Finite Group Databasebeta

← #24911026

Subgroups — #24911026

Order 10624.

632 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581778C211yesmaximal, characteristic, complemented
2C2×#14581769C211yesmaximal, characteristic, complemented
3C2×#14581764C211yesmaximal, characteristic
4C2×C4×Dic166C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127617C211yesmaximal, characteristic, complemented
8#19127615C211yesmaximal, characteristic, complemented
9#19127601C211yesmaximal, characteristic, complemented
10#19127601C211yesmaximal, characteristic, complemented
11#19127596C211yesmaximal, characteristic, complemented
12#19127594C211yesmaximal, characteristic, complemented
13#19127592C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
15#390×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C22×C22⋊D83C2211yescharacteristic
19C23×Dic83C2211yescharacteristic
20#14581778—24no—
21#14581769—24no—
22#14581764—24no—
23C4×Dic166—24no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×C22⋊Dic83C2211yescharacteristic
34C2×C22⋊Dic83C2211yescharacteristic
35C2×C22⋊Dic83C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic
40C2×(C2×D83)⋊C4C2211yescharacteristic
41C2×C4×Dic83C2211yescharacteristic
42C2×C4×Dic83C2211yescharacteristic
43C2×C4×Dic83C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C83⋊#95C2211yescharacteristic
48C22×D4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic
53C2×C22⋊C4×C83C2211yescharacteristic
54C2×C42×C83C2211yescharacteristic
55C23×D83C2311yescharacteristic
56C2×Dic166—22no—
57C2×Dic166—22no—
58C2×Dic166D414yescomplemented
59C2×C22⋊D83—22no—
60C2×C22⋊D83—22no—
61C2×C22⋊D83—22no—
62C2×C22⋊D83—22no—
63C2×C22⋊D83—22no—
64C2×C22⋊D83—41no—
65C22×Dic83—22no—
66C22×Dic83—22no—
67C22×Dic83—22no—
68C22×Dic83C2311yescharacteristic
69C22×Dic83C2311yescharacteristic
70C22×Dic83C2311yescharacteristic
71C22×Dic83C2311yescharacteristic
72C22×Dic83C2311yescharacteristic
73C22×Dic83C2311yescharacteristic
74C22×Dic83C2311yescharacteristic
75C4⋊Dic83—22no—
76C4⋊Dic83—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80Dic83⋊C4—41no—
81Dic83⋊C4D414yes—
82C22⋊Dic83—22no—
83C22⋊Dic83—22no—
84C22⋊Dic83—22no—
85C22⋊Dic83—22no—
86C22⋊Dic83—41no—
87(C2×D83)⋊C4—22no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—22no—
90(C2×D83)⋊C4—22no—
91(C2×D83)⋊C4—41no—
92C4×Dic83—22no—
93C4×Dic83—22no—
94C4×Dic83—22no—
95C24×C83C2311yescharacteristic
96C24×C83C2311yescharacteristic
97C2×D4×C83—22no—
98C2×D4×C83—22no—
99C2×D4×C83—41no—
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C4⋊C4×C83—41no—
106C22⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C22⋊C4×C83—22no—
109C22⋊C4×C83—22no—
110C42×C83—22no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—21no—
120C23×C83—21no—
121C23×C83—21no—
122C23×C83—21no—
123C23×C83—21no—
124C23×C83#4112yes—
125C23×C83C2411yescharacteristic
126C22×D83—21no—
127C22×D83—21no—
128C22×D83—21no—
129C22×D83—21no—
130C22×D83—21no—
131C22×D83—21no—
132C22×D83—21no—
133D4×C83—44no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83C2×D412yes—
147C2×C4×C83C2×D412yes—
148C2×C4×C83#4112yes—
149C22⋊D83—44no—
150C22⋊D83—44no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83—21no—
156C2×Dic83—21no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159C2×Dic83—21no—
160C2×Dic83—21no—
161C2×Dic83—21no—
162C2×Dic83—21no—
163C2×Dic83—21no—
164C2×Dic83—21no—
165C2×Dic83—21no—
166C2×Dic83—21no—
167C2×Dic83—44no—
168C2×Dic83C2×D412yes—
169C2×Dic83C2×D412yes—
170C2×Dic83C2×D412yes—
171C2×Dic83C2×D412yes—
172C2×Dic83#4112yes—
173Dic166—28no—
174C22×C83—22no—
175C22×C83—22no—
176C22×C83—22no—
177C22×C83—41no—
178C22×C83—41no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—41no—
183C22×C83—41no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—41no—
188C22×C83C22×D411yescharacteristic
189C22×C83C22×D411yescharacteristic
190C22×C83C2×#4111yescharacteristic
191C22×C83C2×#4111yescharacteristic
192C22×C83C2×#4111yescharacteristic
193C22×C83H(5,𝔽2)11yescharacteristic
194C22×C832−1+411yescharacteristic
195C2×D83—41no—
196C2×D83—41no—
197C2×D83—41no—
198C2×D83—41no—
199C2×D83—41no—
200C2×D83—41no—
201C2×D83—41no—
202C4×C83—22no—
203C4×C83—22no—
204C4×C83—22no—
205C4×C83—41no—
206C4×C83—41no—
207Dic83—22no—
208Dic83—22no—
209Dic83—41no—
210Dic83—41no—
211Dic83—41no—
212Dic83—41no—
213Dic83#12114yes—
214C2×C83—42no—
215C2×C83—81no—
216C2×C83C2×#12111yescharacteristic
217C2×C83#52911yescharacteristic
218C2×C83#54211yescharacteristic
219C2×C83#54611yescharacteristic
220C2×C83#54811yescharacteristic
221C2×C83#54811yescharacteristic
222C2×C83#55011yescharacteristic
223D83—81no—
224#2426—831nomaximal, Sylow
225C83#242611yescharacteristic, complemented, Sylow
226C2×#124—831no—
227C2×C22⋊Q8—831no—
228C2×#121—831no—
229C2×C4×Q8—831no—
230#396—831no—
231#396—831no—
232#394—831no—
233#394—831no—
234#390—831no—
235#390—831no—
236#390D8311yescharacteristic, complemented
237#389—831no—
238#388—831no—
239#388—831no—
240C4×C22⋊C4—831no—
241C22×Q8—831no—
242C22×D4—831no—
243C22×D4—831no—
244C22×D4C2×D8311yescharacteristic
245C23×C4—831no—
246#124—1664no—
247C22⋊Q8—1664no—
248#121—1664no—
249C4×Q8—1664no—
250C2×C4⋊C4—831no—
251C2×C4⋊C4—831no—
252C2×C4⋊C4—831no—
253C2×C4⋊C4—831no—
254C2×C4⋊C4—831no—
255C2×C4⋊C4—831no—
256C2×C4⋊C4—831no—
257C2×C4⋊C4C2×D8311yescharacteristic
258C2×C22⋊C4—831no—
259C2×C22⋊C4—831no—
260C2×C22⋊C4—831no—
261C2×C22⋊C4—831no—
262C2×C22⋊C4—831no—
263C2×C22⋊C4—831no—
264C2×C22⋊C4—831no—
265C2×C22⋊C4—831no—
266C2×C22⋊C4—831no—
267C2×C22⋊C4—831no—
268C2×C22⋊C4C2×D8311yescharacteristic
269C2×C22⋊C4C2×D8311yescharacteristic
270C2×C22⋊C4C2×D8311yescharacteristic
271C2×C22⋊C4C2×D8311yescharacteristic
272C2×C42—831no—
273C2×C42—831no—
274C2×C42—831no—
275C2×C42C2×D8311yescharacteristic
276#95—831no—
277#95—831no—
278#95—831no—
279#95—831no—
280C24—831no—
281C24C22×D8311yescharacteristic
282C24C22×D8311yescharacteristic
283C2×Q8—834no—
284C2×Q8—1662no—
285C2×Q8—1662no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—41no—
289C2×D4—3321no—
290C2×D4—1662no—
291C2×D4—1662no—
292C2×D4—1662no—
293C2×D4—1662no—
294C2×D4—1662no—
295C22×C4—831no—
296C22×C4—831no—
297C22×C4—831no—
298C22×C4—831no—
299C22×C4—831no—
300C22×C4—831no—
301C22×C4—831no—
302C22×C4—1662no—
303C22×C4—1662no—
304C22×C4—1662no—
305C22×C4C22×D8311yescharacteristic
306C22×C4C22×D8311yescharacteristic
307C22×C4C22×D8311yescharacteristic
308C22×C4C22×D8311yescharacteristic
309C22×C4C22×D8311yescharacteristic
310C4⋊C4—41no—
311C4⋊C4—3321no—
312C4⋊C4—834no—
313C4⋊C4—1662no—
314C4⋊C4—1662no—
315C4⋊C4—1662no—
316C4⋊C4—1662no—
317C4⋊C4—1662no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—3321no—
323C22⋊C4—3321no—
324C22⋊C4—1662no—
325C22⋊C4—1662no—
326C22⋊C4—1662no—
327C22⋊C4—1662no—
328C22⋊C4—1662no—
329C22⋊C4—1662no—
330C22⋊C4—1662no—
331C22⋊C4—1662no—
332C42—22no—
333C42—1662no—
334C42—1662no—
335C42—1662no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—1661no—
350C23—1661no—
351C23—1661no—
352C23—1661no—
353C23—1661no—
354C23—1661no—
355C23—1661no—
356C23#1305059012yes—
357C23C23×D8311yescharacteristic
358Q8—1668no—
359D4—44no—
360D4—3324no—
361D4—3324no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—3324no—
375C2×C4—832no—
376C2×C4—832no—
377C2×C4—832no—
378C2×C4—832no—
379C2×C4—832no—
380C2×C4—1661no—
381C2×C4—1661no—
382C2×C4—1661no—
383C2×C4—1661no—
384C2×C4—1661no—
385C2×C4—1661no—
386C2×C4—1661no—
387C2×C4—1661no—
388C2×C4—1661no—
389C2×C4—1661no—
390C2×C4—1661no—
391C2×C4—1661no—
392C2×C4—1661no—
393C2×C4—1661no—
394C2×C4—1661no—
395C2×C4—1661no—
396C2×C4D4×D8312yes—
397C2×C4D4×D8312yes—
398C2×C4#1305058912yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3321no—
414C22—3321no—
415C22—3321no—
416C22—3321no—
417C22—3321no—
418C22—3321no—
419C22—3321no—
420C22C2×D4×D8311yescharacteristic
421C22C2×D4×D8311yescharacteristic
422C22#41×D8311yescharacteristic
423C22C2×#1305058911yescharacteristic
424C22C2×#1305059011yescharacteristic
425C22#1458183711yescharacteristic
426C22#1458183911yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—3321no—
433C4—3321no—
434C4—3321no—
435C4—3321no—
436C4—834no—
437C4—1662no—
438C4—1662no—
439C2—42no—
440C2—81no—
441C2—6641no—
442C2#121×D8311yescharacteristic
443C2#1912823111yescharacteristic
444C2#1912831511yescharacteristic
445C2#1912834111yescharacteristic
446C2#1912835711yescharacteristic
447C2#1912836011yescharacteristic
448C2#1912837111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC2×#14581764C2
hypercenter#390D83
hypocenterC83#2426
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581778C2semidirect
2. C2×#14581769C2semidirect
4. C2×C4×Dic166C2semidirect
5. #19127624C2semidirect
6. #19127623C2semidirect
7. #19127617C2semidirect
8. #19127615C2semidirect
9. #19127601C2semidirect
10. #19127601C2semidirect
11. #19127596C2semidirect
12. #19127594C2semidirect
13. #19127592C2semidirect
14. C22⋊C4×Dic83C2semidirect
15. #390×C83C2semidirect
58. C2×Dic166D4semidirect
225. C83#2426semidirect
236. #390D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817692
C2×#145817642
C2×C4×Dic1662
#191276242
#191276232
#191276172
#191276152
#191276012
#191275962
#191275942
#191275922
C22⋊C4×Dic832
#390×C832
#242683