Finite Group Databasebeta

← #24913708

Subgroups — #24913708

Order 10624.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581779C211yesmaximal, characteristic, complemented
2C2×#14581773C211yesmaximal, characteristic, complemented
3C2×#14581769C211yesmaximal, characteristic, complemented
4#19127637C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic, complemented
6#19127621C211yesmaximal, characteristic, complemented
7#19127615C211yesmaximal, characteristic, complemented
8#19127615C211yesmaximal, characteristic, complemented
9#19127615C211yesmaximal, characteristic, complemented
10#19127596C211yesmaximal, characteristic, complemented
11#19127594C211yesmaximal, characteristic, complemented
12#19127581C211yesmaximal, characteristic, complemented
13#19127574C211yesmaximal, characteristic, complemented
14#19127561C211yesmaximal, characteristic, complemented
15C2×#121×C83C211yesmaximal, characteristic
16C22×Dic166C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18#14581779—24no—
19#14581773—24no—
20#14581769—24no—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊Dic83C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×C22⋊Dic83C2211yescharacteristic
34C2×C22⋊Dic83C2211yescharacteristic
35C2×C22⋊Dic83C2211yescharacteristic
36C2×C22⋊Dic83C2211yescharacteristic
37C2×C22⋊Dic83C2211yescharacteristic
38C2×C22⋊Dic83C2211yescharacteristic
39C2×C22⋊Dic83C2211yescharacteristic
40C2×C4×Dic83C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C22×D4×C83C2211yescharacteristic
48C22×D4×C83C2211yescharacteristic
49C22×D4×C83C2211yescharacteristic
50C23×C4×C83C2211yescharacteristic
51#121×C83—24no—
52C2×C4⋊C4×C83C2211yescharacteristic
53C2×C22⋊C4×C83C2211yescharacteristic
54C2×C22⋊C4×C83C2211yescharacteristic
55C2×Dic166—22no—
56C2×Dic166—22no—
57C2×Dic166—41no—
58C22×Dic83—22no—
59C22×Dic83—22no—
60C22×Dic83—41no—
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C4⋊Dic83—22no—
70C4⋊Dic83—41no—
71Dic83⋊C4—22no—
72Dic83⋊C4—22no—
73Dic83⋊C4—22no—
74Dic83⋊C4—22no—
75Dic83⋊C4—22no—
76Dic83⋊C4—41no—
77C22⋊Dic83—22no—
78C22⋊Dic83—22no—
79C22⋊Dic83—22no—
80C22⋊Dic83—22no—
81C22⋊Dic83—22no—
82C22⋊Dic83—22no—
83C22⋊Dic83—41no—
84C22⋊Dic83—41no—
85C22⋊Dic83—41no—
86C22⋊Dic83—41no—
87C22⋊Dic83D414yescomplemented
88C4×Dic83—22no—
89C24×C83C2311yescharacteristic
90C24×C83C2311yescharacteristic
91C24×C83C2311yescharacteristic
92C2×D4×C83—22no—
93C2×D4×C83—22no—
94C2×D4×C83—22no—
95C2×D4×C83—22no—
96C2×D4×C83—22no—
97C2×D4×C83—22no—
98C2×D4×C83—41no—
99C2×D4×C83—41no—
100C2×D4×C83—41no—
101C22×C4×C83—22no—
102C22×C4×C83—22no—
103C22×C4×C83—41no—
104C22×C4×C83C2311yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C22×C4×C83C2311yescharacteristic
108C4⋊C4×C83—22no—
109C22⋊C4×C83—22no—
110C22⋊C4×C83D414yes—
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—21no—
125C23×C83—21no—
126C23×C83—21no—
127C23×C83—21no—
128C23×C83—21no—
129C23×C83—21no—
130C23×C83C2×D412yes—
131C23×C83C2×D412yes—
132C23×C83C2411yescharacteristic
133D4×C83—44no—
134D4×C83—44no—
135D4×C83—44no—
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—44no—
146C2×C4×C83C2×D412yes—
147C2×C4×C83C2×D412yes—
148C2×C4×C83#4112yes—
149C2×Dic83—21no—
150C2×Dic83—21no—
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—21no—
168C2×Dic83—21no—
169C2×Dic83—21no—
170C2×Dic83—21no—
171C2×Dic83—44no—
172C2×Dic83C2×D412yes—
173C2×Dic83C2×D412yes—
174Dic166—44no—
175C22×C83—22no—
176C22×C83—22no—
177C22×C83—22no—
178C22×C83—22no—
179C22×C83—22no—
180C22×C83—22no—
181C22×C83—41no—
182C22×C83—41no—
183C22×C83—41no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—41no—
188C22×C83—41no—
189C22×C83—41no—
190C22×C83—41no—
191C22×C83—41no—
192C22×C83—41no—
193C22×C83—41no—
194C22×C83—41no—
195C22×C83—41no—
196C22×C83C22×D411yescharacteristic
197C22×C83C22×D411yescharacteristic
198C22×C83C2×#4111yescharacteristic
199C22×C83H(5,𝔽2)11yescharacteristic
200C22×C83H(5,𝔽2)11yescharacteristic
201C22×C832−1+411yescharacteristic
202C22×C832−1+411yescharacteristic
203C4×C83—22no—
204C4×C83—22no—
205C4×C83—22no—
206C4×C83—41no—
207Dic83—22no—
208Dic83—22no—
209Dic83—41no—
210Dic83—41no—
211Dic83—41no—
212Dic83—41no—
213Dic83—41no—
214Dic83—41no—
215C2×C83—42no—
216C2×C83—42no—
217C2×C83—81no—
218C2×C83#53611yescharacteristic
219C2×C83#53711yescharacteristic
220C2×C83#54211yescharacteristic
221C2×C83D4211yescharacteristic
222C2×C83#54611yescharacteristic
223C2×C83#54711yescharacteristic
224C2×C83#56211yescharacteristic
225#2672—831nomaximal, Sylow
226C83#267211yescharacteristic, complemented, Sylow
227C2×#124—831no—
228C2×#123—831no—
229C2×C22⋊Q8—831no—
230C2×#121D8311yescharacteristic
231#399—831no—
232#396—831no—
233#395—831no—
234#394—831no—
235#394—831no—
236#394—831no—
237#389—831no—
238#388—831no—
239#386—831no—
240#385—831no—
241#380—831no—
242C22×Q8—831no—
243C22×D4C2×D8311yescharacteristic
244C22×D4C2×D8311yescharacteristic
245C22×D4C2×D8311yescharacteristic
246C23×C4—831no—
247C23×C4C2×D8311yescharacteristic
248#124—1664no—
249#123—1664no—
250C22⋊Q8—1664no—
251#121—24no—
252C2×C4⋊C4—831no—
253C2×C4⋊C4—831no—
254C2×C4⋊C4—831no—
255C2×C4⋊C4—831no—
256C2×C4⋊C4—831no—
257C2×C4⋊C4—831no—
258C2×C4⋊C4—831no—
259C2×C4⋊C4—831no—
260C2×C4⋊C4C2×D8311yescharacteristic
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⋊C4—831no—
269C2×C22⋊C4—831no—
270C2×C22⋊C4—831no—
271C2×C22⋊C4—831no—
272C2×C22⋊C4C2×D8311yescharacteristic
273C2×C22⋊C4C2×D8311yescharacteristic
274C2×C42—831no—
275#95—831no—
276#95—831no—
277#95—831no—
278#95—831no—
279#95—831no—
280#95—831no—
281C24C22×D8311yescharacteristic
282C24C22×D8311yescharacteristic
283C24C22×D8311yescharacteristic
284C2×Q8—3321no—
285C2×Q8—1662no—
286C2×Q8—1662no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—3321no—
300C22×C4—831no—
301C22×C4—831no—
302C22×C4—831no—
303C22×C4—831no—
304C22×C4—831no—
305C22×C4—831no—
306C22×C4—831no—
307C22×C4—831no—
308C22×C4—1662no—
309C22×C4—1662no—
310C22×C4C22×D8311yescharacteristic
311C22×C4C22×D8311yescharacteristic
312C22×C4C22×D8311yescharacteristic
313C22×C4C22×D8311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—3321no—
316C4⋊C4—3321no—
317C4⋊C4—1662no—
318C4⋊C4—1662no—
319C4⋊C4—1662no—
320C4⋊C4—1662no—
321C4⋊C4—1662no—
322C4⋊C4—1662no—
323C22⋊C4—22no—
324C22⋊C4—3321no—
325C22⋊C4—3321no—
326C22⋊C4—3321no—
327C22⋊C4—3321no—
328C22⋊C4—834no—
329C22⋊C4—1662no—
330C22⋊C4—1662no—
331C22⋊C4—1662no—
332C22⋊C4—1662no—
333C22⋊C4—1662no—
334C22⋊C4—1662no—
335C22⋊C4C22⋊D8314yes—
336C42—1662no—
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—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23D4×D8312yes—
357C23C2×C22⋊D8312yes—
358C23C23×D8311yescharacteristic
359Q8—3324no—
360D4—44no—
361D4—44no—
362D4—44no—
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—44no—
373C2×C4—3324no—
374C2×C4—832no—
375C2×C4—832no—
376C2×C4—1661no—
377C2×C4—1661no—
378C2×C4—1661no—
379C2×C4—1661no—
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×C4—1661no—
397C2×C4—1661no—
398C2×C4C2×C22⋊D8312yes—
399C2×C4C2×C22⋊D8312yes—
400C2×C4#1305058912yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22C2×D4×D8311yescharacteristic
423C22C22×C22⋊D8311yescharacteristic
424C22C2×#1305058911yescharacteristic
425C22#1458183711yescharacteristic
426C22#1458183711yescharacteristic
427C22#1458183911yescharacteristic
428C22#1458183911yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—3321no—
434C4—3321no—
435C4—3321no—
436C4—3321no—
437C4—3321no—
438C4—3321no—
439C4—1662no—
440C4—1662no—
441C2—42no—
442C2—42no—
443C2—81no—
444C2#1912827211yescharacteristic
445C2#1912827811yescharacteristic
446C2#1912831811yescharacteristic
447C2D4×C22⋊D8311yescharacteristic
448C2#1912834511yescharacteristic
449C2#1912835111yescharacteristic
450C2#1912844611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2672
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14581779C2semidirect
2. C2×#14581773C2semidirect
3. C2×#14581769C2semidirect
4. #19127637C2semidirect
5. #19127624C2semidirect
6. #19127621C2semidirect
7. #19127615C2semidirect
8. #19127615C2semidirect
9. #19127615C2semidirect
10. #19127596C2semidirect
11. #19127594C2semidirect
12. #19127581C2semidirect
13. #19127574C2semidirect
14. #19127561C2semidirect
87. C22⋊Dic83D4semidirect
226. C83#2672semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
C2×#145817732
C2×#145817692
#191276372
#191276242
#191276212
#191276152
#191275962
#191275942
#191275812
#191275742
#191275612
C2×#121×C832
#267283