Finite Group Databasebeta

← #35320464

Subgroups — #35320464

Order 21248.

422 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916234C211yesmaximal, characteristic
2C2×Dic83⋊C16C211yesmaximal, characteristic, complemented
3C2×#19127509C211yesmaximal, characteristic
4C2×#19127507C211yesmaximal, characteristic
5#24907633C211yesmaximal, characteristic, complemented
6#24905558C211yesmaximal, characteristic, complemented
7C83×#1382C211yesmaximal, characteristic, complemented
8C2×#14581814C2211yescharacteristic
9#19128169—21no—
10C22×C83⋊C16C2211yescharacteristic
11#19127795C411yescharacteristic
12#19127795C411yescharacteristic
13#19127788—21no—
14#19127788—21no—
15#19127783C411yescharacteristic
16#19127783C411yescharacteristic
17C2×#14581728C2211yescharacteristic
18C2×#14581727—21no—
19#19127724—21no—
20#19127695—21no—
21C2×C8×Dic83C2211yescharacteristic
22Dic83⋊C16—21no—
23Dic83⋊C16—21no—
24#19127509—22no—
25#19127507—22no—
26C22×C16×C83C2211yescharacteristic
27C2×#108×C83C2211yescharacteristic
28#427×C83C2211yescharacteristic
29C2×#13050590—21no—
30#14581814—21no—
31#14581814—21no—
32#14581814D412yes—
33#14581814Q812yes—
34C2×Dic332C2×C411yescharacteristic
35C2×#13050579—21no—
36C2×D664C2×C411yescharacteristic
37C2×#13050574—21no—
38C22×C83⋊C8D411yescharacteristic
39#14581757—21no—
40#14581757—21no—
41#14581754—21no—
42#14581754—21no—
43(C22⋊D83)⋊C4—41no—
44(C22⋊D83)⋊C4—41no—
45#14581749—41no—
46C2×C4×Dic83D411yescharacteristic
47C42×D83—41no—
48C2×C83⋊C16—21no—
49C2×C83⋊C16—21no—
50C2×C83⋊C16—21no—
51#14581728—21no—
52#14581728—21no—
53#14581727—42no—
54Dic83⋊C8—41no—
55Dic83⋊C8—41no—
56#14581715—21no—
57#14581715—21no—
58#14581712—21no—
59#14581712—21no—
60C8×Dic83C2×C411yescharacteristic
61C8×Dic83C2×C411yescharacteristic
62C8×Dic83D411yescharacteristic
63C8×Dic83D411yescharacteristic
64C2×M16×C83—21no—
65C22×C8×C83C2311yescharacteristic
66#117×C83—21no—
67C2×C16×C83—21no—
68C2×C16×C83—21no—
69C2×C16×C83—21no—
70#108×C83—21no—
71#108×C83—21no—
72#107×C83C2×C411yescharacteristic
73#107×C83C2×C411yescharacteristic
74#106×C83—21no—
75#13050590—22no—
76#13050590—22no—
77#13050590—41no—
78#13050590—41no—
79C2×Dic166—21no—
80C2×D332—21no—
81C2×C22⋊D83—41no—
82C22×Dic83SD1611yescharacteristic
83C22×Dic83SD1611yescharacteristic
84C2×C4×D83—41no—
85C2×C4×D83—41no—
86C2×C4×D83—41no—
87Dic332—21no—
88Dic332C4⋊C412yes—
89#13050579—22no—
90#13050579—22no—
91D664—21no—
92D664C4⋊C412yes—
93#13050574—41no—
94#13050574—41no—
95C2×C83⋊C8—21no—
96C2×C83⋊C8—21no—
97C2×C83⋊C8—41no—
98C2×C83⋊C8C22⋊C411yescharacteristic
99C2×C83⋊C8C22⋊C411yescharacteristic
100Dic83⋊C4—41no—
101Dic83⋊C4—41no—
102(C2×D83)⋊C4—41no—
103(C2×D83)⋊C4—41no—
104C4×Dic83—41no—
105C4×Dic83—41no—
106C4×Dic83C22⋊C411yescharacteristic
107C4×Dic83C22⋊C411yescharacteristic
108C4×Dic83D811yescharacteristic
109C4×Dic83D811yescharacteristic
110C83⋊C16—41no—
111C22×C4×C83C2×D411yescharacteristic
112M16×C83—41no—
113M16×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83C22×C411yescharacteristic
116C2×C8×C83C2×D411yescharacteristic
117C2×C8×C83C2×D411yescharacteristic
118C2×C8×C83C2×Q811yescharacteristic
119C2×C8×C83#4111yescharacteristic
120C2×C8×C83#4111yescharacteristic
121C4⋊C4×C83—21no—
122C4⋊C4×C83—21no—
123C22⋊C4×C83—41no—
124C42×C83—41no—
125C16×C83—41no—
126C23×C83C2×SD1611yescharacteristic
127C22×D83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83C2×C22⋊C411yescharacteristic
131C2×C4×C83#12111yescharacteristic
132C2×C4×C83C22⋊Q811yescharacteristic
133C2×C4×C83C2×D811yescharacteristic
134C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C83#13711yescharacteristic
136C22⋊D83—42no—
137C22⋊D83—42no—
138C2×Dic83—22no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83#10211yescharacteristic
142C2×Dic83#10211yescharacteristic
143C2×Dic83#10211yescharacteristic
144C2×Dic83#10211yescharacteristic
145D332—22no—
146D332—41no—
147C4×D83—42no—
148C4×D83—42no—
149C4×D83—82no—
150C4×D83—82no—
151Dic166—22no—
152Dic166—41no—
153C8×C83—81no—
154C8×C83C2×C4⋊C411yescharacteristic
155C8×C83#11711yescharacteristic
156C8×C83#12111yescharacteristic
157C8×C83C22⋊Q811yescharacteristic
158C83⋊C8—21no—
159C83⋊C8—21no—
160C83⋊C8—81no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83C2×#10211yescharacteristic
164C22×C83#46111yescharacteristic
165C22×C83#47611yescharacteristic
166C2×D83—42no—
167C2×D83—81no—
168C2×D83—81no—
169C4×C83—81no—
170C4×C83—81no—
171C4×C83#38011yescharacteristic
172C4×C83#41711yescharacteristic
173C4×C83#45911yescharacteristic
174C4×C83#47411yescharacteristic
175Dic83—21no—
176Dic83—21no—
177Dic83—21no—
178Dic83—42no—
179Dic83#35912yescomplemented
180#12815—831nomaximal, Sylow
181C2×C83—21no—
182C2×C83—21no—
183C2×C83#180211yescharacteristic
184C2×C83C2×#35911yescharacteristic
185C2×C83#214311yescharacteristic
186D83—82no—
187C4×#135—831no—
188C2×C4⋊C16—831no—
189C2×#359—831no—
190C2×#359—831no—
191#1848—831no—
192#1382—831no—
193#1382D8311yescharacteristic, complemented
194C83#1281511yescharacteristic, complemented, Sylow
195C2×#135—831no—
196C4×#41—1661no—
197C22×C16—831no—
198C22×C16C2×D8311yescharacteristic
199C4×Q16—831no—
200C4×Q16—831no—
201C4×SD16—1661no—
202C4×SD16—1661no—
203C4×D8—831no—
204C4×D8—831no—
205C2×#108—831no—
206C2×#108—1661no—
207C2×#108C2×D8311yescharacteristic
208#427C2×D8311yescharacteristic
209#423—1661no—
210#416—1661no—
211C2×C4×C8—831no—
212C4⋊C16—1661no—
213C4⋊C16—1661no—
214#359—1662no—
215#359—1662no—
216C2×#41—1661no—
217#135—832no—
218#135—832no—
219#135—1661no—
220#135—1661no—
221C2×Q16—831no—
222C2×SD16—1661no—
223C2×D8—831no—
224C2×M16—21no—
225C2×M16—1661no—
226C22×C8—831no—
227C22×C8C22×D8311yescharacteristic
228C4×Q8—1661no—
229C4×Q8—1661no—
230C4×D4—3321no—
231C4×D4—3321no—
232C4×D4—1661no—
233C4×D4—1661no—
234#117—21no—
235#117—3321no—
236C2×C42—3321no—
237C2×C42—831no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—1661no—
242C2×C16—1661no—
243C2×C16—1661no—
244#108—21no—
245#108—21no—
246#108—3322no—
247#108—1661no—
248#108—1661no—
249#107C4×D8311yescharacteristic
250#107C4×D8311yescharacteristic
251#106—21no—
252C4⋊C8—3321no—
253C4⋊C8—3321no—
254#103—1661no—
255#103—1661no—
256#102—1661no—
257#102—1661no—
258C4×C8—831no—
259C4×C8—831no—
260C4×C8—831no—
261C4×C8—831no—
262#41—3321no—
263#41—3321no—
264#41—1662no—
265#41—1662no—
266C2×Q8—1661no—
267C2×D4—3321no—
268C2×D4—1661no—
269C22×C4—3321no—
270C22×C4—3321no—
271C22×C4—3321no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4D4×D8311yescharacteristic
275Q16—832no—
276Q16—1661no—
277SD16—1662no—
278SD16—1662no—
279D8—832no—
280D8—1661no—
281M16—41no—
282M16—41no—
283M16—3321no—
284M16—3321no—
285C2×C8—41no—
286C2×C8—3321no—
287C2×C8—831no—
288C2×C8—831no—
289C2×C8—1661no—
290C2×C8—1661no—
291C2×C8C2×C4×D8311yescharacteristic
292C2×C8D4×D8311yescharacteristic
293C2×C8D4×D8311yescharacteristic
294C2×C8Q8×D8311yescharacteristic
295C2×C8#1305059011yescharacteristic
296C2×C8#1305059011yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—3321no—
300C4⋊C4—3321no—
301C22⋊C4—41no—
302C22⋊C4—3321no—
303C22⋊C4—3321no—
304C42—41no—
305C42—3321no—
306C42—3321no—
307C42—831no—
308C42—831no—
309C42—831no—
310C42—831no—
311C16—41no—
312C16—3321no—
313C23—3321no—
314C23SD16×D8311yescharacteristic
315Q8—3321no—
316Q8—1662no—
317D4—3322no—
318D4—3322no—
319D4—3321no—
320D4—1662no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—3321no—
324C2×C4—3322no—
325C2×C4—3322no—
326C2×C4—3321no—
327C2×C4—831no—
328C2×C4—831no—
329C2×C4—831no—
330C2×C4—831no—
331C2×C4—6642no—
332C2×C4—6642no—
333C2×C4—1662no—
334C2×C4C22⋊C4×D8311yescharacteristic
335C2×C4#1458176511yescharacteristic
336C2×C4#1458177011yescharacteristic
337C2×C4D8×D8311yescharacteristic
338C2×C4#1458181911yescharacteristic
339C2×C4#1458182411yescharacteristic
340C8—81no—
341C8—6641no—
342C8—1661no—
343C8—1661no—
344C8C4⋊C4×D8311yescharacteristic
345C8#1458174911yescharacteristic
346C8#1458176511yescharacteristic
347C8#1458177011yescharacteristic
348C22—21no—
349C22—21no—
350C22—3322no—
351C22—6641no—
352C22—6641no—
353C22#102×D8311yescharacteristic
354C22#1912790511yescharacteristic
355C22#1912799011yescharacteristic
356C4—81no—
357C4—81no—
358C4—3322no—
359C4—832no—
360C4—1661no—
361C4—1661no—
362C4—1661no—
363C4#1912756011yescharacteristic
364C4#1912770111yescharacteristic
365C4#1912789511yescharacteristic
366C4#1912798011yescharacteristic
367C2—21no—
368C2—21no—
369C2—6642no—
370C2#2490746211yescharacteristic
371C2#359×D8311yescharacteristic
372C2#2490918911yescharacteristic
subgroupG/H
centerC2×C4D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1382C2
socleC22×C83C2×#102
prime radicalC2×#14581814C22
cosocleC22×C8×C83C23
hypercenter#1382D83
hypocenterC83#12815
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #1382

Complements

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

NH ≅ G/Ntype
2. C2×Dic83⋊C16C2semidirect
5. #24907633C2semidirect
6. #24905558C2semidirect
7. C83×#1382C2semidirect
179. Dic83#359semidirect
193. #1382D83semidirect
194. C83#12815semidirect

Maximal subgroups

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

subgroupindex
#249162342
C2×Dic83⋊C162
C2×#191275092
C2×#191275072
#249076332
#249055582
C83×#13822
#1281583