Finite Group Databasebeta

← #34053972

Subgroups — #34053972

Order 19072.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011959C211yesmaximal, characteristic, complemented
2C2×#18011938C211yesmaximal, characteristic, complemented
3C2×SD16×D149C211yesmaximal, characteristic, complemented
4#23750191C211yesmaximal, characteristic, complemented
5#23749773C211yesmaximal, characteristic, complemented
6#23749772C211yesmaximal, characteristic, complemented
7#23749769C211yesmaximal, characteristic, complemented
8#23749768C211yesmaximal, characteristic, complemented
9#23749765C211yesmaximal, characteristic, complemented
10#23749764C211yesmaximal, characteristic, complemented
11#23749762C211yesmaximal, characteristic, complemented
12#23749632C211yesmaximal, characteristic, complemented
13#23749620C211yesmaximal, characteristic, complemented
14#23749590C211yesmaximal, characteristic, complemented
15#448×C149C211yesmaximal, characteristic, complemented
16#18011959—24no—
17#18011959—24no—
18C2×#14558737—21no—
19C2×#14558737C2211yescharacteristic
20C2×#14558737C2211yescharacteristic, complemented
21C2×#14558736—21no—
22C2×#14558736—21no—
23C2×#14558736C2211yescharacteristic
24C2×#14558736C2211yescharacteristic, complemented
25C2×Q8×D149—21no—
26C2×Q8×D149C2211yescharacteristic
27C2×Q8×D149C2211yescharacteristic, complemented
28C2×D4×D149C2211yescharacteristic
29#18011938—24no—
30C2×#14558725C2211yescharacteristic
31C2×#14558724C2211yescharacteristic
32C2×#14558723C2211yescharacteristic
33C2×#14558723C2211yescharacteristic
34SD16×D149—24no—
35C2×D1192C2211yescharacteristic
36C2×#14558721C2211yescharacteristic
37C2×#14558719C2211yescharacteristic
38C2×C8×D149C2211yescharacteristic
39#18011907—21no—
40#18011897—21no—
41#18011886C2211yescharacteristic
42#18011885C2211yescharacteristic
43#18011882C2211yescharacteristic
44#18011881—21no—
45#18011879—21no—
46#18011869C2211yescharacteristic
47#18011836C2211yescharacteristic, complemented
48#18011835C2211yescharacteristic, complemented
49#18011835C2211yescharacteristic, complemented
50#18011834C2211yescharacteristic, complemented
51#18011833C2211yescharacteristic, complemented
52#18011831C2211yescharacteristic, complemented
53C22⋊#11924746C2211yescharacteristic, complemented
54#18011823C2211yescharacteristic, complemented
55#18011823C2211yescharacteristic, complemented
56C22×Q8×C149C2211yescharacteristic, complemented
57C2×SD16×C149C2211yescharacteristic
58C2×SD16×C149C2211yescharacteristic
59#121×C149C2211yescharacteristic
60#103×C149C2211yescharacteristic, complemented
61#103×C149C2211yescharacteristic, complemented
62C22⋊C8×C149C2211yescharacteristic, complemented
63C23×D149—21no—
64#14558737—28no—
65#14558737—22no—
66#14558737—22no—
67#14558737—22no—
68#14558737D414yescomplemented
69#14558736—28no—
70#14558736—28no—
71#14558736—22no—
72#14558736—22no—
73#14558736—22no—
74#14558736—22no—
75C2×Dic298—21no—
76C2×Dic298—21no—
77C2×Dic298D411yescharacteristic
78C2×Dic298C2311yescharacteristic
79Q8×D149—28no—
80Q8×D149—22no—
81Q8×D149—22no—
82Q8×D149—22no—
83Q8×D149D414yescomplemented
84C2×D596—21no—
85C2×D596—21no—
86C2×D596—21no—
87C2×D596D411yescharacteristic
88C2×D596C2311yescharacteristic
89C2×D596C2311yescharacteristic
90C2×C22⋊D149—21no—
91C2×C22⋊D149—21no—
92C2×C22⋊D149—21no—
93C2×C22⋊D149—21no—
94C2×C22⋊D149D411yescharacteristic
95C2×C22⋊D149D411yescharacteristic
96D4×D149—22no—
97D4×D149—22no—
98C2×C4×D149—21no—
99C2×C4×D149—21no—
100C2×C4×D149—21no—
101C2×C4×D149—21no—
102C2×C4×D149—21no—
103C2×C4×D149C2311yescharacteristic
104C2×C4×D149C2311yescharacteristic
105#14558725—22no—
106#14558724—22no—
107#14558723—22no—
108#14558723—22no—
109D1192—22no—
110#14558721—22no—
111#14558719—22no—
112C2×#11924746C2311yescharacteristic
113C2×#11924746C2311yescharacteristic
114C8×D149—22no—
115C4⋊Dic149C2311yescharacteristic
116C22⋊Dic149—21no—
117#14558713—21no—
118#14558713—21no—
119#14558713—21no—
120C4×Dic149—21no—
121C2×Q8×C149—22no—
122C2×Q8×C149—22no—
123C2×Q8×C149—21no—
124C2×Q8×C149C2311yescharacteristic
125C2×Q8×C149C2311yescharacteristic
126C2×D4×C149—21no—
127C2×D4×C149C2311yescharacteristic
128C22×C4×C149—21no—
129C22×C4×C149C2311yescharacteristic
130SD16×C149—22no—
131SD16×C149—22no—
132C2×C8×C149C2311yescharacteristic
133C2×C8×C149C2311yescharacteristic
134C4⋊C4×C149C2311yescharacteristic
135C22⋊C4×C149—21no—
136C22×D149—21no—
137C22×D149—21no—
138C22×D149—42no—
139C22×D149—81no—
140C22×D149—81no—
141C22×D149C2×D411yescharacteristic
142C22×D149C2×D411yescharacteristic
143Dic298—24no—
144Dic298—21no—
145Dic298—22no—
146Dic298—24no—
147Dic298C2×D412yes—
148D596—24no—
149D596—24no—
150D596—22no—
151D596—21no—
152D596—21no—
153D596—21no—
154D596—81no—
155D596C2×D412yes—
156C22⋊D149—24no—
157C22⋊D149—22no—
158C22⋊D149—22no—
159C22⋊D149—42no—
160C22⋊D149—81no—
161C22⋊D149—81no—
162C2×Dic149—21no—
163C2×Dic149—21no—
164C2×Dic149C2×D411yescharacteristic
165C2×Dic149C2×D411yescharacteristic
166C4×D149—24no—
167C4×D149—24no—
168C4×D149—24no—
169C4×D149—24no—
170C4×D149—21no—
171C4×D149—21no—
172C4×D149—24no—
173C4×D149C2×D412yes—
174C4×D149C2×D412yes—
175#11924746—21no—
176#11924746—21no—
177C23×C149—21no—
178C23×C149C2×D411yescharacteristic
179Q8×C149—24no—
180Q8×C149—21no—
181Q8×C149—21no—
182Q8×C149C2×D412yes—
183Q8×C149C2×D412yes—
184D4×C149—21no—
185D4×C149—21no—
186D4×C149—81no—
187C2×C4×C149—24no—
188C2×C4×C149—21no—
189C2×C4×C149—21no—
190C2×C4×C149—21no—
191C2×C4×C149—21no—
192C2×C4×C149—21no—
193C2×C4×C149C2×D411yescharacteristic
194C2×C4×C149C2411yescharacteristic
195C8×C149—21no—
196C8×C149—21no—
197C22×C149—21no—
198C22×C149—21no—
199C22×C149—21no—
200C22×C149—41no—
201C22×C149—41no—
202C22×C149—41no—
203C22×C149C22×D411yescharacteristic
204C2×D149—21no—
205C2×D149—21no—
206C2×D149—21no—
207C2×D149—22no—
208C2×D149—21no—
209C2×D149—21no—
210C2×D149—41no—
211C2×D149—41no—
212C2×D149—41no—
213C2×D149—41no—
214C2×D149—41no—
215C2×D149—84no—
216C2×D149#12012yes—
217C4×C149—22no—
218C4×C149—22no—
219C4×C149—21no—
220C4×C149—41no—
221C4×C149C22×D411yescharacteristic
222C4×C149C22×D411yescharacteristic
223Dic149—22no—
224Dic149—21no—
225Dic149—41no—
226Dic149#12012yes—
227C2×C149—41no—
228C2×C149—81no—
229C2×C149C2×#12011yescharacteristic
230C2×C149#57711yescharacteristic
231C2×C149#57711yescharacteristic
232D149—22no—
233D149—41no—
234D149—42no—
235D149—81no—
236C149#301011yescharacteristic, complemented, Sylow
237#3010—1491nomaximal, Sylow
238C2×2−1+4—1491no—
239C2×AGL(1,ℤ/8)—1491no—
240C22×SD16—1491no—
241#535—1491no—
242#449—1491no—
243#449—1491no—
244#449—1491no—
245#449—1491no—
246#448—1491no—
247#448—1491no—
248#448—1491no—
249#448D14911yescharacteristic, complemented
250#419—1491no—
251#416—1491no—
252#408—1491no—
2532−1+4—2984no—
2542−1+4—2984no—
255C2×#41—2981no—
256C2×#41—2981no—
257C2×#41—2981no—
258C2×#41—1491no—
259C2×#41—1491no—
260C2×#41—1491no—
261C2×#41—1491no—
262C22×Q8—2981no—
263C22×Q8—1491no—
264C22×Q8—1491no—
265C22×Q8C2×D14911yescharacteristic, complemented
266C22×D4—1491no—
267AGL(1,ℤ/8)—2984no—
268C2×SD16—2984no—
269C2×SD16—1491no—
270C2×SD16—1491no—
271C2×SD16—1491no—
272C2×SD16—1491no—
273C2×SD16C2×D14911yescharacteristic
274C2×SD16C2×D14911yescharacteristic
275C2×D8—1491no—
276C2×D8—1491no—
277C2×M16—1491no—
278C22×C8—1491no—
279#127—2981no—
280#124—2981no—
281#121—1491no—
282#121—1491no—
283#121—1491no—
284#121C2×D14911yescharacteristic
285#120—2981no—
286#120—2981no—
287#117—1491no—
288#103—1491no—
289#103—1491no—
290#103—1491no—
291#103—1491no—
292#103C2×D14911yescharacteristic, complemented
293#103C2×D14911yescharacteristic, complemented
294#102—1491no—
295#102—1491no—
296C22⋊C8—1491no—
297C22⋊C8—1491no—
298C22⋊C8—1491no—
299C22⋊C8C2×D14911yescharacteristic, complemented
300C24—2981no—
301#41—2988no—
302#41—2988no—
303#41—2982no—
304#41—2982no—
305#41—2982no—
306#41—2982no—
307#41—2988no—
308#41—2982no—
309#41—2982no—
310#41—2982no—
311#41—1494no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—2981no—
316C2×Q8—2981no—
317C2×Q8—2988no—
318C2×Q8—2982no—
319C2×Q8—2982no—
320C2×Q8—2982no—
321C2×Q8—1494no—
322C2×Q8—1491no—
323C2×Q8—1491no—
324C2×Q8C22×D14911yescharacteristic
325C2×Q8C22×D14911yescharacteristic
326C2×D4—21no—
327C2×D4—2981no—
328C2×D4—2981no—
329C2×D4—2981no—
330C2×D4—2981no—
331C2×D4—2981no—
332C2×D4—2981no—
333C2×D4—2982no—
334C2×D4—2982no—
335C2×D4—2981no—
336C2×D4—1491no—
337C2×D4—1491no—
338C2×D4—1491no—
339C2×D4—1491no—
340C2×D4—1491no—
341C2×D4C22×D14911yescharacteristic
342C22×C4—21no—
343C22×C4—2981no—
344C22×C4—2981no—
345C22×C4—2981no—
346C22×C4—2981no—
347C22×C4—2981no—
348C22×C4—1491no—
349C22×C4—1491no—
350C22×C4C22×D14911yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—2982no—
354SD16—2982no—
355SD16—2982no—
356SD16—2982no—
357D8—2982no—
358D8—2982no—
359M16—2982no—
360C2×C8—2982no—
361C2×C8—1491no—
362C2×C8—1491no—
363C2×C8C22×D14911yescharacteristic
364C2×C8C22×D14911yescharacteristic
365C4⋊C4—1491no—
366C4⋊C4C22×D14911yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—2981no—
369C22⋊C4—2981no—
370C22⋊C4—2981no—
371C22⋊C4—2981no—
372C42—2981no—
373C23—21no—
374C23—2981no—
375C23—2981no—
376C23—5962no—
377C23—1491no—
378C23—1491no—
379C23—11921no—
380C23—11921no—
381C23D4×D14911yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—2984no—
386Q8—2984no—
387Q8—2982no—
388Q8—2981no—
389Q8—1492no—
390Q8D4×D14912yes—
391Q8D4×D14912yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—2984no—
396D4—2984no—
397D4—2982no—
398D4—2984no—
399D4—2982no—
400D4—2982no—
401D4—2981no—
402D4—2981no—
403D4—2981no—
404D4—5962no—
405D4—1492no—
406D4—11921no—
407D4—11921no—
408D4—11921no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—2984no—
416C2×C4—2984no—
417C2×C4—2984no—
418C2×C4—2984no—
419C2×C4—2984no—
420C2×C4—2981no—
421C2×C4—2981no—
422C2×C4—2981no—
423C2×C4—2981no—
424C2×C4—1491no—
425C2×C4—1491no—
426C2×C4—1492no—
427C2×C4—1492no—
428C2×C4D4×D14911yescharacteristic
429C2×C4C23×D14911yescharacteristic
430C8—21no—
431C8—21no—
432C8—2981no—
433C8—2981no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—2981no—
441C22—2981no—
442C22—2981no—
443C22—2981no—
444C22—2981no—
445C22—2982no—
446C22—5961no—
447C22—5961no—
448C22—5961no—
449C22—5961no—
450C22—5961no—
451C22—1492no—
452C22—11924no—
453C22C2×D4×D14911yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—2982no—
459C4—2981no—
460C4—5961no—
461C4—1492no—
462C4C2×D4×D14911yescharacteristic
463C4C2×D4×D14911yescharacteristic
464C2—41no—
465C2—81no—
466C2—2982no—
467C2—5961no—
468C2—5962no—
469C2—11921no—
470C2#120×D14911yescharacteristic
471C2#2375043111yescharacteristic
472C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#448×C149C2
socleC22×C149C22×D4
hypercenter#448D149
hypocenterC149#3010
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#18011959C2semidirect
2. C2×#18011938C2semidirect
3. C2×SD16×D149C2semidirect
4. #23750191C2semidirect
5. #23749773C2semidirect
6. #23749772C2semidirect
7. #23749769C2semidirect
8. #23749768C2semidirect
9. #23749765C2semidirect
10. #23749764C2semidirect
11. #23749762C2semidirect
12. #23749632C2semidirect
13. #23749620C2semidirect
14. #23749590C2semidirect
15. #448×C149C2semidirect
20. C2×#14558737C22semidirect
24. C2×#14558736C22semidirect
27. C2×Q8×D149C22semidirect
47. #18011836C22semidirect
48. #18011835C22semidirect
49. #18011835C22semidirect
50. #18011834C22semidirect
51. #18011833C22semidirect
52. #18011831C22semidirect
53. C22⋊#11924746C22semidirect
54. #18011823C22semidirect
55. #18011823C22semidirect
56. C22×Q8×C149C22semidirect
60. #103×C149C22semidirect
61. #103×C149C22semidirect
62. C22⋊C8×C149C22semidirect
68. #14558737D4semidirect
83. Q8×D149D4semidirect
236. C149#3010semidirect
249. #448D149semidirect
265. C22×Q8C2×D149semidirect
292. #103C2×D149semidirect
293. #103C2×D149semidirect
299. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119592
C2×#180119382
C2×SD16×D1492
#237501912
#237497732
#237497722
#237497692
#237497682
#237497652
#237497642
#237497622
#237496322
#237496202
#237495902
#448×C1492
#3010149