Finite Group Databasebeta

← #35301771

Subgroups — #35301771

Order 21248.

506 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916543C211yesmaximal, characteristic
2#24912452C211yesmaximal, characteristic
3#24907456C211yesmaximal, characteristic
4#24906135C211yesmaximal, characteristic
5#24905365C211yesmaximal, characteristic
6#24905364C211yesmaximal, characteristic
7C83×#1546C211yesmaximal, characteristic
8C2×#14581790—21no—
9C2×C4×Dic166C2211yescharacteristic
10C8×Dic166—22no—
11#19127825—22no—
12#19127754C2211yescharacteristic
13#19127751—22no—
14#19127725C2211yescharacteristic
15C2×#14581710—21no—
16#19127632C414yes—
17#19127608—21no—
18#19127597—22no—
19C4×C4⋊Dic83C2211yescharacteristic
20#19127469—21no—
21#19127447—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25#332×C83—22no—
26C22×Dic166D411yescharacteristic
27C2×#13050574—21no—
28#14581790—42no—
29#14581790—42no—
30C4×Dic166—21no—
31C4×Dic166—21no—
32C4×Dic166—21no—
33C4×Dic166—21no—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83D411yescharacteristic
36C2×Dic83⋊C4—24no—
37C2×Dic83⋊C4C2×C412yes—
38C2×C4×Dic83—21no—
39C2×C4×Dic83D412yes—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)D412yes—
43Dic83⋊C8—24no—
44#14581710—42no—
45C22⋊(C83⋊C8)—21no—
46C4×C83⋊C8—21no—
47C8×Dic83—24no—
48C83⋊#95—24no—
49C83⋊#95C2×C412yes—
50C22×Q8×C83—21no—
51C2×M16×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83C2×C412yes—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83D412yes—
62C22⋊C8×C83—21no—
63C4×C8×C83—21no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166—21no—
67C2×Dic166—21no—
68C2×Dic166D812yes—
69C2×Dic166SD1612yes—
70C22×Dic83—22no—
71C22×Dic83C22⋊C412yes—
72C22×Dic83C22⋊C412yes—
73#13050574—41no—
74#13050574—41no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—41no—
77C4⋊Dic83—24no—
78C4⋊Dic83—21no—
79C4⋊Dic83—21no—
80Dic83⋊C4—22no—
81Dic83⋊C4—22no—
82Dic83⋊C4—48no—
83C4×Dic83—22no—
84C4×Dic83—22no—
85C4×Dic83—41no—
86C4×Dic83—41no—
87C2×Q8×C83—41no—
88C2×Q8×C83—41no—
89C2×Q8×C83—41no—
90C2×Q8×C83—41no—
91C2×Q8×C83—41no—
92C2×Q8×C83—41no—
93C22×C4×C83—22no—
94C22×C4×C83C22×C411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97M16×C83—41no—
98M16×C83—41no—
99C2×C8×C83—22no—
100C2×C8×C83—41no—
101C4⋊C4×C83—22no—
102C4⋊C4×C83—22no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C42×C83C2×D411yescharacteristic
106C42×C83C2×D411yescharacteristic
107C42×C83#4111yescharacteristic
108C42×C83#4111yescharacteristic
109C23×C83C2×C22⋊C411yescharacteristic
110Q8×C83—81no—
111Q8×C83—81no—
112Q8×C83—81no—
113Q8×C83—81no—
114C2×C4×C83—22no—
115C2×C4×C83—41no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83C4×D411yescharacteristic
120C2×C4×C83C4×D411yescharacteristic
121C2×C4×C83#12011yescharacteristic
122C2×C4×C83#12111yescharacteristic
123C2×C4×C83#12111yescharacteristic
124C2×C4×C83#12311yescharacteristic
125C2×C4×C83C2×D811yescharacteristic
126C2×C4×C83C2×SD1611yescharacteristic
127C2×C4×C83#13511yescharacteristic
128C2×C4×C83#13511yescharacteristic
129C2×Dic83—22no—
130C2×Dic83—22no—
131C2×Dic83—24no—
132C2×Dic83—22no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83#10214yes—
137Dic166—22no—
138Dic166—22no—
139Dic166—22no—
140Dic166—22no—
141C8×C83—42no—
142C8×C83—81no—
143C83⋊C8—42no—
144C83⋊C8—81no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83#38611yescharacteristic
148C22×C83C2×#10211yescharacteristic
149C22×C83#41611yescharacteristic
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—82no—
154C4×C83#45612yes—
155C4×C83#46511yescharacteristic
156C4×C83#46511yescharacteristic
157C4×C83#46611yescharacteristic
158C4×C83#46611yescharacteristic
159Dic83—24no—
160Dic83—44no—
161Dic83—42no—
162#9394—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83#188611yescharacteristic
166C2×C83#189311yescharacteristic
167C2×C83#191511yescharacteristic
168#2982—831no—
169#2546—831no—
170#1801—831no—
171#1546—831no—
172#1546D8311yescharacteristic
173#1303—831no—
174#1303—831no—
175C83#939411yescharacteristic, complemented, Sylow
176C2×C4⋊Q8—1661no—
177C2×C4⋊Q8C2×D8311yescharacteristic
178C2×C4×Q8—831no—
179C8×Q8—1662no—
180C8×Q8—1662no—
181#431—831no—
182#431—1662no—
183#431C2×D8311yescharacteristic
184#423—831no—
185#423C2×D8311yescharacteristic
186C2×#101—1661no—
187#398—834no—
188#391—1661no—
189#389—1662no—
190C4×C4⋊C4—831no—
191#343—1661no—
192#332—22no—
193#332—1662no—
194C22×Q8—21no—
195C22×Q8—831no—
196C2×M16—21no—
197C2×M16—1661no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—3322no—
203C4⋊Q8—3322no—
204C4×Q8—1661no—
205C4×Q8—1661no—
206C4×Q8—1661no—
207C4×Q8—1661no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—832no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—1664no—
212C2×C4⋊C4—1661no—
213C2×C4⋊C4C4×D8312yes—
214C2×C42—832no—
215C2×C42—1661no—
216C2×C42C22×D8311yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—832no—
220C4⋊C8—1664no—
221C4⋊C8—1661no—
222C4⋊C8—1661no—
223C4⋊C8C22⋊D8312yes—
224#101—3322no—
225C22⋊C8—21no—
226C22⋊C8—1661no—
227C4×C8—21no—
228C4×C8—1664no—
229C4×C8—1661no—
230#95—832no—
231#95—1664no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—832no—
239C2×Q8—832no—
240C2×Q8—1661no—
241C2×Q8—1661no—
242C2×Q8—1661no—
243C2×Q8—1661no—
244C22×C4—22no—
245C22×C4—832no—
246C22×C4—832no—
247C22×C4—1662no—
248C22×C4C2×C4×D8311yescharacteristic
249C22×C4D4×D8311yescharacteristic
250C22×C4D4×D8311yescharacteristic
251M16—41no—
252M16—41no—
253M16—3321no—
254M16—3321no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—3321no—
258C2×C8—1662no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—3328no—
264C4⋊C4—1664no—
265C4⋊C4—1661no—
266C4⋊C4—1661no—
267C4⋊C4—1662no—
268C4⋊C4—1662no—
269C42—3321no—
270C42—3321no—
271C42—1662no—
272C42—1662no—
273C42D4×D8311yescharacteristic
274C42C2×C22⋊D8311yescharacteristic
275C42#1305058911yescharacteristic
276C42#1305059011yescharacteristic
277C23C22⋊C4×D8311yescharacteristic
278Q8—81no—
279Q8—81no—
280Q8—81no—
281Q8—81no—
282Q8—1662no—
283Q8—1662no—
284Q8—1662no—
285Q8—1662no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—834no—
295C2×C4—1664no—
296C2×C4—1662no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4(C22⋊D83)⋊C411yescharacteristic
300C2×C4C4×C22⋊D8311yescharacteristic
301C2×C4#1458176011yescharacteristic
302C2×C4#1458176311yescharacteristic
303C2×C4#1458176511yescharacteristic
304C2×C4#1458177411yescharacteristic
305C2×C4D8×D8311yescharacteristic
306C2×C4SD16×D8311yescharacteristic
307C2×C4#1458181111yescharacteristic
308C2×C4#1458181311yescharacteristic
309C8—42no—
310C8—81no—
311C8—3322no—
312C8—6641no—
313C22—21no—
314C22—21no—
315C22#1912757711yescharacteristic
316C22#102×D8311yescharacteristic
317C22#1912769411yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—82no—
322C4—3324no—
323C4—3322no—
324C4—1664no—
325C4#1912788212yes—
326C4#1912792711yescharacteristic
327C4#1912792911yescharacteristic
328C4#1912793411yescharacteristic
329C4#1912793611yescharacteristic
330C2—21no—
331C2—21no—
332C2#2490785311yescharacteristic
333C2#2490788811yescharacteristic
334C2#2490801211yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1546C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1546D83
hypocenterC83#9394
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
175. C83#9394semidirect

Maximal subgroups

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

subgroupindex
#249165432
#249124522
#249074562
#249061352
#249053652
#249053642
C83×#15462
#939483