yes—
234C2×C4⋊C4C2×Dic8312yes—
235C2×C22⋊C4—1662no—
236C2×C42—1662no—
237#106—3324no—
238C22⋊C8—22no—
239C22⋊C8C2×Dic8314yes—
240#95—3322no—
241#95—3322no—
242#95—1662no—
243#95—1662no—
244#95—1662no—
245C24C2×D33211yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—3321no—
257C22×C4—3321no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—3321no—
263C22×C4—3321no—
264C22×C4—3321no—
265C22×C4—3321no—
266C22×C4—3321no—
267C22×C4—3321no—
268C22×C4—1662no—
269C22×C4—1662no—
270C22×C4—1662no—
271C22×C4—1662no—
272C22×C4C4⋊Dic8312yes—
273C22×C4C4⋊Dic8312yes—
274C22×C4#1305057912yes—
275C22×C4#1305057912yes—
276C22×C4C22×Dic8311yescharacteristic
277C22×C4D4×D8312yes—
278C22×C4C2×Dic16611yescharacteristic
279C22×C4#1305058912yes—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—3322no—
286C4⋊C4—3322no—
287C4⋊C4—3322no—
288C4⋊C4—3322no—
289C4⋊C4—3322no—
290C4⋊C4—6641no—
291C4⋊C4—6641no—
292C4⋊C4—1664no—
293C22⋊C4—3322no—
294C22⋊C4—3322no—
295C42—3324no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×C4⋊Dic8311yescharacteristic
301C23#1458176211yescharacteristic
302C23#1458177611yescharacteristic
303C23C2×#1305057911yescharacteristic
304C23C2×#1305057911yescharacteristic
305C23#1458182011yescharacteristic
306C23#1458182511yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—3322no—
324C2×C4—3322no—
325C2×C4—3322no—
326C2×C4—3322no—
327C2×C4—3322no—
328C2×C4—3322no—
329C2×C4—3322no—
330C2×C4—3322no—
331C2×C4—3322no—
332C2×C4—3322no—
333C2×C4—3322no—
334C2×C4—3322no—
335C2×C4—6641no—
336C2×C4—6641no—
337C2×C4—6641no—
338C2×C4—6641no—
339C2×C4—6641no—
340C2×C4—6641no—
341C2×C4—6641no—
342C2×C4—6641no—
343C2×C4#1458172414yescomplemented
344C2×C4D4×Dic8312yes—
345C2×C4#1458176912yes—
346C8—44no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1912757611yescharacteristic
360C22C2×#1458172411yescharacteristic
361C22#1912774311yescharacteristic
362C22#1912784211yescharacteristic
363C22#1912785411yescharacteristic
364C22#1912787611yescharacteristic
365C22#1912788611yescharacteristic
366C22#1912788911yescharacteristic
367C22#1912789211yescharacteristic
368C22#1912802011yescharacteristic
369C22#1912803211yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—6642no—
375C4—6642no—
376C4—6642no—
377C4—6642no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2490640511yescharacteristic
383C2#2490643411yescharacteristic
384C2#2490647311yescharacteristic
385C2#2490650211yescharacteristic
386C2#2490765811yescharacteristic
387C2#2490768011yescharacteristic
388C2#2490774611yescharacteristic
subgroupG/H
centerC23#14581762
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC2×#324×C83C2
socleC23×C83#120
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenterC2×#324D83
hypocenterC83#8857
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
19. #324×C83C4semidirect
128. C2×C4×C83#106semidirect
200. C83#8857semidirect
214. #324Dic83semidirect
343. C2×C4#14581724semidirect

Maximal subgroups

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

subgroupindex
#249100812
#249076582
#249057162
#249053432
C2×#324×C832
#885783