Finite Group Databasebeta

← #35323089

Subgroups — #35323089

Order 21248.

420 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 93 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#24912673C211yesmaximal, characteristic
2#24907457C211yesmaximal, characteristic
3#101×Dic83C211yesmaximal, characteristic
4#24905590C211yesmaximal, characteristic
5#24905589C211yesmaximal, characteristic
6#24905589C211yesmaximal, characteristic
7C83×#1394C211yesmaximal, characteristic
8C2×#14581791C411yescharacteristic
9C2×Q8×Dic83C2211yescharacteristic
10#19127724—21no—
11C2×#14581711C2211yescharacteristic
12C2×#14581711C2211yescharacteristic
13M16×Dic83—21no—
14#19127622—21no—
15#19127622—21no—
16#19127608C2211yescharacteristic
17#19127598—21no—
18#19127598—21no—
19#19127571C411yescharacteristic
20#19127570C411yescharacteristic
21C4×Dic83⋊C4C411yescharacteristic
22#19127469—21no—
23#19127469—21no—
24C2×#101×C83C2211yescharacteristic
25C2×#101×C83C2211yescharacteristic
26#391×C83C2211yescharacteristic
27C22×Dic166—21no—
28C2×#13050574—21no—
29C2×#13050574—21no—
30#14581791—41no—
31#14581791—41no—
32Q8×Dic83—41no—
33Q8×Dic83—41no—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83—21no—
36C2×C4⋊Dic83C2×C411yescharacteristic
37C2×C4⋊Dic83C2×C411yescharacteristic
38C2×C4⋊Dic83D411yescharacteristic
39C2×C4⋊Dic83D411yescharacteristic
40C2×Dic83⋊C4—21no—
41C2×Dic83⋊C4—21no—
42C2×Dic83⋊C4—21no—
43C2×Dic83⋊C4—21no—
44C2×C4×Dic83C2×C411yescharacteristic
45C2×C4×Dic83C2×C411yescharacteristic
46C2×C4×Dic83D411yescharacteristic
47C2×C4×Dic83D411yescharacteristic
48Dic83⋊C8—41no—
49Dic83⋊C8—41no—
50#14581711—21no—
51#14581711—21no—
52#14581711—21no—
53#14581711Q812yescomplemented
54#14581702—41no—
55C8×Dic83—41no—
56C83⋊#95—21no—
57C83⋊#95—21no—
58C83⋊#95—21no—
59C22×Q8×C83C2311yescharacteristic
60C2×M16×C83—21no—
61C2×M16×C83—21no—
62C2×C4⋊C4×C83C2×C411yescharacteristic
63C2×C42×C83C2×C411yescharacteristic
64#101×C83—21no—
65#101×C83—21no—
66#101×C83—21no—
67#101×C83Q812yescomplemented
68#95×C83—21no—
69#95×C83—21no—
70C2×Dic166—42no—
71C2×Dic166—81no—
72C2×Dic166—81no—
73C22×Dic83—21no—
74C22×Dic83—21no—
75C22×Dic83C22⋊C411yescharacteristic
76C22×Dic83C22⋊C411yescharacteristic
77C22×Dic83C22⋊C411yescharacteristic
78C22×Dic83C22⋊C411yescharacteristic
79#13050574—22no—
80#13050574—41no—
81#13050574—41no—
82#13050574—41no—
83C2×C83⋊C8—41no—
84C2×C83⋊C8—41no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87C4⋊Dic83—41no—
88C4⋊Dic83—41no—
89C4⋊Dic83—81no—
90C4⋊Dic83—81no—
91Dic83⋊C4—24no—
92Dic83⋊C4—41no—
93Dic83⋊C4—41no—
94Dic83⋊C4—81no—
95Dic83⋊C4—81no—
96C4×Dic83—21no—
97C4×Dic83—21no—
98C4×Dic83—21no—
99C4×Dic83—21no—
100C4×Dic83—21no—
101C4×Dic83—21no—
102C4×Dic83—41no—
103C2×Q8×C83—41no—
104C2×Q8×C83—41no—
105C2×Q8×C83C2×Q811yescharacteristic
106C2×Q8×C83C2×Q811yescharacteristic
107C2×Q8×C83#4111yescharacteristic
108C2×Q8×C83#4111yescharacteristic
109C22×C4×C83—21no—
110C22×C4×C83—21no—
111C22×C4×C83C22×C411yescharacteristic
112C22×C4×C83C2×D411yescharacteristic
113C22×C4×C83C2×D411yescharacteristic
114M16×C83—22no—
115M16×C83—41no—
116M16×C83—41no—
117M16×C83—41no—
118C2×C8×C83—41no—
119C2×C8×C83—41no—
120C4⋊C4×C83—41no—
121C42×C83—41no—
122C23×C83C2×C22⋊C411yescharacteristic
123Q8×C83—41no—
124Q8×C83—41no—
125Q8×C83—41no—
126Q8×C83—41no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—22no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83C4×Q811yescharacteristic
140C2×C4×C83C4×Q811yescharacteristic
141C2×C4×C83C22⋊Q811yescharacteristic
142C2×C4×C83C22⋊Q811yescharacteristic
143C2×C4×C83#12411yescharacteristic
144C2×C4×C83C4⋊Q811yescharacteristic
145C2×Dic83—22no—
146C2×Dic83—22no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—41no—
151C2×Dic83—41no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83#9911yescharacteristic
159C2×Dic83#9911yescharacteristic
160Dic166—84no—
161C8×C83—42no—
162C8×C83—81no—
163C83⋊C8—42no—
164C83⋊C8—81no—
165C22×C83—21no—
166C22×C83—21no—
167C22×C83#39111yescharacteristic
168C22×C83C2×#9911yescharacteristic
169C22×C83#41011yescharacteristic
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—42no—
177C4×C83—81no—
178Dic83—21no—
179Dic83—42no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183Dic83—81no—
184Dic83#35612yes—
185#13302—831nomaximal, Sylow
186C2×C83—21no—
187C2×C83—21no—
188C2×C83#199611yescharacteristic
189C2×C83C2×#35611yescharacteristic
190C2×C83#213411yescharacteristic
191#2567—831no—
192#1801—831no—
193C4×#101—831no—
194#1394—831no—
195#1394—831no—
196#1394—831no—
197#1394D8311yescharacteristic
198C83#1330211yescharacteristic, complemented, Sylow
199C2×C4⋊Q8—831no—
200C2×C4×Q8—831no—
201#423—1661no—
202C2×#101—831no—
203C2×#101—831no—
204C2×#101C2×D8311yescharacteristic
205C2×#101C2×D8311yescharacteristic
206C4×M16—1661no—
207#395—1661no—
208#395—1661no—
209#391—831no—
210#391C2×D8311yescharacteristic
211#389—1661no—
212#389—1661no—
213#384—831no—
214#384—831no—
215C4×C4⋊C4—831no—
216#343—1661no—
217#343—1661no—
218C22×Q8—1661no—
219C22×Q8C22×D8311yescharacteristic
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—1661no—
223C2×M16—1661no—
224C4⋊Q8—3321no—
225C4⋊Q8—3321no—
226C4×Q8—3321no—
227C4×Q8—3321no—
228C2×C4⋊C4—831no—
229C2×C4⋊C4—831no—
230C2×C4⋊C4—831no—
231C2×C4⋊C4—831no—
232C2×C4⋊C4—1661no—
233C2×C4⋊C4—1661no—
234C2×C4⋊C4—1661no—
235C2×C4⋊C4—1661no—
236C2×C4⋊C4—1661no—
237C2×C4⋊C4—1661no—
238C2×C4⋊C4C4×D8311yescharacteristic
239C2×C42—831no—
240C2×C42—831no—
241C2×C42—831no—
242C2×C42—831no—
243C2×C42C4×D8311yescharacteristic
244C4⋊C8—3321no—
245C4⋊C8—3321no—
246#101—21no—
247#101—21no—
248#101—21no—
249#101—832no—
250#101—1661no—
251#101—1661no—
252#101—1661no—
253#101Dic16612yescomplemented
254#97—3321no—
255C4×C8—3321no—
256#95—21no—
257#95—21no—
258#95—1661no—
259#95—1661no—
260#95—1661no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—3322no—
264C2×Q8—6641no—
265C2×Q8—6641no—
266C2×Q8Q8×D8311yescharacteristic
267C2×Q8C2×Dic16611yescharacteristic
268C2×Q8#1305058911yescharacteristic
269C2×Q8#1305059011yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—1661no—
277C22×C4—1661no—
278C22×C4C2×C4×D8311yescharacteristic
279C22×C4D4×D8311yescharacteristic
280C22×C4D4×D8311yescharacteristic
281M16—22no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—3321no—
286M16—3321no—
287M16—3321no—
288M16—1662no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—3321no—
292C2×C8—3321no—
293C4⋊C4—41no—
294C4⋊C4—3321no—
295C4⋊C4—3321no—
296C4⋊C4—3321no—
297C4⋊C4—3321no—
298C4⋊C4—3321no—
299C4⋊C4—3321no—
300C4⋊C4—6641no—
301C4⋊C4—6641no—
302C4⋊C4—6641no—
303C4⋊C4—6641no—
304C4⋊C4—1664no—
305C42—41no—
306C42—3321no—
307C42—1661no—
308C42—1661no—
309C42—1661no—
310C42—1661no—
311C42—1661no—
312C42—1661no—
313C23C22⋊C4×D8311yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318Q8—6644no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—3321no—
332C2×C4—3321no—
333C2×C4—3321no—
334C2×C4—3321no—
335C2×C4—3321no—
336C2×C4—3321no—
337C2×C4—3321no—
338C2×C4—3321no—
339C2×C4—831no—
340C2×C4—831no—
341C2×C4—1662no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1662no—
346C2×C4#1458175711yescharacteristic
347C2×C4C4×Dic16611yescharacteristic
348C2×C4#1458176911yescharacteristic
349C2×C4#1458177011yescharacteristic
350C2×C4#1458177811yescharacteristic
351C2×C4#1458179111yescharacteristic
352C8—42no—
353C8—81no—
354C8—3322no—
355C8—6641no—
356C22—21no—
357C22—21no—
358C22#1912760511yescharacteristic
359C22#99×D8311yescharacteristic
360C22#1912767311yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—81no—
369C4—3322no—
370C4—3321no—
371C4—3321no—
372C4—3321no—
373C4—832no—
374C4—6641no—
375C4—1661no—
376C2—21no—
377C2—21no—
378C2#2490844611yescharacteristic
379C2#356×D8311yescharacteristic
380C2#2490915311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#1394C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenter#1394D83
hypocenterC83#13302
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
53. #14581711Q8semidirect
67. #101×C83Q8semidirect
198. C83#13302semidirect
253. #101Dic166semidirect

Maximal subgroups

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

subgroupindex
#249126732
#249074572
#101×Dic832
#249055902
#249055892
C83×#13942
#1330283