Finite Group Databasebeta

← #27184778

Subgroups — #27184778

Order 12928.

1492 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×#15692302C211yesmaximal, characteristic, complemented
2C22×Q8×D101C211yesmaximal, characteristic, complemented
3Q8×D404C211yesmaximal, characteristic, complemented
4D4×Dic202C213yesmaximal, complemented
5Q8×C22⋊D101C211yesmaximal, characteristic, complemented
6#20265840C213yesmaximal, complemented
7#20265835C213yesmaximal, complemented
8#20265834C211yesmaximal, characteristic, complemented
9#20265833C213yesmaximal, complemented
10#20265750C213yesmaximal, complemented
11#20265747C213yesmaximal, complemented
12#20265687C213yesmaximal, complemented
13#20265686C213yesmaximal, complemented
14#20265653C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16#15692302—28no—
17C2×#13108074C2213yes—
18C2×#13108073C2213yes—
19C2×#13108073C2213yescomplemented
20C22×Dic202C2213yes—
21C22×Dic202C2213yescomplemented
22C2×Q8×D101—28no—
23C2×Q8×D101C2213yes—
24C2×Q8×D101C2211yescharacteristic
25C2×Q8×D101C2213yes—
26C2×Q8×D101C2211yescharacteristic
27C2×Q8×D101C2211yescharacteristic, complemented
28C22×C4×D101C2213yes—
29#15692255C2213yescomplemented
30#15692254C2216yescomplemented
31#15692253C2213yescomplemented
32#15692242C2213yescomplemented
33#15692241C2216yescomplemented
34#15692240C2213yes—
35#15692237C2213yes—
36#15692237C2216yes—
37#15692235C2213yes—
38#15692235C2213yescomplemented
39#15692234C2213yes—
40#15692234C2213yescomplemented
41#15692233C2213yes—
42#15692232C2213yes—
43#15692232C2216yescomplemented
44#15692232C2216yescomplemented
45#15692231C2213yescomplemented
46#15692228C2213yes—
47#15692227C2213yes—
48#15692226C2213yes—
49#15692225C2211yescharacteristic
50#15692223C2213yes—
51Q8×Dic101C2211yescharacteristic, complemented
52C4×Dic202C2213yescomplemented
53#15692220C2213yescomplemented
54C4×D404C2213yes—
55D4×Dic101C2216yes—
56C4×C22⋊D101C2213yes—
57C4×C22⋊D101C2213yes—
58#15692215C2216yes—
59#15692213C2213yes—
60#15692212C2213yes—
61C2×#41×C101C2211yescharacteristic, complemented
62C22×Q8×C101C2211yescharacteristic
63#124×C101C2213yescomplemented
64C22⋊Q8×C101C2213yescomplemented
65#121×C101C2213yes—
66C4×Q8×C101C2211yescharacteristic, complemented
67C4×D4×C101C2213yes—
68C23×D101C2311yescharacteristic
69#13108074—212no—
70#13108073—212no—
71#13108073—212no—
72C2×Dic202—212no—
73C2×Dic202—212no—
74C2×Dic202C2316yes—
75C2×Dic202C2313yes—
76C2×Dic202C2313yes—
77C2×Dic202C2313yes—
78C2×Dic202C2313yes—
79Q8×D101—24no—
80Q8×D101—24no—
81Q8×D101—212no—
82Q8×D101—212no—
83Q8×D101D418yescomplemented
84C2×D404C2311yescharacteristic
85C2×D404C2311yescharacteristic
86C2×C22⋊D101C2313yes—
87C2×C22⋊D101C2313yes—
88C2×C22⋊D101C2313yes—
89C2×C22⋊D101C2311yescharacteristic
90C22×Dic101C2313yes—
91C22×Dic101C2316yes—
92C2×C4×D101—212no—
93C2×C4×D101C2313yes—
94C2×C4×D101C2313yes—
95C2×C4×D101C2313yes—
96C2×C4×D101C2313yes—
97C2×C4×D101C2313yes—
98C4⋊Dic101C2313yes—
99C4⋊Dic101C2316yes—
100C4⋊Dic101C2313yes—
101#13108053C2313yes—
102#13108053C2313yes—
103#13108053C2316yes—
104#13108053C2313yes—
105#13108053C2313yes—
106C22⋊Dic101C2313yes—
107C22⋊Dic101C2313yes—
108C22⋊Dic101C2316yes—
109#13108051C2311yescharacteristic
110#13108051C2313yes—
111#13108051C2313yes—
112#13108051C2311yescharacteristic
113#13108051C2313yes—
114#13108051C2313yes—
115C4×Dic101C2313yes—
116C4×Dic101C2316yes—
117#41×C101—24no—
118C2×Q8×C101—24no—
119C2×Q8×C101C2313yes—
120C2×Q8×C101C2311yescharacteristic
121C2×D4×C101C2313yes—
122C2×D4×C101C2313yes—
123C22×C4×C101C2313yes—
124C22×C4×C101C2313yes—
125C4⋊C4×C101C2313yes—
126C4⋊C4×C101C2313yes—
127C22⋊C4×C101C2311yescharacteristic
128C22⋊C4×C101C2313yes—
129C22⋊C4×C101C2316yes—
130C42×C101C2313yes—
131C23×C101C2413yes—
132C23×C101C2411yescharacteristic
133C22×D101—22no—
134C22×D101—22no—
135C22×D101—22no—
136C22×D101C2411yescharacteristic
137C22×D101C2411yescharacteristic
138C22×D101C2411yescharacteristic
139Q8×C101—26no—
140Q8×C101C2×D414yes—
141D4×C101—26no—
142D4×C101—26no—
143C2×C4×C101—26no—
144C2×C4×C101—26no—
145C2×C4×C101C2411yescharacteristic
146C2×C4×C101C2413yes—
147C2×C4×C101C2413yes—
148C2×C4×C101C2411yescharacteristic
149C2×C4×C101C2413yes—
150C22⋊D101—22no—
151C22⋊D101—26no—
152C22⋊D101—26no—
153C22⋊D101—26no—
154C2×Dic101—26no—
155C2×Dic101—212no—
156C2×Dic101C2411yescharacteristic
157C2×Dic101C2413yes—
158C2×Dic101C2413yes—
159C2×Dic101C2413yes—
160C2×Dic101C2413yes—
161D404—22no—
162D404—22no—
163C4×D101—26no—
164C4×D101—26no—
165C4×D101—26no—
166C4×D101—26no—
167C4×D101C2×D4112yes—
168Dic202—26no—
169Dic202—26no—
170Dic202—212no—
171Dic202—26no—
172Dic202C2×D4112yes—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—23no—
176C22×C101—23no—
177C22×C101—23no—
178C22×C1012−1+412yescomplemented
179C22×C101C2511yescharacteristic
180C2×D101—28no—
181C2×D101—21no—
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—21no—
188C2×D101C22×D412yes—
189C2×D1012−1+412yescomplemented
190C4×C101—21no—
191C4×C101—23no—
192C4×C101—21no—
193C4×C101—23no—
194C4×C101C22×D416yes—
195Dic101—23no—
196Dic101—23no—
197Dic101—23no—
198Dic101—21no—
199Dic101C22×D416yes—
200C2×C101—22no—
201C2×C101—43no—
202C2×C101C23×D411yescharacteristic
203C2×C101C2×2−1+411yescharacteristic
204C2×C101C2×2−1+411yescharacteristic
205D101—22no—
206D101—22no—
207D101—41no—
208#3486—1011nomaximal, Sylow
209C101#348611yescharacteristic, complemented, Sylow
210C2×2−1+4—1011no—
211C23×Q8—1011no—
212D4×Q8—1011no—
213D4×Q8—1013no—
214D4×Q8—1011no—
215D4×Q8—1013no—
216#548—1011no—
217#548—1013no—
218#548—1013no—
219#548D10111yescharacteristic, complemented
220#536—1013no—
221#536—1013no—
222#525—1013no—
223#525—1013no—
224#519—1011no—
2252−1+4—2028no—
226C2×#41—1013no—
227C2×#41—1013no—
228C2×#41—1013no—
229C2×#41C2×D10111yescharacteristic, complemented
230C22×Q8—1011no—
231C22×Q8—1013no—
232C22×Q8—1013no—
233C22×Q8—1011no—
234C22×Q8—1011no—
235C22×Q8—1013no—
236C22×Q8—1013no—
237C22×Q8—2028no—
238C22×Q8C2×D10111yescharacteristic
239C23×C4—1013no—
240C4⋊Q8—1013no—
241C4⋊Q8—1016no—
242C4⋊Q8—1013no—
243#124—1013no—
244#124—1016no—
245#124C2×D10113yescomplemented
246#123—1013no—
247#123—1016no—
248#123—1013no—
249C22⋊Q8—1013no—
250C22⋊Q8—1013no—
251C22⋊Q8—1013no—
252C22⋊Q8—1013no—
253C22⋊Q8—1013no—
254C22⋊Q8—1013no—
255C22⋊Q8—1013no—
256C22⋊Q8—1016no—
257C22⋊Q8—1016no—
258C22⋊Q8C2×D10113yescomplemented
259#121—1013no—
260#121—1013no—
261#121—1013no—
262#121C2×D10113yes—
263#120—1013no—
264#120—1011no—
265C4×Q8—1011no—
266C4×Q8—1013no—
267C4×Q8—1013no—
268C4×Q8C2×D10111yescharacteristic, complemented
269C4×D4—1016no—
270C4×D4—1013no—
271C4×D4—1013no—
272C4×D4—1016no—
273C4×D4—1013no—
274C4×D4C2×D10113yes—
275#117—1013no—
276#117—1013no—
277C24—1011no—
278#41—24no—
279#41—20212no—
280#41—20212no—
281#41—20212no—
282C2×Q8—24no—
283C2×Q8—1018no—
284C2×Q8—1013no—
285C2×Q8—1016no—
286C2×Q8—1013no—
287C2×Q8—1013no—
288C2×Q8—1013no—
289C2×Q8—20212no—
290C2×Q8—20212no—
291C2×Q8—20212no—
292C2×Q8—20212no—
293C2×Q8—2024no—
294C2×Q8—2024no—
295C2×Q8C22×D10113yes—
296C2×Q8C22×D10111yescharacteristic
297C2×D4—1013no—
298C2×D4—1011no—
299C2×D4—1013no—
300C2×D4—1013no—
301C2×D4—1011no—
302C2×D4—1011no—
303C2×D4C22×D10113yes—
304C2×D4C22×D10113yes—
305C22×C4—1013no—
306C22×C4—1013no—
307C22×C4—1016no—
308C22×C4—1013no—
309C22×C4—1013no—
310C22×C4—1013no—
311C22×C4—1013no—
312C22×C4—20212no—
313C22×C4C22×D10113yes—
314C22×C4C22×D10113yes—
315C4⋊C4—1013no—
316C4⋊C4—1013no—
317C4⋊C4—1013no—
318C4⋊C4—1016no—
319C4⋊C4—1013no—
320C4⋊C4—1013no—
321C4⋊C4—1016no—
322C4⋊C4—1013no—
323C4⋊C4C22×D10113yes—
324C4⋊C4C22×D10113yes—
325C22⋊C4—1013no—
326C22⋊C4—1013no—
327C22⋊C4—1016no—
328C22⋊C4—1013no—
329C22⋊C4—1011no—
330C22⋊C4—1011no—
331C22⋊C4—1013no—
332C22⋊C4—1013no—
333C22⋊C4—1013no—
334C22⋊C4C22×D10116yes—
335C22⋊C4C22×D10111yescharacteristic
336C22⋊C4C22×D10113yes—
337C42—1016no—
338C42—1013no—
339C42C22×D10113yes—
340C23—1011no—
341C23—1011no—
342C23—1011no—
343C23—2022no—
344C23—2022no—
345C23—2022no—
346C23C23×D10113yes—
347C23C23×D10111yescharacteristic
348Q8—26no—
349Q8—10112no—
350Q8—20212no—
351Q8—2026no—
352Q8—2026no—
353Q8—2026no—
354Q8D4×D10114yes—
355D4—26no—
356D4—26no—
357D4—2022no—
358D4—2022no—
359D4—2026no—
360D4—2026no—
361D4—2026no—
362D4—2022no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—1013no—
366C2×C4—10112no—
367C2×C4—1013no—
368C2×C4—1011no—
369C2×C4—1013no—
370C2×C4—1013no—
371C2×C4—2026no—
372C2×C4—2026no—
373C2×C4—2026no—
374C2×C4—20212no—
375C2×C4—2026no—
376C2×C4—2026no—
377C2×C4C23×D10113yes—
378C2×C4C23×D10111yescharacteristic
379C2×C4C23×D10113yes—
380C2×C4C23×D10111yescharacteristic
381C2×C4C23×D10113yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—1012no—
388C22—1012no—
389C22—2028no—
390C22—2021no—
391C22—2021no—
392C22—2021no—
393C22—2021no—
394C22—2021no—
395C22—2021no—
396C22—2021no—
397C22#1569230212yescomplemented
398C22C24×D10111yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—1016no—
404C4—2023no—
405C4—2023no—
406C4—2023no—
407C4—2021no—
408C4C2×D4×D10116yes—
409C2—22no—
410C2—43no—
411C2—4041no—
412C2—2022no—
413C2—2022no—
414C2C22×D4×D10111yescharacteristic
415C22−1+4×D10111yescharacteristic
416C2C2×#1569230211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#548×C101C2
hypercenter#548D101
hypocenterC101#3486
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15692302C2semidirect
2. C22×Q8×D101C2semidirect
3. Q8×D404C2semidirect
4. D4×Dic202C2semidirect
5. Q8×C22⋊D101C2semidirect
6. #20265840C2semidirect
7. #20265835C2semidirect
8. #20265834C2semidirect
9. #20265833C2semidirect
10. #20265750C2semidirect
11. #20265747C2semidirect
12. #20265687C2semidirect
13. #20265686C2semidirect
14. #20265653C2semidirect
15. #548×C101C2semidirect
19. C2×#13108073C22semidirect
21. C22×Dic202C22semidirect
27. C2×Q8×D101C22semidirect
29. #15692255C22semidirect
30. #15692254C22semidirect
31. #15692253C22semidirect
32. #15692242C22semidirect
33. #15692241C22semidirect
38. #15692235C22semidirect
40. #15692234C22semidirect
43. #15692232C22semidirect
44. #15692232C22semidirect
45. #15692231C22semidirect
51. Q8×Dic101C22semidirect
52. C4×Dic202C22semidirect
53. #15692220C22semidirect
61. C2×#41×C101C22semidirect
63. #124×C101C22semidirect
64. C22⋊Q8×C101C22semidirect
66. C4×Q8×C101C22semidirect
83. Q8×D101D4semidirect
178. C22×C1012−1+4semidirect
189. C2×D1012−1+4semidirect
209. C101#3486semidirect
219. #548D101semidirect
229. C2×#41C2×D101semidirect
245. #124C2×D101semidirect
258. C22⋊Q8C2×D101semidirect
268. C4×Q8C2×D101semidirect
397. C22#15692302semidirect

Maximal subgroups

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

subgroupindex
C2×#156923022
C22×Q8×D1012
Q8×D4042
D4×Dic2022
Q8×C22⋊D1012
#202658402
#202658352
#202658342
#202658332
#202657502
#202657472
#202656872
#202656862
#202656532
#548×C1012
#3486101