Finite Group Databasebeta

← #27182839

Subgroups — #27182839

Order 12928.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692276C211yesmaximal, characteristic, complemented
2#20265838C211yesmaximal, characteristic, complemented
3#20265821C211yesmaximal, characteristic, complemented
4#20265558C211yesmaximal, characteristic, complemented
5#20265406C211yesmaximal, characteristic, complemented
6#20265403C211yesmaximal, characteristic, complemented
7#20265397C211yesmaximal, characteristic, complemented
8#20265394C211yesmaximal, characteristic, complemented
9#20265336C211yesmaximal, characteristic, complemented
10#20265333C211yesmaximal, characteristic, complemented
11#20265321C211yesmaximal, characteristic, complemented
12#20265311C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14C8×C22⋊D101C211yesmaximal, characteristic, complemented
15C22×SD16×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic, complemented
17C2×#13108073C2211yescharacteristic
18#15692276—24no—
19C2×#13108064C2211yescharacteristic, complemented
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108059C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×C8×D101C2211yescharacteristic, complemented
25#15692244—21no—
26#15692244C2211yescharacteristic, complemented
27#15692242C2211yescharacteristic
28#15692236—21no—
29#15692234C2211yescharacteristic
30#15692231—21no—
31#15692231C2211yescharacteristic, complemented
32#15692229—21no—
33#15692229C2211yescharacteristic
34#15692227—21no—
35#15692226C2211yescharacteristic
36#15692224—21no—
37Q8×Dic101C2211yescharacteristic, complemented
38D4×Dic101C2211yescharacteristic
39C4×C22⋊D101—21no—
40C4×C22⋊D101C2211yescharacteristic
41C2×C22⋊Dic101—21no—
42#15692187C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50C22⋊#11717579C2211yescharacteristic
51#15692167C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C22×Q8×C101C2211yescharacteristic
54C22×D4×C101C2211yescharacteristic
55C2×SD16×C101—24no—
56C2×SD16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic
59C2×SD16×C101C2211yescharacteristic
60C22×C8×C101C2211yescharacteristic
61#13108075—22no—
62#13108075—41no—
63#13108073—22no—
64#13108073—41no—
65C2×Dic202C2311yescharacteristic
66C2×D404—21no—
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101D411yescharacteristic
72C22×Dic101—21no—
73C22×Dic101—21no—
74C2×C4×D101—21no—
75C2×C4×D101C2311yescharacteristic
76#13108064—22no—
77#13108064—41no—
78#13108063—22no—
79#13108059—22no—
80#13108059—41no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C8×D101—22no—
84C4⋊Dic101—21no—
85C4⋊Dic101C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053—21no—
89#13108053D411yescharacteristic
90C22⋊Dic101—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101D411yescharacteristic
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051D411yescharacteristic
101C4×Dic101—21no—
102C4×Dic101C2311yescharacteristic
103C24×C101—21no—
104C2×Q8×C101—21no—
105C2×Q8×C101—22no—
106C2×Q8×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101—22no—
110C2×D4×C101—21no—
111C2×D4×C101—41no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116SD16×C101—22no—
117SD16×C101—22no—
118SD16×C101—22no—
119SD16×C101D414yescomplemented
120C2×C8×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—42no—
126C23×C101—42no—
127C23×C101—81no—
128C23×C101C2×D411yescharacteristic
129C22×D101—21no—
130C22×D101C2×D411yescharacteristic
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—42no—
135Q8×C101C2×D412yes—
136D4×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—42no—
140D4×C101C2×D412yes—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—42no—
146C2×C4×C101C2×D411yescharacteristic
147C2×C4×C101C2411yescharacteristic
148C22⋊D101—22no—
149C22⋊D101—42no—
150C22⋊D101—42no—
151C22⋊D101—81no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—42no—
156C2×Dic101—81no—
157C2×Dic101C2×D411yescharacteristic
158C2×Dic101C2×D411yescharacteristic
159C2×Dic101C2×D411yescharacteristic
160D404—21no—
161D404—21no—
162D404—42no—
163C4×D101—21no—
164C4×D101—21no—
165C4×D101—42no—
166Dic202—21no—
167Dic202—21no—
168C8×C101—21no—
169C8×C101C2×D412yes—
170#11717579—21no—
171#11717579—21no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—22no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—84no—
181C22×C101#13512yescomplemented
182C22×C101C22×D411yescharacteristic
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—21no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—41no—
189C4×C101—22no—
190C4×C101—21no—
191C4×C101—41no—
192C4×C101C22×D411yescharacteristic
193C4×C101H(5,𝔽2)11yescharacteristic
194Dic101—21no—
195Dic101—21no—
196Dic101—21no—
197Dic101—41no—
198Dic101—41no—
199Dic101—41no—
200C2×C101—22no—
201C2×C101—42no—
202C2×C101—81no—
203C2×C101D4211yescharacteristic
204C2×C101C2×#13511yescharacteristic
205C2×C101#57711yescharacteristic
206D101—41no—
207D101—81no—
208#3282—1011nomaximal, Sylow
209C101#328211yescharacteristic, complemented, Sylow
210C2×#135—1011no—
211C22×SD16D10111yescharacteristic, complemented
212#548—1011no—
213#546—1011no—
214#495—1011no—
215#465—1011no—
216#465—1011no—
217#464—1011no—
218#463—1011no—
219#452—1011no—
220#451—1011no—
221#449—1011no—
222#447—1011no—
223C4×SD16—1011no—
224C8×D4—1011no—
225C2×#41—1011no—
226C2×#41—1011no—
227C22×Q8C2×D10111yescharacteristic
228C22×D4C2×D10111yescharacteristic
229#135—2024no—
230C2×Q16—1011no—
231C2×Q16—1011no—
232C2×SD16—24no—
233C2×SD16—1011no—
234C2×SD16C2×D10111yescharacteristic
235C2×SD16C2×D10111yescharacteristic
236C2×SD16C2×D10111yescharacteristic
237C2×SD16C2×D10111yescharacteristic
238C2×D8—1011no—
239C2×D8—1011no—
240C22×C8—1011no—
241C22×C8C2×D10111yescharacteristic
242#124—1011no—
243#124—1011no—
244#124—2021no—
245#123—2021no—
246C22⋊Q8—1011no—
247C22⋊Q8—1011no—
248C22⋊Q8—2021no—
249#121—1011no—
250#121—1011no—
251#121—2021no—
252#121—2021no—
253#120—2021no—
254C4×Q8—1011no—
255C4×D4—1011no—
256C4×D4—1011no—
257C4×D4—2021no—
258C2×C22⋊C4—2021no—
259#106—1011no—
260C4⋊C8—1011no—
261#103—1011no—
262#103—1011no—
263#103—1011no—
264#102—1011no—
265#102—1011no—
266#102—1011no—
267C22⋊C8—1011no—
268C22⋊C8—1011no—
269C4×C8—1011no—
270C24—21no—
271#41—4041no—
272#41—4041no—
273#41—2022no—
274#41—2022no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1011no—
279C2×Q8C22×D10111yescharacteristic
280C2×Q8C22×D10111yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—1011no—
285C2×D4—1011no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4C22×D10111yescharacteristic
291C2×D4C22×D10111yescharacteristic
292C22×C4—21no—
293C22×C4—1011no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4—2021no—
297C22×C4C22×D10111yescharacteristic
298Q16—4041no—
299Q16—2022no—
300SD16—22no—
301SD16—22no—
302SD16—22no—
303SD16—2022no—
304SD16C22⋊D10114yescomplemented
305D8—4041no—
306D8—2022no—
307C2×C8—22no—
308C2×C8—1011no—
309C2×C8—1011no—
310C2×C8—2022no—
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—1011no—
314C4⋊C4—1011no—
315C4⋊C4—1011no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4—2021no—
319C22⋊C4—1011no—
320C22⋊C4—1011no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4—2021no—
330C42—1011no—
331C42—2021no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—1011no—
338C23—2021no—
339C23D4×D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—2021no—
345Q8—2021no—
346Q8C2×C22⋊D10112yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—8081no—
352D4—4042no—
353D4—4042no—
354D4—4042no—
355D4—2022no—
356D4—2021no—
357D4—2021no—
358D4C2×C22⋊D10112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—8081no—
365C2×C4—1011no—
366C2×C4—1011no—
367C2×C4—1011no—
368C2×C4—4042no—
369C2×C4—4042no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4—2021no—
375C2×C4D4×D10111yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—21no—
378C8—2021no—
379C8—2021no—
380C8C2×C22⋊D10112yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—84no—
390C22—4041no—
391C22—4041no—
392C22—4041no—
393C22—2021no—
394C22—2021no—
395C22—2021no—
396C22#1569227612yescomplemented
397C22C2×D4×D10111yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—4041no—
402C4—4041no—
403C4—4041no—
404C4—2021no—
405C4—2021no—
406C4—2021no—
407C4C22×C22⋊D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—8081no—
413C2—4041no—
414C2D4×C22⋊D10111yescharacteristic
415C2C2×#1569227611yescharacteristic
416C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC22×C101C22×D4
hypercenterC22×SD16D101
hypocenterC101#3282
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15692276C2semidirect
2. #20265838C2semidirect
3. #20265821C2semidirect
4. #20265558C2semidirect
5. #20265406C2semidirect
6. #20265403C2semidirect
7. #20265397C2semidirect
8. #20265394C2semidirect
9. #20265336C2semidirect
10. #20265333C2semidirect
11. #20265321C2semidirect
12. #20265311C2semidirect
13. SD16×Dic101C2semidirect
14. C8×C22⋊D101C2semidirect
15. C22×SD16×C101C2semidirect
16. C2×#13108075C22semidirect
19. C2×#13108064C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108063C22semidirect
24. C2×C8×D101C22semidirect
26. #15692244C22semidirect
31. #15692231C22semidirect
37. Q8×Dic101C22semidirect
42. #15692187C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692179C22semidirect
47. #15692177C22semidirect
51. #15692167C22semidirect
52. C8×Dic101C22semidirect
119. SD16×C101D4semidirect
181. C22×C101#135semidirect
209. C101#3282semidirect
211. C22×SD16D101semidirect
304. SD16C22⋊D101semidirect
396. C22#15692276semidirect

Maximal subgroups

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

subgroupindex
C2×#156922762
#202658382
#202658212
#202655582
#202654062
#202654032
#202653972
#202653942
#202653362
#202653332
#202653212
#202653112
SD16×Dic1012
C8×C22⋊D1012
C22×SD16×C1012
#3282101