Finite Group Databasebeta

← #24910504

Subgroups — #24910504

Order 10624.

688 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×#14581764C211yesmaximal, characteristic
2C2×C4×C22⋊D83C211yesmaximal, characteristic, complemented
3C2×#14581748C211yesmaximal, characteristic, complemented
4#19127604C211yesmaximal, characteristic, complemented
5#19127601C211yesmaximal, characteristic, complemented
6#19127601C211yesmaximal, characteristic, complemented
7#19127598C211yesmaximal, characteristic, complemented
8#19127595C211yesmaximal, characteristic, complemented
9#19127593C211yesmaximal, characteristic, complemented
10#19127577C211yesmaximal, characteristic, complemented
11#19127552C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
14#19127548C211yesmaximal, characteristic, complemented
15#386×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C23×Dic83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20#14581764C418yescomplemented
21C4×C22⋊D83—24no—
22#14581748—24no—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×C4×Dic83C2211yescharacteristic
38C2×C4×Dic83C2211yescharacteristic
39C2×C4×Dic83C2211yescharacteristic
40C2×C4×Dic83C2211yescharacteristic
41C2×C4×Dic83C2211yescharacteristic
42C2×C4×Dic83C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C22×D4×C83C2211yescharacteristic
48C23×C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53#95×C83C2211yescharacteristic
54C23×D83C2311yescharacteristic
55C2×C22⋊D83—22no—
56C2×C22⋊D83—22no—
57C2×C22⋊D83—41no—
58C2×C22⋊D83—41no—
59C2×C22⋊D83C2×C414yes—
60C2×C22⋊D83C2×C414yescomplemented
61C22×Dic83—22no—
62C22×Dic83—22no—
63C22×Dic83C2×C414yescomplemented
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C22×Dic83C2311yescharacteristic
70C22×Dic83C2311yescharacteristic
71C2×C4×D83—22no—
72C2×C4×D83—22no—
73C2×C4×D83—41no—
74C4⋊Dic83—41no—
75Dic83⋊C4—22no—
76Dic83⋊C4—41no—
77Dic83⋊C4C2×C414yescomplemented
78Dic83⋊C4D414yescomplemented
79C22⋊Dic83—22no—
80C22⋊Dic83—22no—
81C22⋊Dic83—22no—
82C22⋊Dic83—22no—
83C22⋊Dic83C2×C414yescomplemented
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—41no—
87(C2×D83)⋊C4C2×C414yescomplemented
88C4×Dic83—22no—
89C4×Dic83—22no—
90C4×Dic83—22no—
91C4×Dic83—22no—
92C4×Dic83—41no—
93C4×Dic83—41no—
94C24×C83C2311yescharacteristic
95C24×C83C2311yescharacteristic
96C2×D4×C83—41no—
97C2×D4×C83—41no—
98C2×D4×C83C2×C414yescomplemented
99C22×C4×C83—22no—
100C22×C4×C83—22no—
101C22×C4×C83—41no—
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C22⋊C4×C83—22no—
108C22⋊C4×C83—22no—
109C22⋊C4×C83—41no—
110C23×C83—21no—
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×C83C22×C412yes—
123C23×C83C22×C412yes—
124C23×C83C2411yescharacteristic
125C22×D83—21no—
126C22×D83—21no—
127C22×D83—21no—
128C22×D83—21no—
129C22×D83—21no—
130C22×D83—21no—
131C22×D83C22×C412yes—
132D4×C83—44no—
133C2×C4×C83—21no—
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—44no—
146C2×C4×C83C22×C412yes—
147C2×C4×C83C2×D412yes—
148C2×C4×C83#4112yes—
149C22⋊D83—28no—
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—28no—
160C2×Dic83—21no—
161C2×Dic83—21no—
162C2×Dic83—21no—
163C2×Dic83—21no—
164C2×Dic83—21no—
165C2×Dic83—21no—
166C2×Dic83C22×C412yes—
167C2×Dic83C22×C412yes—
168C2×Dic83C22×C412yes—
169C2×Dic83C2×D412yes—
170C2×Dic83C2×D412yes—
171C2×Dic83#4112yes—
172C2×Dic83#4112yes—
173C4×D83—44no—
174C22×C83—22no—
175C22×C83—22no—
176C22×C83—22no—
177C22×C83—22no—
178C22×C83—22no—
179C22×C83—22no—
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×C83C23×C411yescharacteristic
189C22×C83C22×D411yescharacteristic
190C22×C83C2×#4111yescharacteristic
191C22×C83C2×#4111yescharacteristic
192C22×C83C2×#4111yescharacteristic
193C22×C83H(5,𝔽2)11yescharacteristic
194C22×C83H(5,𝔽2)11yescharacteristic
195C2×D83—22no—
196C2×D83—22no—
197C2×D83—22no—
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—22no—
210Dic83—22no—
211Dic83—41no—
212Dic83—41no—
213Dic83C4×D414yes—
214C2×C83—42no—
215C2×C83—42no—
216C2×C83C2×C4×D411yescharacteristic
217C2×C83C4×#4111yescharacteristic
218C2×C83#51811yescharacteristic
219C2×C83#54611yescharacteristic
220C2×C83#55011yescharacteristic
221C2×C83#55311yescharacteristic
222C2×C83#55511yescharacteristic
223D83—42no—
224#2363—831nomaximal, Sylow
225C83#236311yescharacteristic, complemented, Sylow
226C2×#121—831no—
227C2×C4×D4—831no—
228C2×#117—831no—
229#390—831no—
230#390—831no—
231#390—831no—
232#389—831no—
233#388—831no—
234#388—831no—
235#386—831no—
236#386D8311yescharacteristic, complemented
237C4×C4⋊C4—831no—
238C4×C22⋊C4—831no—
239C4×C22⋊C4—831no—
240C4×C22⋊C4—831no—
241C22×D4—831no—
242C22×D4—831no—
243C22×D4C2×D8311yescharacteristic
244C23×C4—831no—
245C23×C4—831no—
246C23×C4C2×D8311yescharacteristic
247#121—838no—
248C4×D4—1664no—
249#117—1664no—
250C2×C4⋊C4—831no—
251C2×C4⋊C4—831no—
252C2×C4⋊C4—831no—
253C2×C4⋊C4—831no—
254C2×C4⋊C4—831no—
255C2×C22⋊C4—831no—
256C2×C22⋊C4—831no—
257C2×C22⋊C4—831no—
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⋊C4C2×D8311yescharacteristic
265C2×C22⋊C4C2×D8311yescharacteristic
266C2×C22⋊C4C2×D8311yescharacteristic
267C2×C42—831no—
268C2×C42—831no—
269C2×C42—831no—
270C2×C42—831no—
271C2×C42—831no—
272C2×C42—831no—
273#95—831no—
274#95—831no—
275#95—831no—
276#95—831no—
277#95C2×D8311yescharacteristic
278#95C2×D8311yescharacteristic
279C24—831no—
280C24C22×D8311yescharacteristic
281C24C22×D8311yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—3321no—
285C2×D4—3321no—
286C2×D4—834no—
287C2×D4—834no—
288C2×D4—1662no—
289C2×D4—1662no—
290C2×D4C4×D8314yescomplemented
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—3321no—
295C22×C4—831no—
296C22×C4—834no—
297C22×C4—831no—
298C22×C4—831no—
299C22×C4—831no—
300C22×C4—831no—
301C22×C4—831no—
302C22×C4—831no—
303C22×C4—1662no—
304C22×C4—1662no—
305C22×C4—1662no—
306C22×C4—1662no—
307C22×C4C22×D8311yescharacteristic
308C22×C4C22×D8311yescharacteristic
309C22×C4C22×D8311yescharacteristic
310C22×C4C22×D8311yescharacteristic
311C22×C4C22×D8311yescharacteristic
312C4⋊C4—3321no—
313C4⋊C4—3321no—
314C4⋊C4—834no—
315C4⋊C4—834no—
316C4⋊C4—1662no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—41no—
320C22⋊C4—3321no—
321C22⋊C4—834no—
322C22⋊C4—834no—
323C22⋊C4—1662no—
324C22⋊C4—1662no—
325C22⋊C4—1662no—
326C22⋊C4—1662no—
327C22⋊C4—1662no—
328C22⋊C4—1662no—
329C42—3321no—
330C42—3321no—
331C42—1662no—
332C42—1662no—
333C42—1662no—
334C42—1662no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—832no—
348C23—1661no—
349C23—1661no—
350C23—1661no—
351C23—1661no—
352C23—1661no—
353C23—1661no—
354C23C2×C4×D8312yes—
355C23C2×C4×D8312yes—
356C23C23×D8311yescharacteristic
357D4—44no—
358D4—3324no—
359D4—1668no—
360C2×C4—21no—
361C2×C4—21no—
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—44no—
373C2×C4—3324no—
374C2×C4—832no—
375C2×C4—832no—
376C2×C4—832no—
377C2×C4—832no—
378C2×C4—832no—
379C2×C4—832no—
380C2×C4—832no—
381C2×C4—1661no—
382C2×C4—1661no—
383C2×C4—1661no—
384C2×C4—1661no—
385C2×C4—1661no—
386C2×C4—1661no—
387C2×C4—1668no—
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×C4C2×C4×D8312yes—
397C2×C4D4×D8312yes—
398C2×C4#1305058912yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
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—1662no—
418C22—1662no—
419C22—1662no—
420C22C22×C4×D8311yescharacteristic
421C22C2×D4×D8311yescharacteristic
422C22#41×D8311yescharacteristic
423C22#41×D8311yescharacteristic
424C22C2×#1305058911yescharacteristic
425C22#1458183711yescharacteristic
426C22#1458183711yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—3321no—
433C4—3321no—
434C4—834no—
435C4—1662no—
436C4—1662no—
437C4—1662no—
438C4—1662no—
439C2—42no—
440C2—42no—
441C2—3322no—
442C2C4×D4×D8311yescharacteristic
443C2C4×#1305058911yescharacteristic
444C2#1912817411yescharacteristic
445C2#1912833811yescharacteristic
446C2#1912836911yescharacteristic
447C2#1912839611yescharacteristic
448C2#1912841511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutatorC22×C83C23×C4
Fitting#386×C83C2
socle, cosocleC23×C83C24
prime radicalC2×#14581764C2
hypercenter#386D83
hypocenterC83#2363
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D83C2semidirect
3. C2×#14581748C2semidirect
4. #19127604C2semidirect
5. #19127601C2semidirect
6. #19127601C2semidirect
7. #19127598C2semidirect
8. #19127595C2semidirect
9. #19127593C2semidirect
10. #19127577C2semidirect
11. #19127552C2semidirect
12. C22⋊C4×Dic83C2semidirect
13. C4×C22⋊Dic83C2semidirect
14. #19127548C2semidirect
15. #386×C83C2semidirect
20. #14581764C4semidirect
60. C2×C22⋊D83C2×C4semidirect
63. C22×Dic83C2×C4semidirect
77. Dic83⋊C4C2×C4semidirect
78. Dic83⋊C4D4semidirect
83. C22⋊Dic83C2×C4semidirect
87. (C2×D83)⋊C4C2×C4semidirect
98. C2×D4×C83C2×C4semidirect
225. C83#2363semidirect
236. #386D83semidirect
290. C2×D4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817642
C2×C4×C22⋊D832
C2×#145817482
#191276042
#191276012
#191275982
#191275952
#191275932
#191275772
#191275522
C22⋊C4×Dic832
C4×C22⋊Dic832
#191275482
#386×C832
#236383