Finite Group Databasebeta

← #35300628

Subgroups — #35300628

Order 21248.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127982C211yesmaximal, characteristic
2C2×#19127778C211yesmaximal, characteristic
3#24908264C211yesmaximal, characteristic, complemented
4#24907460C211yesmaximal, characteristic
5C2×#19127448C211yesmaximal, characteristic, complemented
6#24905297C211yesmaximal, characteristic, complemented
7C83×#1296C211yesmaximal, characteristic, complemented
8C22×#13050573—21no—
9C2×#14581792C2211yescharacteristic
10C2×C4×D332C2211yescharacteristic
11#19127982—22no—
12#19127982C414yescomplemented
13#19127778—22no—
14#19127778—22no—
15C2×#14581726—21no—
16C2×C4⋊(C83⋊C8)C2211yescharacteristic
17C2×#14581715—21no—
18C2×#14581715—21no—
19C2×#14581714—21no—
20C2×#14581712—21no—
21C2×(C2×D83)⋊C8—21no—
22C2×C4×C83⋊C8C2211yescharacteristic
23#19127463—21no—
24#19127448—22no—
25C2×C4⋊C8×C83C2211yescharacteristic
26C2×C4⋊C8×C83C2211yescharacteristic
27#384×C83C2211yescharacteristic
28C22×Dic166—21no—
29C22×D332D411yescharacteristic
30C22×C4×D83—21no—
31C2×#13050573—24no—
32C2×#13050573—42no—
33C2×#13050573—42no—
34C22×C83⋊C8—21no—
35C22×C83⋊C8D411yescharacteristic
36C22×C83⋊C8D411yescharacteristic
37#14581792—21no—
38#14581792—21no—
39#14581792—21no—
40#14581792C2×C412yes—
41C4×D332—21no—
42C4×D332—21no—
43C4×D332C2×C412yes—
44C4×D332D412yes—
45C2×C4⋊Dic83—21no—
46C2×C4⋊Dic83—21no—
47C2×C4⋊Dic83D411yescharacteristic
48C2×(C2×D83)⋊C4—21no—
49#14581726—24no—
50C4⋊(C83⋊C8)—21no—
51C4⋊(C83⋊C8)—21no—
52#14581715—42no—
53#14581715—42no—
54#14581714—24no—
55#14581712—42no—
56(C2×D83)⋊C8—24no—
57C4×C83⋊C8—21no—
58C4×C83⋊C8—21no—
59C22×C8×C83—21no—
60C22×C8×C83—21no—
61C2×C4⋊C4×C83—21no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C42×C83C2311yescharacteristic
65C4⋊C8×C83—21no—
66C4⋊C8×C83—21no—
67C4⋊C8×C83C2×C412yes—
68C4⋊C8×C83D412yes—
69C23×D83—21no—
70C2×Dic166—22no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×Dic166—41no—
74C2×Dic166—41no—
75C2×Dic166—41no—
76C2×D332—21no—
77C2×D332—21no—
78C2×D332—21no—
79C2×D332—21no—
80C2×D332—21no—
81C2×D332C22⋊C412yes—
82C22×Dic83—21no—
83C22×Dic83—41no—
84C2×C4×D83—22no—
85C2×C4×D83—22no—
86C2×C4×D83—41no—
87C2×C4×D83—41no—
88C2×C4×D83—41no—
89C2×C4×D83—41no—
90#13050573—48no—
91C2×C83⋊C8—21no—
92C2×C83⋊C8—21no—
93C2×C83⋊C8—21no—
94C2×C83⋊C8—21no—
95C2×C83⋊C8—21no—
96C2×C83⋊C8—41no—
97C2×C83⋊C8—41no—
98C2×C83⋊C8—41no—
99C2×C83⋊C8D812yes—
100C4⋊Dic83—21no—
101C4⋊Dic83—22no—
102C4⋊Dic83—22no—
103C4⋊Dic83—41no—
104C4⋊Dic83—41no—
105C4⋊Dic83C22⋊C412yes—
106(C2×D83)⋊C4—22no—
107(C2×D83)⋊C4—22no—
108C22×C4×C83—41no—
109C22×C4×C83C2×D411yescharacteristic
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83#4111yescharacteristic
112C2×C8×C83—22no—
113C2×C8×C83—22no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C2×C8×C83—41no—
117C2×C8×C83—41no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83—41no—
120C4⋊C4×C83—41no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83—81no—
123C42×C83C22×C411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C42×C83C2×D411yescharacteristic
126C42×C83#4111yescharacteristic
127C23×C83#12111yescharacteristic
128C22×D83—22no—
129C22×D83—41no—
130C22×D83—41no—
131C22×D83—41no—
132C22×D83—41no—
133C22×D83—41no—
134C22×D83—41no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—81no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83C2×C22⋊C411yescharacteristic
141C2×C4×C83C4×D411yescharacteristic
142C2×C4×C83C4×D411yescharacteristic
143C2×C4×C83#12011yescharacteristic
144C2×C4×C83#12111yescharacteristic
145C2×C4×C83#12311yescharacteristic
146C2×C4×C83C2×D811yescharacteristic
147C2×C4×C83#13511yescharacteristic
148C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C83#13711yescharacteristic
152C2×C4×C83#13711yescharacteristic
153C2×C4×C83#13711yescharacteristic
154C2×Dic83—22no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—42no—
158C2×Dic83—81no—
159C2×Dic83—81no—
160D332—22no—
161D332—22no—
162D332—41no—
163D332—41no—
164C4×D83—42no—
165C4×D83—42no—
166C4×D83—42no—
167C4×D83—42no—
168Dic166—42no—
169Dic166—42no—
170Dic166—81no—
171Dic166—81no—
172C8×C83—42no—
173C8×C83—42no—
174C83⋊C8—22no—
175C83⋊C8—41no—
176C83⋊C8—81no—
177C22×C83#38611yescharacteristic
178C22×C83#45911yescharacteristic
179C22×C83#46311yescharacteristic
180C22×C83#46811yescharacteristic
181C22×C83#47011yescharacteristic
182C22×C83#47111yescharacteristic
183C22×C83#47311yescharacteristic
184C2×D83—42no—
185C2×D83—42no—
186C2×D83—42no—
187C2×D83—81no—
188C2×D83—81no—
189C2×D83—81no—
190C2×D83—81no—
191C4×C83—21no—
192C4×C83—21no—
193C4×C83—161no—
194C4×C83#41711yescharacteristic
195C4×C83#41911yescharacteristic
196C4×C83C4×D811yescharacteristic
197C4×C83#44211yescharacteristic
198C4×C83#44711yescharacteristic
199C4×C83#44911yescharacteristic
200C4×C83#46811yescharacteristic
201C4×C83#47011yescharacteristic
202Dic83—42no—
203Dic83—82no—
204#9227—831nomaximal, Sylow
205C2×C83#168511yescharacteristic
206C2×C83#168911yescharacteristic
207C2×C83#169911yescharacteristic
208C2×C83#170311yescharacteristic
209C2×C83#187911yescharacteristic
210C2×C83#189411yescharacteristic
211C2×C83#190811yescharacteristic
212D83—82no—
213C2×#474—831no—
214C2×#436—831no—
215#1956—831no—
216#1802—831no—
217C2×#332—831no—
218#1296D8311yescharacteristic, complemented
219#1277—831no—
220C83#922711yescharacteristic, complemented, Sylow
221C22×M16—1661no—
222C2×C4⋊Q8—831no—
223C2×C4×D4—831no—
224#474—834no—
225#474—1662no—
226#436—1662no—
227#436—1662no—
228C2×#107—1661no—
229C2×C4⋊C8—831no—
230C2×C4⋊C8C2×D8311yescharacteristic
231C2×C4⋊C8C2×D8311yescharacteristic
232C2×#103—1661no—
233C2×#103—1661no—
234C2×#102—1661no—
235C2×#102—1661no—
236C2×C22⋊C8—1661no—
237C2×C4×C8—831no—
238#384C2×D8311yescharacteristic
239#340—1661no—
240#332—1662no—
241C22×Q8—1661no—
242C22×D4—831no—
243C23×C4—1661no—
244C2×M16—3322no—
245C2×M16—3322no—
246C2×M16—1664no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—831no—
250C22×C8—831no—
251C22×C8—1661no—
252C4⋊Q8—832no—
253C4⋊Q8—1661no—
254C4⋊Q8—1661no—
255C4⋊Q8—1661no—
256C4×D4—832no—
257C4×D4—832no—
258C4×D4—1661no—
259C4×D4—1661no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—831no—
264C2×C4⋊C4—1661no—
265C2×C4⋊C4—1661no—
266C2×C22⋊C4—1661no—
267C2×C42C22×D8311yescharacteristic
268#107—1664no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1661no—
272C4⋊C8—1661no—
273C4⋊C8C4×D8312yes—
274C4⋊C8D33212yes—
275#103—3322no—
276#103—3322no—
277#102—3322no—
278#102—1664no—
279C22⋊C8—1664no—
280C4×C8—1661no—
281C4×C8—1661no—
282C24—1661no—
283C2×Q8—3321no—
284C2×Q8—3321no—
285C2×Q8—3321no—
286C2×Q8—3321no—
287C2×Q8—3321no—
288C2×Q8—1662no—
289C2×D4—832no—
290C2×D4—1661no—
291C2×D4—1661no—
292C2×D4—1661no—
293C2×D4—1661no—
294C2×D4—1661no—
295C22×C4—41no—
296C22×C4—3321no—
297C22×C4—3321no—
298C22×C4—3321no—
299C22×C4—3321no—
300C22×C4—3321no—
301C22×C4—1661no—
302C22×C4—1662no—
303C22×C4—1662no—
304C22×C4D4×D8311yescharacteristic
305C22×C4D4×D8311yescharacteristic
306C22×C4#1305059011yescharacteristic
307M16—3328no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—3321no—
315C2×C8—3321no—
316C2×C8—3321no—
317C2×C8—832no—
318C2×C8—1661no—
319C2×C8—1661no—
320C2×C8—1661no—
321C2×C8—1661no—
322C2×C8—1661no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—81no—
328C4⋊C4—3321no—
329C4⋊C4—3321no—
330C4⋊C4—832no—
331C4⋊C4—1662no—
332C4⋊C4—1662no—
333C4⋊C4—1661no—
334C22⋊C4—1662no—
335C22⋊C4—1662no—
336C42C2×C4×D8311yescharacteristic
337C42D4×D8311yescharacteristic
338C42C2×D33211yescharacteristic
339C42#1305059111yescharacteristic
340C23—3321no—
341C23—3321no—
342C23—3321no—
343C23—3321no—
344C23—3321no—
345C23—3321no—
346C23—1662no—
347C23#1458176511yescharacteristic
348Q8—3322no—
349Q8—3322no—
350Q8—6641no—
351Q8—6641no—
352D4—3321no—
353D4—3321no—
354D4—1662no—
355D4—1662no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—3321no—
362C2×C4—3321no—
363C2×C4—3322no—
364C2×C4—3322no—
365C2×C4—3322no—
366C2×C4—3322no—
367C2×C4—3322no—
368C2×C4—6641no—
369C2×C4—6641no—
370C2×C4—1662no—
371C2×C4C22⋊C4×D8311yescharacteristic
372C2×C4#1458175411yescharacteristic
373C2×C4C4×D33211yescharacteristic
374C2×C4#1458176211yescharacteristic
375C2×C4#1458176711yescharacteristic
376C2×C4#1458177711yescharacteristic
377C2×C4D8×D8311yescharacteristic
378C2×C4#1458181511yescharacteristic
379C2×C4#1458181611yescharacteristic
380C2×C4#1458181911yescharacteristic
381C2×C4#1458181911yescharacteristic
382C2×C4#1458182111yescharacteristic
383C2×C4#1458182111yescharacteristic
384C2×C4#1458182411yescharacteristic
385C8—42no—
386C8—42no—
387C8—3321no—
388C8—6641no—
389C8—1662no—
390C22—3322no—
391C22—3322no—
392C22—3322no—
393C22—6641no—
394C22—6641no—
395C22—6641no—
396C22—6641no—
397C22#1912757911yescharacteristic
398C22#1912789511yescharacteristic
399C22#1912791911yescharacteristic
400C22#1912794611yescharacteristic
401C22#1912795811yescharacteristic
402C22#1912796311yescharacteristic
403C22#1912797511yescharacteristic
404C4—21no—
405C4—21no—
406C4—161no—
407C4—3322no—
408C4—6642no—
409C4#1912769911yescharacteristic
410C4#1912770811yescharacteristic
411C4#1912778311yescharacteristic
412C4#1912781211yescharacteristic
413C4#1912783611yescharacteristic
414C4#1912784911yescharacteristic
415C4#1912794811yescharacteristic
416C4#1912796011yescharacteristic
417C2—6642no—
418C2#2490692711yescharacteristic
419C2#2490695111yescharacteristic
420C2#2490701311yescharacteristic
421C2#2490704111yescharacteristic
422C2#2490781211yescharacteristic
423C2#2490789511yescharacteristic
424C2#2490797911yescharacteristic
subgroupG/H
centerC23#14581765
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1296C2
socleC23×C83#121
prime radicalC22×D332D4
cosocleC2×C42×C83C23
hypercenter#1296D83
hypocenterC83#9227
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #24908264C2semidirect
5. C2×#19127448C2semidirect
6. #24905297C2semidirect
7. C83×#1296C2semidirect
12. #19127982C4semidirect
218. #1296D83semidirect
220. C83#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#191279822
C2×#191277782
#249082642
#249074602
C2×#191274482
#249052972
C83×#12962
#922783