Finite Group Databasebeta

← #35321491

Subgroups — #35321491

Order 21248.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916380C211yesmaximal, characteristic, complemented
2#24909199C211yesmaximal, characteristic, complemented
3#24909191C211yesmaximal, characteristic, complemented
4#24908362C211yesmaximal, characteristic, complemented
5#24905559C211yesmaximal, characteristic, complemented
6#24905551C211yesmaximal, characteristic, complemented
7C2×#508×C83C211yesmaximal, characteristic
8#19128468—21no—
9C2×#14581732C2211yescharacteristic
10#19127823—22no—
11#19127818—22no—
12C2×#14581728C2211yescharacteristic
13#19127739C2211yescharacteristic
14#19127718—21no—
15#19127718—21no—
16#19127675—21no—
17#19127656C2211yescharacteristic
18#19127511—22no—
19#19127508—22no—
20C2×#135×C83C2211yescharacteristic
21C2×#135×C83C2211yescharacteristic
22#508×C83—22no—
23#508×C83—22no—
24C2×Q32×C83C411yescharacteristic, complemented
25C2×SD32×C83C411yescharacteristic, complemented
26C2×SD32×C83C411yescharacteristic, complemented
27C2×D16×C83C411yescharacteristic, complemented
28C22×C16×C83C2211yescharacteristic
29#14581799—24no—
30#14581799—42no—
31C2×#13050574—21no—
32C2×#13050574—42no—
33C2×#13050574D411yescharacteristic
34#14581751—21no—
35#14581751D411yescharacteristic
36#14581732—21no—
37#14581732—21no—
38C2×C83⋊C16—21no—
39#14581728—21no—
40#14581728D412yes—
41#14581726—21no—
42#14581724—21no—
43#14581719—24no—
44#14581719—42no—
45#14581707—42no—
46#14581702—21no—
47C8×Dic83—21no—
48C2×#41×C83—21no—
49C2×#41×C83—21no—
50#135×C83—21no—
51#135×C83—21no—
52#135×C83—21no—
53#135×C83—21no—
54#135×C83—21no—
55#135×C83—21no—
56#135×C83D412yes—
57#135×C83D412yes—
58C2×Q16×C83C2×C411yescharacteristic
59C2×Q16×C83C2×C411yescharacteristic
60C2×SD16×C83—21no—
61C2×SD16×C83—21no—
62C2×D8×C83C2×C411yescharacteristic
63C2×D8×C83C2×C411yescharacteristic
64C22×C8×C83C2311yescharacteristic
65Q32×C83—22no—
66SD32×C83—22no—
67SD32×C83—22no—
68D16×C83—22no—
69C2×C16×C83—21no—
70C2×C16×C83—21no—
71C2×C16×C83C2×C411yescharacteristic
72C2×C16×C83C2×C411yescharacteristic
73#13050574—21no—
74#13050574—22no—
75#13050574—44no—
76#13050574—41no—
77#13050574D812yes—
78C2×C83⋊C8—21no—
79C2×C83⋊C8—44no—
80C2×C83⋊C8—41no—
81C4⋊Dic83—21no—
82C4⋊Dic83—41no—
83C22⋊Dic83—21no—
84C22⋊Dic83—41no—
85C4×Dic83—21no—
86C4×Dic83—41no—
87C83⋊C16—41no—
88#41×C83—22no—
89#41×C83—22no—
90#41×C83—41no—
91#41×C83—41no—
92#41×C83—41no—
93#41×C83—41no—
94#41×C83—41no—
95#41×C83—41no—
96C2×Q8×C83—21no—
97C2×Q8×C83—21no—
98C2×D4×C83—21no—
99C2×D4×C83—21no—
100C2×D4×C83—41no—
101C2×D4×C83—41no—
102C22×C4×C83—41no—
103C22×C4×C83—41no—
104C22×C4×C83C2×D411yescharacteristic
105Q16×C83—21no—
106Q16×C83—21no—
107Q16×C83—21no—
108Q16×C83C22⋊C412yes—
109SD16×C83—22no—
110SD16×C83—22no—
111SD16×C83—41no—
112SD16×C83—41no—
113D8×C83—21no—
114D8×C83—21no—
115D8×C83—21no—
116D8×C83C22⋊C412yes—
117C2×C8×C83C22×C411yescharacteristic
118C2×C8×C83C2×D411yescharacteristic
119C2×C8×C83C2×D411yescharacteristic
120C2×C8×C83C2×D411yescharacteristic
121C2×C8×C83#4111yescharacteristic
122C2×C8×C83#4111yescharacteristic
123C16×C83—21no—
124C16×C83—21no—
125C23×C83—41no—
126C23×C83—41no—
127C23×C83AGL(1,ℤ/8)11yescharacteristic
128Q8×C83—22no—
129Q8×C83—41no—
130Q8×C83—41no—
131Q8×C83—41no—
132D4×C83—22no—
133D4×C83—42no—
134D4×C83—42no—
135D4×C83—41no—
136D4×C83—41no—
137D4×C83—41no—
138D4×C83—81no—
139D4×C83—81no—
140C2×C4×C83—41no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83—41no—
144C2×C4×C83—81no—
145C2×C4×C83—81no—
146C2×C4×C83C4×D411yescharacteristic
147C2×C4×C83#12011yescharacteristic
148C2×C4×C83#12111yescharacteristic
149C2×C4×C83C2×D811yescharacteristic
150C2×C4×C83#13511yescharacteristic
151C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
152C2×Dic83—41no—
153C2×Dic83—81no—
154C8×C83C2×C22⋊C411yescharacteristic
155C8×C83C4×D411yescharacteristic
156C8×C83#12111yescharacteristic
157C8×C83#12311yescharacteristic
158C83⋊C8—22no—
159C83⋊C8—42no—
160C22×C83—41no—
161C22×C83—42no—
162C22×C83—81no—
163C22×C83—161no—
164C22×C83—161no—
165C22×C83#44211yescharacteristic
166C22×C83#44911yescharacteristic
167C22×C83#45911yescharacteristic
168C4×C83—42no—
169C4×C83—81no—
170C4×C83#38611yescharacteristic
171C4×C83C4×D811yescharacteristic
172C4×C83#44711yescharacteristic
173C4×C83#46311yescharacteristic
174Dic83—81no—
175Dic83—161no—
176#13016—831nomaximal, Sylow
177C2×C83—21no—
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—82no—
181C2×C83—161no—
182C2×C83#187911yescharacteristic
183C2×#508D8311yescharacteristic
184#2957—831no—
185#2145—831no—
186#2143—831no—
187#1976—831no—
188#1382—831no—
189#1379—831no—
190C83#1301611yescharacteristic, complemented, Sylow
191C2×#135C2×D8311yescharacteristic
192C2×#135C2×D8311yescharacteristic
193#568—1661no—
194#508—22no—
195#508—22no—
196C2×Q32Dic8311yescharacteristic, complemented
197C2×SD32Dic8311yescharacteristic, complemented
198C2×SD32Dic8311yescharacteristic, complemented
199C2×D16Dic8311yescharacteristic, complemented
200C2×M32—831no—
201C22×C16C2×D8311yescharacteristic
202#444—1662no—
203#443—1662no—
204C2×#108—831no—
205#427—831no—
206#421—1661no—
207#421—1661no—
208#410—1661no—
209#405—831no—
210#360—1662no—
211#359—1662no—
212C2×#41—21no—
213C2×#41—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135C22⋊D8312yes—
221#135C22⋊D8312yes—
222C2×Q16C2×Dic8311yescharacteristic
223C2×Q16C2×Dic8311yescharacteristic
224C2×SD16—21no—
225C2×SD16—21no—
226C2×D8C2×Dic8311yescharacteristic
227C2×D8C2×Dic8311yescharacteristic
228#131—3322no—
229#131—1664no—
230C2×M16—3322no—
231C2×M16—831no—
232C2×M16—1661no—
233C22×C8C22×D8311yescharacteristic
234#117—831no—
235#117—1661no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—1661no—
241M32—1661no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—1661no—
245C2×C16C2×Dic8311yescharacteristic
246C2×C16C2×Dic8311yescharacteristic
247#108—832no—
248#108—1661no—
249#107—1661no—
250#106—1661no—
251#104—3322no—
252#104—1664no—
253#99—3322no—
254#97—1661no—
255C4×C8—1661no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4D4×D8311yescharacteristic
273Q16—21no—
274Q16—21no—
275Q16—21no—
276Q16C22⋊Dic8312yes—
277SD16—22no—
278SD16—22no—
279SD16—41no—
280SD16—41no—
281D8—21no—
282D8—21no—
283D8—21no—
284D8C22⋊Dic8312yes—
285M16—3321no—
286M16—3324no—
287M16—832no—
288M16—1662no—
289M16—1661no—
290C2×C8—3321no—
291C2×C8—3324no—
292C2×C8—1661no—
293C2×C8C22×Dic8311yescharacteristic
294C2×C8D4×D8311yescharacteristic
295C2×C8C2×C22⋊D8311yescharacteristic
296C2×C8C2×C22⋊D8311yescharacteristic
297C2×C8#1305058911yescharacteristic
298C2×C8#1305058911yescharacteristic
299C4⋊C4—3321no—
300C4⋊C4—1661no—
301C22⋊C4—3321no—
302C22⋊C4—1661no—
303C42—3321no—
304C42—1661no—
305C16—21no—
306C16—21no—
307C16—3321no—
308C23—41no—
309C23—41no—
310C23#1458181611yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3321no—
330C2×C4—6641no—
331C2×C4D4×Dic8311yescharacteristic
332C2×C4#1458176011yescharacteristic
333C2×C4#1458176411yescharacteristic
334C2×C4D8×D8311yescharacteristic
335C2×C4#1458181111yescharacteristic
336C2×C4#1458181611yescharacteristic
337C8—3322no—
338C8—1662no—
339C8C2×C22⋊Dic8311yescharacteristic
340C8D4×Dic8311yescharacteristic
341C8#1458176411yescharacteristic
342C8#1458177311yescharacteristic
343C22—41no—
344C22—42no—
345C22—81no—
346C22—161no—
347C22—161no—
348C22#1912781311yescharacteristic
349C22#1912784311yescharacteristic
350C22#1912789411yescharacteristic
351C4—42no—
352C4—81no—
353C4—13281no—
354C4—6641no—
355C4#1912758111yescharacteristic
356C4D8×Dic8311yescharacteristic
357C4#1912783311yescharacteristic
358C4#1912791411yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—161no—
364C2#2490781311yescharacteristic
subgroupG/H
centerC4#19127833
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
Fitting, prime radicalC2×#508×C83C2
socleC2×C83#1879
cosocleC22×C8×C83C23
hypercenterC2×#508D83
hypocenterC83#13016
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. #24916380C2semidirect
2. #24909199C2semidirect
3. #24909191C2semidirect
4. #24908362C2semidirect
5. #24905559C2semidirect
6. #24905551C2semidirect
24. C2×Q32×C83C4semidirect
25. C2×SD32×C83C4semidirect
26. C2×SD32×C83C4semidirect
27. C2×D16×C83C4semidirect
190. C83#13016semidirect
196. C2×Q32Dic83semidirect
197. C2×SD32Dic83semidirect
198. C2×SD32Dic83semidirect
199. C2×D16Dic83semidirect

Maximal subgroups

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

subgroupindex
#249163802
#249091992
#249091912
#249083622
#249055592
#249055512
C2×#508×C832
#1301683