Finite Group Databasebeta

← #35300489

Subgroups — #35300489

Order 21248.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C83⋊C8C211yesmaximal, characteristic
2#24907483C211yesmaximal, characteristic
3#24905962C211yesmaximal, characteristic, complemented
4#24905918C211yesmaximal, characteristic, complemented
5#24905360C211yesmaximal, characteristic, complemented
6#24905302C211yesmaximal, characteristic, complemented
7C83×#3076C211yesmaximal, characteristic
8Q8×C83⋊C8—21no—
9D4×C83⋊C8—22no—
10D4×C83⋊C8—21no—
11#19127754—21no—
12#19127754C2211yescharacteristic
13#19127725C2211yescharacteristic
14C2×#14581719—21no—
15C2×C4×C83⋊C8—21no—
16C2×C4×C83⋊C8C2211yescharacteristic
17C4×C4⋊Dic83C2211yescharacteristic
18#19127460—21no—
19#19127438—22no—
20Q8⋊(C83⋊C8)—21no—
21Q8⋊(C83⋊C8)—21no—
22#533×C83C2211yescharacteristic
23C4×#41×C83C2211yescharacteristic
24#479×C83—21no—
25#478×C83—21no—
26#477×C83C411yescharacteristic, complemented
27#476×C83C411yescharacteristic, complemented
28#475×C83C411yescharacteristic, complemented
29#474×C83C411yescharacteristic, complemented
30#427×C83—21no—
31C2×C4⋊C8×C83C2211yescharacteristic
32#416×C83—21no—
33C2×#13050574—21no—
34C22×C83⋊C8—22no—
35C22×C83⋊C8—21no—
36C2×C4⋊Dic83—21no—
37C2×C4×Dic83—21no—
38C4⋊(C83⋊C8)—22no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42#14581719—42no—
43C22⋊(C83⋊C8)—22no—
44C22⋊(C83⋊C8)D412yes—
45C4×C83⋊C8—22no—
46C4×C83⋊C8—21no—
47C4×C83⋊C8—21no—
48C4×C83⋊C8D411yescharacteristic
49C4×C83⋊C8D411yescharacteristic
50C83⋊#95—21no—
51C2×#41×C83D411yescharacteristic
52C22×C8×C83—21no—
53C4⋊Q8×C83C2×C411yescharacteristic
54C4⋊Q8×C83C2×C411yescharacteristic
55#125×C83—21no—
56C22⋊Q8×C83—21no—
57C22⋊Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×Q8×C83C2×C411yescharacteristic
61C4×Q8×C83D411yescharacteristic
62C4×D4×C83—21no—
63C4×D4×C83—21no—
64C4×D4×C83C2×C411yescharacteristic
65C4×D4×C83D411yescharacteristic
66#117×C83—21no—
67#117×C83—21no—
68#117×C83D411yescharacteristic
69C2×C42×C83—21no—
70C2×C42×C83C2311yescharacteristic
71#107×C83—21no—
72#107×C83—21no—
73#106×C83—21no—
74#106×C83—21no—
75C4⋊C8×C83—21no—
76C4⋊C8×C83C2×C411yescharacteristic
77C4⋊C8×C83C2×C411yescharacteristic
78#103×C83—21no—
79#103×C83—21no—
80#102×C83—21no—
81#102×C83—21no—
82C22×Dic83—41no—
83#13050574—41no—
84#13050574—41no—
85C2×C83⋊C8—21no—
86C2×C83⋊C8—22no—
87C2×C83⋊C8—21no—
88C2×C83⋊C8—42no—
89C2×C83⋊C8—44no—
90C2×C83⋊C8—41no—
91C4⋊Dic83—22no—
92C4⋊Dic83—41no—
93C4×Dic83—41no—
94C4×Dic83—41no—
95#41×C83—21no—
96#41×C83—21no—
97#41×C83SD1612yes—
98#41×C83Q1612yes—
99C2×Q8×C83—21no—
100C2×Q8×C83—21no—
101C2×Q8×C83C22⋊C411yescharacteristic
102C2×D4×C83—21no—
103C2×D4×C83C22⋊C411yescharacteristic
104C22×C4×C83—21no—
105C22×C4×C83—21no—
106C22×C4×C83—21no—
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83C2×D411yescharacteristic
109C22×C4×C83#4111yescharacteristic
110C2×C8×C83—21no—
111C2×C8×C83—21no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C4⋊C4×C83—21no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—21no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—41no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83C22⋊C411yescharacteristic
123C4⋊C4×C83C22⋊C411yescharacteristic
124C22⋊C4×C83—21no—
125C22⋊C4×C83—21no—
126C22⋊C4×C83—21no—
127C22⋊C4×C83—41no—
128C42×C83—21no—
129C42×C83—21no—
130C42×C83—21no—
131C42×C83—41no—
132C42×C83C22×C411yescharacteristic
133C42×C83C2×D411yescharacteristic
134C42×C83C2×D411yescharacteristic
135C42×C83#4111yescharacteristic
136C23×C83—21no—
137C23×C83#12111yescharacteristic
138Q8×C83—21no—
139Q8×C83—41no—
140Q8×C83—41no—
141Q8×C83—81no—
142Q8×C83#10312yes—
143D4×C83—22no—
144D4×C83—22no—
145D4×C83—21no—
146D4×C83#10312yes—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—22no—
151C2×C4×C83—22no—
152C2×C4×C83—21no—
153C2×C4×C83—21no—
154C2×C4×C83—41no—
155C2×C4×C83—42no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—41no—
159C2×C4×C83—41no—
160C2×C4×C83C2×C22⋊C411yescharacteristic
161C2×C4×C83C4×D411yescharacteristic
162C2×C4×C83C4×D411yescharacteristic
163C2×C4×C83#12011yescharacteristic
164C2×C4×C83#12111yescharacteristic
165C2×C4×C83#12311yescharacteristic
166C2×C4×C83C2×SD1611yescharacteristic
167C2×C4×C83C2×Q1611yescharacteristic
168C2×C4×C83#13511yescharacteristic
169C2×C4×C83#13511yescharacteristic
170C2×Dic83—42no—
171C2×Dic83—81no—
172C2×Dic83—81no—
173C8×C83—41no—
174C8×C83—41no—
175C83⋊C8—22no—
176C83⋊C8—42no—
177C83⋊C8—41no—
178C83⋊C8—81no—
179C22×C83—21no—
180C22×C83—21no—
181C22×C83—22no—
182C22×C83—41no—
183C22×C83—41no—
184C22×C83#38611yescharacteristic
185C22×C83#46511yescharacteristic
186C22×C83#46711yescharacteristic
187C4×C83—21no—
188C4×C83—22no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—41no—
192C4×C83—41no—
193C4×C83—81no—
194C4×C83—81no—
195C4×C83C2×#10311yescharacteristic
196C4×C83#41611yescharacteristic
197C4×C83#44311yescharacteristic
198C4×C83#44411yescharacteristic
199C4×C83#46511yescharacteristic
200C4×C83#46711yescharacteristic
201Dic83—82no—
202#9207—831nomaximal, Sylow
203C2×C83—21no—
204C2×C83—21no—
205C2×C83—42no—
206C2×C83#187611yescharacteristic
207C2×C83#189211yescharacteristic
208C2×C83#190611yescharacteristic
209#3076D8311yescharacteristic
210C8×#41—831no—
211#1806—831no—
212#1503—831no—
213#1493—831no—
214#1301—831no—
215#1279—831no—
216C83#920711yescharacteristic, complemented, Sylow
217#533C2×D8311yescharacteristic
218C4×#41C2×D8311yescharacteristic
219#479—21no—
220#478—21no—
221#477Dic8311yescharacteristic, complemented
222#476Dic8311yescharacteristic, complemented
223#475Dic8311yescharacteristic, complemented
224#474Dic8311yescharacteristic, complemented
225C8×Q8—1661no—
226C8×D4—1662no—
227C8×D4—1661no—
228#431—831no—
229#431—1661no—
230#427—21no—
231#423—831no—
232C2×C4⋊C8C2×D8311yescharacteristic
233C2×#104—1661no—
234#416—21no—
235C2×C4×C8—831no—
236C2×C4×C8—1661no—
237C4×C4⋊C4—831no—
238#339—1661no—
239#328—1662no—
240Q8⋊C8—1661no—
241Q8⋊C8—1661no—
242C2×#41C22⋊D8311yescharacteristic
243C2×M16—1661no—
244C22×C8—21no—
245C22×C8—1662no—
246C22×C8—1661no—
247C4⋊Q8C2×Dic8311yescharacteristic
248C4⋊Q8C2×Dic8311yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C2×Dic8311yescharacteristic
255C4×Q8C22⋊D8311yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C2×Dic8311yescharacteristic
259C4×D4C22⋊D8311yescharacteristic
260#117—21no—
261#117—21no—
262#117C22⋊D8311yescharacteristic
263C2×C4⋊C4—1661no—
264C2×C42—21no—
265C2×C42—1661no—
266C2×C42C22×D8311yescharacteristic
267#107—21no—
268#107—21no—
269#106—21no—
270#106—21no—
271C4⋊C8—21no—
272C4⋊C8—1662no—
273C4⋊C8—1661no—
274C4⋊C8—1661no—
275C4⋊C8—1661no—
276C4⋊C8C2×Dic8311yescharacteristic
277C4⋊C8C2×Dic8311yescharacteristic
278#104—3322no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283C22⋊C8—832no—
284C22⋊C8—1662no—
285C4×C8—831no—
286C4×C8—831no—
287C4×C8—1662no—
288C4×C8—1661no—
289C4×C8—1661no—
290#95—1661no—
291#41—21no—
292#41—21no—
293#41#1305057712yes—
294#41#1305058012yes—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8C22⋊Dic8311yescharacteristic
298C2×D4—21no—
299C2×D4C22⋊Dic8311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—3321no—
304C22×C4D4×D8311yescharacteristic
305C22×C4C2×C22⋊D8311yescharacteristic
306C22×C4#1305058911yescharacteristic
307M16—3321no—
308M16—3321no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3322no—
314C2×C8—3324no—
315C2×C8—3321no—
316C2×C8—1661no—
317C2×C8—1662no—
318C2×C8—1661no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—3321no—
328C4⋊C4—1662no—
329C4⋊C4C22⋊Dic8311yescharacteristic
330C4⋊C4C22⋊Dic8311yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—3321no—
340C42—3321no—
341C42C22×Dic8311yescharacteristic
342C42D4×D8311yescharacteristic
343C42C2×C22⋊D8311yescharacteristic
344C42#1305058911yescharacteristic
345C23—21no—
346C23#1458176411yescharacteristic
347Q8—21no—
348Q8—41no—
349Q8—41no—
350Q8—81no—
351Q8#1458171612yes—
352D4—22no—
353D4—22no—
354D4—21no—
355D4#1458171612yes—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—3322no—
370C2×C4—6641no—
371C2×C4—6641no—
372C2×C4C2×C22⋊Dic8311yescharacteristic
373C2×C4D4×Dic8311yescharacteristic
374C2×C4D4×Dic8311yescharacteristic
375C2×C4#1458176011yescharacteristic
376C2×C4#1458176411yescharacteristic
377C2×C4#1458177311yescharacteristic
378C2×C4C2×#1305057711yescharacteristic
379C2×C4C2×#1305058011yescharacteristic
380C2×C4#1458181211yescharacteristic
381C2×C4#1458181211yescharacteristic
382C8—41no—
383C8—41no—
384C8—3321no—
385C8—3322no—
386C8—6641no—
387C8—1662no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22#1912758111yescharacteristic
394C22#1912793211yescharacteristic
395C22#1912794011yescharacteristic
396C4—21no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—6642no—
405C4C2×#1458171611yescharacteristic
406C4#1912769711yescharacteristic
407C4#1912781811yescharacteristic
408C4#1912782311yescharacteristic
409C4#1912793211yescharacteristic
410C4#1912794011yescharacteristic
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2490779211yescharacteristic
415C2#2490788711yescharacteristic
416C2#2490796611yescharacteristic
subgroupG/H
centerC2×C4#14581764
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3076C2
socleC22×C83#386
prime radicalC2×#41×C83D4
cosocleC2×C42×C83C23
hypercenter#3076D83
hypocenterC83#9207
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
3. #24905962C2semidirect
4. #24905918C2semidirect
5. #24905360C2semidirect
6. #24905302C2semidirect
26. #477×C83C4semidirect
27. #476×C83C4semidirect
28. #475×C83C4semidirect
29. #474×C83C4semidirect
216. C83#9207semidirect
221. #477Dic83semidirect
222. #476Dic83semidirect
223. #475Dic83semidirect
224. #474Dic83semidirect

Maximal subgroups

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

subgroupindex
#41×C83⋊C82
#249074832
#249059622
#249059182
#249053602
#249053022
C83×#30762
#920783