Finite Group Databasebeta

← #35292980

Subgroups — #35292980

Order 21248.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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#24915814C211yesmaximal, characteristic
2#24907530C211yesmaximal, characteristic
3#24907224C211yesmaximal, characteristic
4#24905287C211yesmaximal, characteristic, complemented
5#24905286C211yesmaximal, characteristic, complemented
6#24905285C211yesmaximal, characteristic, complemented
7C83×#1271C211yesmaximal, characteristic, complemented
8C2×#14581791C411yescharacteristic
9C2×#14581770—21no—
10C2×C42×D83C2211yescharacteristic
11C2×(C2×D83)⋊C8—21no—
12C2×(C2×D83)⋊C8—21no—
13C2×#14581702—21no—
14#19127605—21no—
15#19127599C2211yescharacteristic
16#19127596—21no—
17#19127570C411yescharacteristic
18#19127560—21no—
19#19127558C411yescharacteristic
20#19127558C411yescharacteristic
21#19127456C2211yescharacteristic
22#19127456C2211yescharacteristic
23#19127455—21no—
24#389×C83C2211yescharacteristic
25#336×C83C2211yescharacteristic
26#336×C83C2211yescharacteristic
27C22×Dic166—21no—
28C22×C4×D83C2×C411yescharacteristic
29C22×C4×D83D411yescharacteristic
30C22×C4×D83D411yescharacteristic
31C22×C83⋊C8—21no—
32C22×C83⋊C8—21no—
33#14581791—42no—
34#14581770—42no—
35#14581770—42no—
36C2×C4⋊Dic83C2×C411yescharacteristic
37C2×C4⋊Dic83C2×C411yescharacteristic
38C2×Dic83⋊C4—21no—
39C2×Dic83⋊C4—21no—
40C2×Dic83⋊C4—21no—
41C2×Dic83⋊C4—21no—
42C2×(C2×D83)⋊C4—21no—
43C2×(C2×D83)⋊C4—21no—
44C2×C4×Dic83C2×C411yescharacteristic
45C2×C4×Dic83D411yescharacteristic
46C2×C4×Dic83D411yescharacteristic
47C42×D83—42no—
48(C2×D83)⋊C8—24no—
49(C2×D83)⋊C8—42no—
50#14581702—42no—
51C83⋊#95—21no—
52C83⋊#95—21no—
53C83⋊#95—21no—
54C22×C8×C83—21no—
55C22×C8×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83C2×C411yescharacteristic
58C2×C4⋊C4×C83C2×C411yescharacteristic
59C2×C42×C83C2311yescharacteristic
60#95×C83—21no—
61C23×D83C22⋊C411yescharacteristic
62C2×Dic166—42no—
63C2×Dic166—42no—
64C2×Dic166—42no—
65C22×Dic83—21no—
66C22×Dic83—21no—
67C22×Dic83C22⋊C411yescharacteristic
68C22×Dic83C22⋊C411yescharacteristic
69C22×Dic83C22⋊C411yescharacteristic
70C2×C4×D83—22no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83—21no—
77C2×C4×D83—22no—
78C2×C4×D83—22no—
79C2×C4×D83—41no—
80C2×C4×D83—41no—
81C2×C4×D83—41no—
82C2×C83⋊C8—22no—
83C2×C83⋊C8—41no—
84C2×C83⋊C8—41no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8—41no—
88C4⋊Dic83—21no—
89C4⋊Dic83—21no—
90C4⋊Dic83—41no—
91Dic83⋊C4—42no—
92Dic83⋊C4—42no—
93Dic83⋊C4—41no—
94Dic83⋊C4—41no—
95Dic83⋊C4—42no—
96(C2×D83)⋊C4—42no—
97(C2×D83)⋊C4—42no—
98C4×Dic83—22no—
99C4×Dic83—22no—
100C4×Dic83—41no—
101C22×C4×C83—21no—
102C22×C4×C83—21no—
103C22×C4×C83C22×C411yescharacteristic
104C22×C4×C83C2×D411yescharacteristic
105C22×C4×C83C2×D411yescharacteristic
106C2×C8×C83—22no—
107C2×C8×C83—41no—
108C2×C8×C83—41no—
109C2×C8×C83—41no—
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83—21no—
114C4⋊C4×C83—41no—
115C4⋊C4×C83—41no—
116C4⋊C4×C83—41no—
117C42×C83—41no—
118C23×C83C2×C22⋊C411yescharacteristic
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—21no—
123C22×D83—21no—
124C22×D83—21no—
125C22×D83#10112yes—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83#13111yescharacteristic
139C2×C4×C83#13111yescharacteristic
140C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C83#13711yescharacteristic
143C2×C4×C83#13711yescharacteristic
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—41no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83#9912yes—
158C2×Dic83#10412yes—
159C4×D83—42no—
160C4×D83—42no—
161C4×D83—42no—
162C4×D83—42no—
163C4×D83—42no—
164C4×D83—42no—
165Dic166—84no—
166C8×C83—42no—
167C8×C83—81no—
168C83⋊C8—42no—
169C83⋊C8—81no—
170C22×C83#40811yescharacteristic
171C22×C83C2×#9911yescharacteristic
172C22×C83C2×#10111yescharacteristic
173C22×C83#41811yescharacteristic
174C22×C83#41911yescharacteristic
175C22×C83C2×#10411yescharacteristic
176C22×C83#42111yescharacteristic
177C2×D83—22no—
178C2×D83—22no—
179C2×D83—22no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—41no—
184C4×C83—21no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—21no—
190C4×C83—81no—
191C4×C83—81no—
192Dic83—22no—
193Dic83—22no—
194Dic83—41no—
195Dic83—81no—
196Dic83—81no—
197#7708—831nomaximal, Sylow
198C2×C83#147211yescharacteristic
199C2×C83#148411yescharacteristic
200C2×C83#149711yescharacteristic
201C2×C83#150811yescharacteristic
202C2×C83#151011yescharacteristic
203C2×C83#153111yescharacteristic
204C2×C83#155611yescharacteristic
205D83—42no—
206#2857—831no—
207#1817—831no—
208#1749—831no—
209#1271—831no—
210#1271—831no—
211#1271—831no—
212#1271D8311yescharacteristic, complemented
213C83#770811yescharacteristic, complemented, Sylow
214C2×C4⋊Q8—831no—
215C2×C22⋊Q8—1661no—
216C22×C42—831no—
217C2×C22⋊C8—1661no—
218C2×C22⋊C8—1661no—
219C2×#97—1661no—
220#391—1661no—
221#389—831no—
222#389—1661no—
223#389C2×D8311yescharacteristic
224#384—831no—
225#380—831no—
226#380—831no—
227#380—1661no—
228#336—831no—
229#336—831no—
230#336—1661no—
231#336C2×D8311yescharacteristic
232#336C2×D8311yescharacteristic
233C22×Q8—1661no—
234C23×C4—831no—
235C23×C4—831no—
236C23×C4—831no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—1661no—
240C22×C8—1661no—
241C4⋊Q8—3322no—
242C22⋊Q8—3322no—
243C22⋊Q8—3322no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—831no—
246C2×C4⋊C4—831no—
247C2×C4⋊C4—1661no—
248C2×C4⋊C4—1661no—
249C2×C4⋊C4—1661no—
250C2×C4⋊C4—1661no—
251C2×C4⋊C4C4×D8311yescharacteristic
252C2×C4⋊C4C4×D8311yescharacteristic
253C2×C22⋊C4—1661no—
254C2×C22⋊C4—1661no—
255C2×C42—3322no—
256C2×C42—831no—
257C2×C42—831no—
258C2×C42—831no—
259C2×C42C22×D8311yescharacteristic
260C22⋊C8—3322no—
261C22⋊C8—1664no—
262#97—3322no—
263#95—21no—
264#95—1661no—
265#95—1661no—
266#95—1661no—
267C24—831no—
268C2×Q8—3322no—
269C2×Q8—3322no—
270C2×Q8—3322no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3321no—
274C22×C4—3321no—
275C22×C4—3321no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—1662no—
280C22×C4—1662no—
281C22×C4—1662no—
282C22×C4—1661no—
283C22×C4—1661no—
284C22×C4—1661no—
285C22×C4—1661no—
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4—1661no—
289C22×C4—1661no—
290C22×C4C2×C4×D8311yescharacteristic
291C22×C4D4×D8311yescharacteristic
292C22×C4D4×D8311yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—3321no—
300C2×C8—3321no—
301C2×C8—3321no—
302C2×C8—3321no—
303C2×C8—3321no—
304C2×C8—1662no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—3321no—
311C4⋊C4—3322no—
312C4⋊C4—3322no—
313C4⋊C4—3322no—
314C4⋊C4—3321no—
315C4⋊C4—3321no—
316C4⋊C4—1661no—
317C4⋊C4—1661no—
318C22⋊C4—3322no—
319C22⋊C4—3322no—
320C42—41no—
321C42—3321no—
322C42—1662no—
323C42—1662no—
324C23—832no—
325C23—1661no—
326C23—1661no—
327C23—1661no—
328C23—1661no—
329C23—1661no—
330C23—1661no—
331C23C22⋊C4×D8311yescharacteristic
332Q8—6644no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—3322no—
346C2×C4—3322no—
347C2×C4—3322no—
348C2×C4—3322no—
349C2×C4—3322no—
350C2×C4—3322no—
351C2×C4—3321no—
352C2×C4—3321no—
353C2×C4—3321no—
354C2×C4—3321no—
355C2×C4—3321no—
356C2×C4—3321no—
357C2×C4—832no—
358C2×C4—832no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4#1458179811yescharacteristic
367C2×C4#1458180011yescharacteristic
368C2×C4#1458181611yescharacteristic
369C2×C4#1458181911yescharacteristic
370C2×C4#1458182111yescharacteristic
371C2×C4#1458182411yescharacteristic
372C8—42no—
373C8—81no—
374C8—3322no—
375C8—6641no—
376C22—3321no—
377C22—3321no—
378C22—3321no—
379C22—3321no—
380C22—1662no—
381C22—1662no—
382C22—1662no—
383C22#1912766611yescharacteristic
384C22#99×D8311yescharacteristic
385C22#101×D8311yescharacteristic
386C22#1912770411yescharacteristic
387C22#1912770811yescharacteristic
388C22#104×D8311yescharacteristic
389C22#1912771611yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—3321no—
399C4—6641no—
400C4—6641no—
401C4—1662no—
402C4—1662no—
403C2—3322no—
404C2#2490582511yescharacteristic
405C2#2490587811yescharacteristic
406C2#2490593411yescharacteristic
407C2#2490598111yescharacteristic
408C2#2490599011yescharacteristic
409C2#2490607611yescharacteristic
410C2#2490617811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D83
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1271C2
socleC23×C83C2×C22⋊C4
prime radicalC23×D83C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1271D83
hypocenterC83#7708
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #24905287C2semidirect
5. #24905286C2semidirect
6. #24905285C2semidirect
7. C83×#1271C2semidirect
212. #1271D83semidirect
213. C83#7708semidirect

Maximal subgroups

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

subgroupindex
#249158142
#249075302
#249072242
#249052872
#249052862
#249052852
C83×#12712
#770883