Finite Group Databasebeta

← C2×#23750304 #34058191

Subgroups — C2×#23750304 #34058191

Order 19072.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750304C2116yesmaximal, direct product factor
2C2×#18011906C211yesmaximal, characteristic, complemented
3C2×#18011905C211yesmaximal, characteristic, complemented
4C2×#18011897C211yesmaximal, characteristic, complemented
5C2×#18011894C211yesmaximal, characteristic, complemented
6C2×#18011893C211yesmaximal, characteristic, complemented
7C2×#18011892C211yesmaximal, characteristic, complemented
8C2×#18011890C211yesmaximal, characteristic, complemented
9C2×#18011889C211yesmaximal, characteristic, complemented
10C2×#18011879C211yesmaximal, characteristic
11C2×C4×C22⋊D149C211yesmaximal, characteristic, complemented
12C2×#18011868C211yesmaximal, characteristic, complemented
13C2×#18011867C211yesmaximal, characteristic, complemented
14C22×#14558713C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D149C211yesmaximal, characteristic, complemented
16C2×C4×D4×C149C211yesmaximal, characteristic, complemented
17C24×D149C2211yescharacteristic
18C22×Dic298C2211yescharacteristic, complemented
19C22×C22⋊D149C2211yescharacteristic
20C22×C22⋊D149C2211yescharacteristic
21C23×Dic149C2211yescharacteristic
22C22×C4×D149C2211yescharacteristic
23C22×C4×D149C2211yescharacteristic
24#18011906C2218yescomplemented
25#18011905C2218yescomplemented
26#18011897C2218yescomplemented
27#18011894C2218yescomplemented
28#18011893C2218yescomplemented
29#18011892C2218yescomplemented
30#18011890C2218yescomplemented
31#18011889C2218yescomplemented
32#18011879C2218yes—
33C4×C22⋊D149C2218yescomplemented
34#18011868C2218yescomplemented
35#18011867C2218yescomplemented
36C2×C4⋊Dic149C2211yescharacteristic, complemented
37C2×#14558715C2211yescharacteristic, complemented
38C2×#14558715C2211yescharacteristic, complemented
39C2×#14558715C2211yescharacteristic, complemented
40C2×#14558715C2211yescharacteristic, complemented
41C2×#14558715C2211yescharacteristic, complemented
42C2×#14558715C2211yescharacteristic, complemented
43C2×C22⋊Dic149C2211yescharacteristic
44C2×C22⋊Dic149C2211yescharacteristic
45C2×C22⋊Dic149C2211yescharacteristic, complemented
46C2×C22⋊Dic149C2211yescharacteristic, complemented
47C2×#14558713C2211yescharacteristic
48C2×#14558713C2211yescharacteristic
49C2×#14558713C2211yescharacteristic
50C2×#14558713C2211yescharacteristic
51C2×#14558713C2211yescharacteristic
52C2×#14558713C2211yescharacteristic
53C2×#14558713C2211yescharacteristic
54C2×#14558713C2211yescharacteristic
55C2×#14558713C2218yescomplemented
56C22⋊C4×D149C2218yescomplemented
57C2×C4×Dic149C2211yescharacteristic, complemented
58C2×C4×Dic149C2211yescharacteristic, complemented
59C22×D4×C149C2211yescharacteristic
60C23×C4×C149C2211yescharacteristic
61C23×C4×C149C2211yescharacteristic, complemented
62C4×D4×C149C2218yescomplemented
63C2×C4⋊C4×C149C2211yescharacteristic, complemented
64C2×C22⋊C4×C149C2211yescharacteristic
65C2×C22⋊C4×C149C2211yescharacteristic, complemented
66C2×C42×C149C2211yescharacteristic, complemented
67C23×D149—24no—
68C23×D149—24no—
69C23×D149—42no—
70C23×D149C2311yescharacteristic
71C23×D149C2314yes—
72C23×D149C2311yescharacteristic
73C2×Dic298—42no—
74C2×Dic298C2314yescomplemented
75C2×C22⋊D149—24no—
76C2×C22⋊D149—42no—
77C2×C22⋊D149C2314yes—
78C2×C22⋊D149C2314yes—
79C22×Dic149—24no—
80C22×Dic149C2311yescharacteristic
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2311yescharacteristic
83C22×Dic149C2311yescharacteristic
84C22×Dic149C2314yes—
85C22×Dic149C2311yescharacteristic
86C22×Dic149C2311yescharacteristic
87C2×C4×D149—24no—
88C2×C4×D149—42no—
89C2×C4×D149C2314yes—
90C2×C4×D149C2314yes—
91C4⋊Dic149C2314yescomplemented
92#14558715C2314yescomplemented
93#14558715C2314yescomplemented
94#14558715C2314yescomplemented
95#14558715C2314yescomplemented
96#14558715C2314yescomplemented
97#14558715C2314yescomplemented
98C22⋊Dic149C2314yes—
99C22⋊Dic149C2314yes—
100C22⋊Dic149C2314yescomplemented
101C22⋊Dic149C2314yescomplemented
102#14558713C2314yes—
103#14558713C2314yes—
104#14558713C2314yes—
105#14558713C2314yes—
106#14558713C2314yes—
107#14558713C2314yes—
108#14558713C2314yes—
109#14558713C2314yes—
110C4×Dic149C2314yescomplemented
111C4×Dic149C2314yescomplemented
112C24×C149C2311yescharacteristic
113C24×C149C2311yescharacteristic
114C2×D4×C149—24no—
115C2×D4×C149C2314yes—
116C22×C4×C149—42no—
117C22×C4×C149—42no—
118C22×C4×C149C2311yescharacteristic
119C22×C4×C149C2311yescharacteristic
120C22×C4×C149C2311yescharacteristic
121C22×C4×C149C2311yescharacteristic
122C22×C4×C149C2311yescharacteristic
123C22×C4×C149C2314yes—
124C22×C4×C149C2314yescomplemented
125C4⋊C4×C149C2314yescomplemented
126C22⋊C4×C149C2314yes—
127C22⋊C4×C149C2314yescomplemented
128C42×C149C2314yescomplemented
129C22×D149—22no—
130C22×D149—22no—
131C22×D149—22no—
132C22×D149—28no—
133C22×D149—22no—
134C22×D149—22no—
135C22×D149—28no—
136C22×D149—44no—
137C22×D149—416no—
138C22×D149#4114yes—
139C22×D149C2412yes—
140C22×D149C2412yes—
141Dic298—44no—
142C22⋊D149—28no—
143C22⋊D149—44no—
144C2×Dic149—22no—
145C2×Dic149—28no—
146C2×Dic149—22no—
147C2×Dic149—22no—
148C2×Dic149—22no—
149C2×Dic149—22no—
150C2×Dic149—22no—
151C2×Dic149C2412yes—
152C2×Dic149C2412yes—
153C2×Dic149C2412yes—
154C2×Dic149C2412yes—
155C2×Dic149C2412yes—
156C2×Dic149C2412yes—
157C4×D149—28no—
158C4×D149—44no—
159C23×C149—22no—
160C23×C149—22no—
161C23×C149—22no—
162C23×C149—22no—
163C23×C149—22no—
164C23×C149#4114yes—
165C23×C149C2412yes—
166C23×C149C2412yes—
167C23×C149C2411yescharacteristic
168D4×C149—28no—
169C2×C4×C149—22no—
170C2×C4×C149—22no—
171C2×C4×C149—22no—
172C2×C4×C149—22no—
173C2×C4×C149—22no—
174C2×C4×C149—44no—
175C2×C4×C149—44no—
176C2×C4×C149C2412yes—
177C2×C4×C149C2412yes—
178C2×C4×C149C2412yes—
179C2×C4×C149C2412yes—
180C2×C4×C149C2412yes—
181C22×C149—28no—
182C22×C149—22no—
183C22×C149—22no—
184C22×C149—22no—
185C22×C149—22no—
186C22×C149—22no—
187C22×C149—44no—
188C22×C149C2×#4112yes—
189C22×C149C2×#4114yes—
190C22×C149C2×#4112yes—
191C22×C149H(5,𝔽2)12yes—
192C22×C149C2511yescharacteristic
193C2×D149—28no—
194C2×D149—22no—
195C2×D149—22no—
196C2×D149—22no—
197C2×D149—22no—
198C2×D149—22no—
199C2×D149—416no—
200C2×D149—44no—
201C2×D149C2×#4114yes—
202C4×C149—22no—
203C4×C149—22no—
204C4×C149—22no—
205C4×C149—22no—
206C4×C149—22no—
207Dic149—22no—
208Dic149—22no—
209Dic149—22no—
210Dic149—22no—
211Dic149—22no—
212Dic149—22no—
213C2×C149—24no—
214C2×C149—42no—
215C2×C149#55114yescomplemented
216C2×C149C22×#4111yescharacteristic
217C2×C149C22×#4111yescharacteristic
218C2×C149C2×H(5,𝔽2)11yescharacteristic
219D149—24no—
220D149—42no—
221C149C2×#55111yescharacteristic, complemented, Sylow
222C2×#551—1491nomaximal, Sylow
223#551—14916no—
224C2×#126—1491no—
225C2×#126—1491no—
226C2×#124—1491no—
227C2×#123—1491no—
228C2×#123—1491no—
229C2×#123—1491no—
230C2×C22⋊Q8—1491no—
231C2×C22⋊Q8—1491no—
232C2×#120—1491no—
233C2×C4×D4—1491no—
234C2×C4×D4D14911yescharacteristic, complemented
235C2×#117—1491no—
236C2×#117—1491no—
237C22×C22⋊C4—1491no—
238C22×C22⋊C4—1491no—
239C25—1491no—
240C22×Q8—1491no—
241C22×D4—1491no—
242C22×D4—1491no—
243C22×D4C2×D14911yescharacteristic
244C23×C4—1491no—
245C23×C4—1491no—
246C23×C4—1491no—
247C23×C4C2×D14911yescharacteristic
248C23×C4C2×D14911yescharacteristic, complemented
249#126—1498no—
250#126—1498no—
251#124—1498no—
252#123—1498no—
253#123—1498no—
254#123—1498no—
255C22⋊Q8—1498no—
256C22⋊Q8—1498no—
257#120—1498no—
258C4×D4—1498no—
259C4×D4C2×D14918yescomplemented
260#117—1498no—
261#117—1498no—
262C2×C4⋊C4—1491no—
263C2×C4⋊C4—1491no—
264C2×C4⋊C4—1491no—
265C2×C4⋊C4—1491no—
266C2×C4⋊C4—1491no—
267C2×C4⋊C4—1491no—
268C2×C4⋊C4—1491no—
269C2×C4⋊C4C2×D14911yescharacteristic, complemented
270C2×C22⋊C4—1491no—
271C2×C22⋊C4—1491no—
272C2×C22⋊C4—1491no—
273C2×C22⋊C4—1491no—
274C2×C22⋊C4—1491no—
275C2×C22⋊C4—1491no—
276C2×C22⋊C4—1491no—
277C2×C22⋊C4—1491no—
278C2×C22⋊C4—1491no—
279C2×C22⋊C4—1491no—
280C2×C22⋊C4—1498no—
281C2×C22⋊C4—1491no—
282C2×C22⋊C4—1491no—
283C2×C22⋊C4—1498no—
284C2×C22⋊C4C2×D14911yescharacteristic
285C2×C22⋊C4C2×D14911yescharacteristic, complemented
286C2×C42—1491no—
287C2×C42—1491no—
288C2×C42C2×D14911yescharacteristic, complemented
289C24—2984no—
290C24—2984no—
291C24—5962no—
292C24—1491no—
293C24—1494no—
294C24—1491no—
295C24C22×D14911yescharacteristic
296C24C22×D14911yescharacteristic
297C2×Q8—5962no—
298C2×Q8—1494no—
299C2×D4—24no—
300C2×D4—2984no—
301C2×D4—5962no—
302C2×D4—1494no—
303C2×D4—1494no—
304C2×D4C22×D14914yes—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—2984no—
308C22×C4—2984no—
309C22×C4—5962no—
310C22×C4—1494no—
311C22×C4—1491no—
312C22×C4—1491no—
313C22×C4—1491no—
314C22×C4—1491no—
315C22×C4—1491no—
316C22×C4—1491no—
317C22×C4—1494no—
318C22×C4—1494no—
319C22×C4C22×D14911yescharacteristic
320C22×C4C22×D14911yescharacteristic
321C22×C4C22×D14911yescharacteristic
322C22×C4C22×D14911yescharacteristic
323C22×C4C22×D14911yescharacteristic
324C22×C4C22×D14914yes—
325C22×C4C22×D14914yescomplemented
326C4⋊C4—1494no—
327C4⋊C4—1494no—
328C4⋊C4—1494no—
329C4⋊C4—1494no—
330C4⋊C4—1494no—
331C4⋊C4—1494no—
332C4⋊C4—1494no—
333C4⋊C4C22×D14914yescomplemented
334C22⋊C4—1494no—
335C22⋊C4—1494no—
336C22⋊C4—1494no—
337C22⋊C4—1494no—
338C22⋊C4—1494no—
339C22⋊C4—1494no—
340C22⋊C4—1494no—
341C22⋊C4—1494no—
342C22⋊C4—1494no—
343C22⋊C4—1494no—
344C22⋊C4—1494no—
345C22⋊C4—1494no—
346C22⋊C4C22×D14914yes—
347C22⋊C4C22×D14914yescomplemented
348C42—1494no—
349C42—1494no—
350C42C22×D14914yescomplemented
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—2988no—
357C23—2982no—
358C23—2982no—
359C23—2982no—
360C23—2988no—
361C23—2982no—
362C23—2982no—
363C23—5964no—
364C23—59616no—
365C23—1492no—
366C23—1494no—
367C23—1492no—
368C23#1455873614yes—
369C23C23×D14912yes—
370C23C23×D14912yes—
371C23C23×D14911yescharacteristic
372Q8—5964no—
373D4—28no—
374D4—2988no—
375D4—5964no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—2988no—
384C2×C4—2982no—
385C2×C4—2982no—
386C2×C4—2988no—
387C2×C4—2982no—
388C2×C4—2982no—
389C2×C4—2982no—
390C2×C4—2982no—
391C2×C4—5964no—
392C2×C4—1492no—
393C2×C4—1492no—
394C2×C4—1492no—
395C2×C4—1492no—
396C2×C4—1492no—
397C2×C4—1492no—
398C2×C4C23×D14912yes—
399C2×C4C23×D14912yes—
400C2×C4C23×D14912yes—
401C2×C4C23×D14912yes—
402C2×C4C23×D14912yes—
403C22—22no—
404C22—22no—
405C22—28no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—44no—
410C22—2982no—
411C22—2982no—
412C22—2988no—
413C22—2982no—
414C22—2982no—
415C22—2982no—
416C22—59616no—
417C22—5964no—
418C22—1494no—
419C22#41×D14912yes—
420C22C2×#1455873614yes—
421C22C2×#1455873612yes—
422C22#1801195612yes—
423C22C24×D14911yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—2982no—
430C4—2982no—
431C4—2982no—
432C4—2982no—
433C4—2982no—
434C4—2982no—
435C2—24no—
436C2—42no—
437C2—2984no—
438C2—5962no—
439C2#2375030414yesdirect product factor
440C2C2×#41×D14911yescharacteristic
441C2C22×#1455873611yescharacteristic
442C2C2×#1801195611yescharacteristic
subgroupG/H
centerC23C23×D149
commutator, cosocleC22×C149C25
FrattiniC22C24×D149
FittingC2×C4×D4×C149C2
socleC23×C149C24
prime radicalC2×#18011879C2
hypercenterC2×C4×D4D149
hypocenterC149C2×#551
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #23750304C2direct (N × H)
2. C2×#18011906C2semidirect
3. C2×#18011905C2semidirect
4. C2×#18011897C2semidirect
5. C2×#18011894C2semidirect
6. C2×#18011893C2semidirect
7. C2×#18011892C2semidirect
8. C2×#18011890C2semidirect
9. C2×#18011889C2semidirect
11. C2×C4×C22⋊D149C2semidirect
12. C2×#18011868C2semidirect
13. C2×#18011867C2semidirect
14. C22×#14558713C2semidirect
15. C2×C22⋊C4×D149C2semidirect
16. C2×C4×D4×C149C2semidirect
18. C22×Dic298C22semidirect
24. #18011906C22semidirect
25. #18011905C22semidirect
26. #18011897C22semidirect
27. #18011894C22semidirect
28. #18011893C22semidirect
29. #18011892C22semidirect
30. #18011890C22semidirect
31. #18011889C22semidirect
33. C4×C22⋊D149C22semidirect
34. #18011868C22semidirect
35. #18011867C22semidirect
36. C2×C4⋊Dic149C22semidirect
37. C2×#14558715C22semidirect
38. C2×#14558715C22semidirect
39. C2×#14558715C22semidirect
40. C2×#14558715C22semidirect
41. C2×#14558715C22semidirect
42. C2×#14558715C22semidirect
45. C2×C22⋊Dic149C22semidirect
46. C2×C22⋊Dic149C22semidirect
55. C2×#14558713C22semidirect
56. C22⋊C4×D149C22semidirect
57. C2×C4×Dic149C22semidirect
58. C2×C4×Dic149C22semidirect
61. C23×C4×C149C22semidirect
62. C4×D4×C149C22semidirect
63. C2×C4⋊C4×C149C22semidirect
65. C2×C22⋊C4×C149C22semidirect
66. C2×C42×C149C22semidirect
74. C2×Dic298C23semidirect
91. C4⋊Dic149C23semidirect
92. #14558715C23semidirect
93. #14558715C23semidirect
94. #14558715C23semidirect
95. #14558715C23semidirect
96. #14558715C23semidirect
97. #14558715C23semidirect
100. C22⋊Dic149C23semidirect
101. C22⋊Dic149C23semidirect
110. C4×Dic149C23semidirect
111. C4×Dic149C23semidirect
124. C22×C4×C149C23semidirect
125. C4⋊C4×C149C23semidirect
127. C22⋊C4×C149C23semidirect
128. C42×C149C23semidirect
215. C2×C149#551semidirect
221. C149C2×#551semidirect
234. C2×C4×D4D149semidirect
248. C23×C4C2×D149semidirect
259. C4×D4C2×D149semidirect
269. C2×C4⋊C4C2×D149semidirect
285. C2×C22⋊C4C2×D149semidirect
288. C2×C42C2×D149semidirect
325. C22×C4C22×D149semidirect
333. C4⋊C4C22×D149semidirect
347. C22⋊C4C22×D149semidirect
350. C42C22×D149semidirect
439. C2#23750304direct (N × H)

Maximal subgroups

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

subgroupindex
#237503042
C2×#180119062
C2×#180119052
C2×#180118972
C2×#180118942
C2×#180118932
C2×#180118922
C2×#180118902
C2×#180118892
C2×#180118792
C2×C4×C22⋊D1492
C2×#180118682
C2×#180118672
C22×#145587132
C2×C22⋊C4×D1492
C2×C4×D4×C1492
C2×#551149