Finite Group Databasebeta

← #34056235

Subgroups — #34056235

Order 19072.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011935C211yesmaximal, characteristic, complemented
2C2×#18011931C211yesmaximal, characteristic, complemented
3#23750293C211yesmaximal, characteristic, complemented
4#23750263C211yesmaximal, characteristic, complemented
5#23750011C211yesmaximal, characteristic, complemented
6#23749874C211yesmaximal, characteristic, complemented
7#23749859C211yesmaximal, characteristic, complemented
8#23749850C211yesmaximal, characteristic, complemented
9#23749818C211yesmaximal, characteristic, complemented
10#23749784C211yesmaximal, characteristic, complemented
11#23749772C211yesmaximal, characteristic, complemented
12#23749760C211yesmaximal, characteristic, complemented
13#23749722C211yesmaximal, characteristic, complemented
14#23749693C211yesmaximal, characteristic, complemented
15#449×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic, complemented
17C2×#14558735C2211yescharacteristic
18C2×D4×D149C2211yescharacteristic
19#18011935—24no—
20#18011931—24no—
21C2×#14558726C2211yescharacteristic, complemented
22C2×#14558725C2211yescharacteristic
23C2×#14558724C2211yescharacteristic
24C2×#14558723C2211yescharacteristic, complemented
25C2×D1192C2211yescharacteristic
26C2×#14558721C2211yescharacteristic
27C2×#14558721C2211yescharacteristic
28C2×#14558719C2211yescharacteristic
29C22×#11924746C2211yescharacteristic, complemented
30#18011907—21no—
31#18011907C2211yescharacteristic, complemented
32#18011900C2211yescharacteristic
33#18011894—21no—
34#18011887C2211yescharacteristic
35#18011886C2211yescharacteristic
36#18011885C2211yescharacteristic
37#18011884—21no—
38#18011883—21no—
39#18011883—21no—
40#18011879—21no—
41#18011876C2211yescharacteristic, complemented
42D4×Dic149C2211yescharacteristic
43C4×C22⋊D149C2211yescharacteristic
44#18011871—21no—
45C2×#14558713—21no—
46#18011844C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011836C2211yescharacteristic, complemented
49#18011833C2211yescharacteristic, complemented
50#18011832C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic
52#18011823C2211yescharacteristic
53#18011821C2211yescharacteristic, complemented
54C2×#41×C149C2211yescharacteristic
55C2×SD16×C149C2211yescharacteristic, complemented
56C2×D8×C149C2211yescharacteristic
57#121×C149C2211yescharacteristic, complemented
58#103×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic
60C22⋊C8×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62#14558736—22no—
63#14558736—41no—
64#14558735—22no—
65#14558735—41no—
66C2×Dic298C2311yescharacteristic
67C2×D596—21no—
68C2×D596C2311yescharacteristic
69C2×D596C2311yescharacteristic
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149D411yescharacteristic
77D4×D149—22no—
78D4×D149—41no—
79C22×Dic149—21no—
80C22×Dic149—21no—
81C2×C4×D149—21no—
82C2×C4×D149C2311yescharacteristic
83#14558726—22no—
84#14558725—22no—
85#14558724D414yescomplemented
86#14558723—22no—
87D1192—41no—
88#14558721—22no—
89#14558721—22no—
90#14558719—22no—
91C2×#11924746—22no—
92C2×#11924746C2311yescharacteristic
93C2×#11924746C2311yescharacteristic
94C4⋊Dic149C2311yescharacteristic
95#14558715—21no—
96#14558715—21no—
97#14558715D411yescharacteristic
98C22⋊Dic149—21no—
99C22⋊Dic149D411yescharacteristic
100#14558713—21no—
101#14558713—21no—
102#14558713—21no—
103#14558713—21no—
104#14558713—21no—
105#14558713D411yescharacteristic
106C4×Dic149—21no—
107C4×Dic149C2311yescharacteristic
108#41×C149—22no—
109#41×C149—41no—
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149—21no—
113C2×D4×C149C2311yescharacteristic
114C2×D4×C149C2311yescharacteristic
115C22×C4×C149—21no—
116C22×C4×C149C2311yescharacteristic
117SD16×C149—41no—
118D8×C149—22no—
119C2×C8×C149C2311yescharacteristic
120C2×C8×C149C2311yescharacteristic
121C4⋊C4×C149C2311yescharacteristic
122C22⋊C4×C149—21no—
123C22×D149—21no—
124C22×D149—21no—
125C22×D149—42no—
126C22×D149—42no—
127C22×D149—81no—
128C22×D149C2×D411yescharacteristic
129Dic298—21no—
130Dic298C2×D412yes—
131D596—21no—
132D596—21no—
133D596—21no—
134D596—21no—
135D596—81no—
136C22⋊D149—41no—
137C22⋊D149—42no—
138C22⋊D149—42no—
139C22⋊D149—42no—
140C22⋊D149—42no—
141C22⋊D149—81no—
142C22⋊D149—81no—
143C2×Dic149—21no—
144C2×Dic149—21no—
145C2×Dic149—42no—
146C2×Dic149—81no—
147C2×Dic149C2×D411yescharacteristic
148C2×Dic149C2×D411yescharacteristic
149C2×Dic149C2×D411yescharacteristic
150C4×D149—21no—
151C4×D149—21no—
152C4×D149—42no—
153#11924746—21no—
154#11924746C2×D412yes—
155C23×C149—21no—
156C23×C149—21no—
157C23×C149C2×D411yescharacteristic
158Q8×C149—21no—
159Q8×C149—21no—
160D4×C149—21no—
161D4×C149—21no—
162D4×C149—21no—
163D4×C149—42no—
164D4×C149—81no—
165D4×C149C2×D412yes—
166C2×C4×C149—21no—
167C2×C4×C149—21no—
168C2×C4×C149—21no—
169C2×C4×C149—21no—
170C2×C4×C149—42no—
171C2×C4×C149C2×D411yescharacteristic
172C2×C4×C149C2411yescharacteristic
173C8×C149—21no—
174C8×C149—21no—
175C22×C149—21no—
176C22×C149—22no—
177C22×C149—21no—
178C22×C149—21no—
179C22×C149—41no—
180C22×C149—41no—
181C22×C149—41no—
182C22×C149—41no—
183C22×C149—41no—
184C22×C149C22×D411yescharacteristic
185C2×D149—21no—
186C2×D149—21no—
187C2×D149—21no—
188C2×D149—41no—
189C2×D149—41no—
190C2×D149—41no—
191C2×D149—41no—
192C2×D149—41no—
193C2×D149—41no—
194C2×D149—84no—
195C4×C149—21no—
196C4×C149—41no—
197C4×C149—41no—
198C4×C149C22×D411yescharacteristic
199C4×C149H(5,𝔽2)11yescharacteristic
200Dic149—21no—
201Dic149—22no—
202Dic149—21no—
203Dic149—21no—
204Dic149—41no—
205C2×C149—42no—
206C2×C149—41no—
207C2×C149—81no—
208C2×C149D4211yescharacteristic
209C2×C149#57511yescharacteristic
210C2×C149#57611yescharacteristic
211D149—41no—
212D149—81no—
213D149—81no—
214C149#326811yescharacteristic, complemented, Sylow
215#3268—1491nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1491no—
217C2×#135—1491no—
218#550—1491no—
219#546—1491no—
220#496—1491no—
221#468—1491no—
222#466—1491no—
223#464—1491no—
224#459—1491no—
225#452—1491no—
226#449—1491no—
227#449D14911yescharacteristic, complemented
228#447—1491no—
229#440—1491no—
230#435—1491no—
231C2×#41—1491no—
232C2×#41—1491no—
233C2×#41C2×D14911yescharacteristic
234C22×D4—1491no—
235AGL(1,ℤ/8)—2984no—
236#135—2984no—
237C2×Q16—1491no—
238C2×SD16—1491no—
239C2×SD16—1491no—
240C2×SD16—1491no—
241C2×SD16C2×D14911yescharacteristic, complemented
242C2×D8—1491no—
243C2×D8—1491no—
244C2×D8—1491no—
245C2×D8C2×D14911yescharacteristic
246C2×M16—1491no—
247C22×C8—1491no—
248#127—2981no—
249#127—1491no—
250#124—1491no—
251#123—2981no—
252C22⋊Q8—1491no—
253#121—2981no—
254#121—2981no—
255#121—2981no—
256#121—1491no—
257#121—1491no—
258#121C2×D14911yescharacteristic, complemented
259#120—2981no—
260C4×Q8—1491no—
261C4×D4—2981no—
262C4×D4—1491no—
263C4×D4—1491no—
264C2×C22⋊C4—2981no—
265#107—1491no—
266C4⋊C8—1491no—
267#103—1491no—
268#103C2×D14911yescharacteristic, complemented
269#102—1491no—
270#102—1491no—
271#102—1491no—
272#102C2×D14911yescharacteristic
273C22⋊C8—1491no—
274C22⋊C8C2×D14911yescharacteristic, complemented
275#97—1491no—
276C24—2981no—
277#41—22no—
278#41—41no—
279#41—2982no—
280#41—2982no—
281#41—5961no—
282#41—5961no—
283C2×Q8—1491no—
284C2×Q8C22×D14911yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2981no—
288C2×D4—2982no—
289C2×D4—2981no—
290C2×D4—2981no—
291C2×D4—2981no—
292C2×D4—2981no—
293C2×D4—2981no—
294C2×D4—2981no—
295C2×D4—5961no—
296C2×D4—1491no—
297C2×D4—1491no—
298C2×D4—1491no—
299C2×D4C22×D14911yescharacteristic
300C2×D4C22×D14911yescharacteristic
301C22×C4—21no—
302C22×C4—2981no—
303C22×C4—2981no—
304C22×C4—2981no—
305C22×C4—1491no—
306C22×C4C22×D14911yescharacteristic
307Q16—2982no—
308SD16—41no—
309SD16—2982no—
310SD16—2982no—
311SD16—1494no—
312D8—22no—
313D8—2982no—
314D8—2982no—
315D8—5961no—
316M16—2982no—
317C2×C8—2982no—
318C2×C8—1491no—
319C2×C8—1491no—
320C2×C8C22×D14911yescharacteristic
321C2×C8C22×D14911yescharacteristic
322C4⋊C4—2981no—
323C4⋊C4—2981no—
324C4⋊C4—1491no—
325C4⋊C4—1491no—
326C4⋊C4C22×D14911yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2981no—
329C22⋊C4—2981no—
330C22⋊C4—2981no—
331C22⋊C4—2981no—
332C22⋊C4—2981no—
333C22⋊C4—2981no—
334C22⋊C4—1491no—
335C22⋊C4—1491no—
336C42—2981no—
337C42—1491no—
338C23—21no—
339C23—21no—
340C23—2981no—
341C23—2981no—
342C23—5962no—
343C23—5962no—
344C23—1491no—
345C23—11921no—
346C23D4×D14911yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2981no—
350Q8—1492no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—81no—
356D4—2981no—
357D4—2981no—
358D4—2981no—
359D4—2981no—
360D4—5961no—
361D4—5962no—
362D4—5962no—
363D4—5962no—
364D4—5962no—
365D4—11921no—
366D4—11921no—
367D4—11921no—
368D4D4×D14912yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2981no—
375C2×C4—2981no—
376C2×C4—2981no—
377C2×C4—2981no—
378C2×C4—5962no—
379C2×C4—5962no—
380C2×C4—1491no—
381C2×C4—1491no—
382C2×C4—1491no—
383C2×C4—11921no—
384C2×C4D4×D14911yescharacteristic
385C2×C4C23×D14911yescharacteristic
386C8—21no—
387C8—21no—
388C8—2981no—
389C8—1492no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—2981no—
400C22—2981no—
401C22—2981no—
402C22—5961no—
403C22—5961no—
404C22—5961no—
405C22—5961no—
406C22—5961no—
407C22—5961no—
408C22—11924no—
409C22C2×D4×D14911yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—2981no—
414C4—2981no—
415C4—2981no—
416C4—2982no—
417C4—5961no—
418C4C2×D4×D14911yescharacteristic
419C4#1801195611yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—5961no—
424C2—11921no—
425C2—11921no—
426C2#2375025711yescharacteristic
427C2#2375042211yescharacteristic
428C2#2375042911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#449×C149C2
socleC22×C149C22×D4
hypercenter#449D149
hypocenterC149#3268
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#18011935C2semidirect
2. C2×#18011931C2semidirect
3. #23750293C2semidirect
4. #23750263C2semidirect
5. #23750011C2semidirect
6. #23749874C2semidirect
7. #23749859C2semidirect
8. #23749850C2semidirect
9. #23749818C2semidirect
10. #23749784C2semidirect
11. #23749772C2semidirect
12. #23749760C2semidirect
13. #23749722C2semidirect
14. #23749693C2semidirect
15. #449×C149C2semidirect
16. C2×#14558736C22semidirect
21. C2×#14558726C22semidirect
24. C2×#14558723C22semidirect
29. C22×#11924746C22semidirect
31. #18011907C22semidirect
41. #18011876C22semidirect
46. #18011844C22semidirect
47. #18011840C22semidirect
48. #18011836C22semidirect
49. #18011833C22semidirect
50. #18011832C22semidirect
53. #18011821C22semidirect
55. C2×SD16×C149C22semidirect
57. #121×C149C22semidirect
58. #103×C149C22semidirect
60. C22⋊C8×C149C22semidirect
85. #14558724D4semidirect
214. C149#3268semidirect
227. #449D149semidirect
241. C2×SD16C2×D149semidirect
258. #121C2×D149semidirect
268. #103C2×D149semidirect
274. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
C2×#180119312
#237502932
#237502632
#237500112
#237498742
#237498592
#237498502
#237498182
#237497842
#237497722
#237497602
#237497222
#237496932
#449×C1492
#3268149