Finite Group Databasebeta

← #34056396

Subgroups — #34056396

Order 19072.

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×#18011931C211yesmaximal, characteristic, complemented
2C2×SD16×D149C211yesmaximal, characteristic, complemented
3#23750285C211yesmaximal, characteristic, complemented
4#23750263C211yesmaximal, characteristic, complemented
5#23750010C211yesmaximal, characteristic, complemented
6#23749857C211yesmaximal, characteristic, complemented
7#23749856C211yesmaximal, characteristic, complemented
8#23749850C211yesmaximal, characteristic, complemented
9#23749842C211yesmaximal, characteristic, complemented
10#23749779C211yesmaximal, characteristic, complemented
11#23749772C211yesmaximal, characteristic, complemented
12#23749760C211yesmaximal, characteristic, complemented
13#23749716C211yesmaximal, characteristic, complemented
14#23749688C211yesmaximal, characteristic, complemented
15#452×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic, complemented
17C2×Q8×D149C2211yescharacteristic
18C2×D4×D149C2211yescharacteristic
19#18011931—24no—
20C2×#14558726C2211yescharacteristic, complemented
21C2×#14558725C2211yescharacteristic
22C2×#14558724C2211yescharacteristic
23C2×#14558723C2211yescharacteristic
24C2×#14558723C2211yescharacteristic, complemented
25SD16×D149—24no—
26C2×D1192C2211yescharacteristic
27C2×#14558721C2211yescharacteristic
28C22×#11924746C2211yescharacteristic, complemented
29C2×C8×D149C2211yescharacteristic
30#18011900C2211yescharacteristic
31#18011900C2211yescharacteristic, complemented
32#18011897—21no—
33#18011894—21no—
34#18011889—21no—
35#18011887C2211yescharacteristic
36#18011886C2211yescharacteristic
37#18011885C2211yescharacteristic
38#18011884—21no—
39#18011883—21no—
40#18011879—21no—
41#18011876C2211yescharacteristic, complemented
42D4×Dic149C2211yescharacteristic
43#18011873—21no—
44C4×C22⋊D149C2211yescharacteristic
45C2×#14558713—21no—
46#18011842C2211yescharacteristic, complemented
47#18011840C2211yescharacteristic, complemented
48#18011836C2211yescharacteristic, complemented
49#18011835C2211yescharacteristic, complemented
50#18011833C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic
52#18011823C2211yescharacteristic
53C8×Dic149C2211yescharacteristic, complemented
54C2×#41×C149C2211yescharacteristic
55C2×Q16×C149C2211yescharacteristic, complemented
56C2×SD16×C149C2211yescharacteristic
57C22⋊Q8×C149C2211yescharacteristic, complemented
58#103×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic
60C22⋊C8×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62#14558736—22no—
63#14558736—41no—
64C2×Dic298—21no—
65C2×Dic298C2311yescharacteristic
66Q8×D149—22no—
67Q8×D149—41no—
68C2×D596—21no—
69C2×D596C2311yescharacteristic
70C2×D596C2311yescharacteristic
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74C2×C22⋊D149D411yescharacteristic
75D4×D149—22no—
76D4×D149—41no—
77C22×Dic149—21no—
78C2×C4×D149—21no—
79C2×C4×D149—21no—
80C2×C4×D149C2311yescharacteristic
81#14558726—22no—
82#14558725—22no—
83#14558724D414yescomplemented
84#14558723—22no—
85#14558723—22no—
86D1192—41no—
87#14558721—22no—
88C2×#11924746—22no—
89C2×#11924746C2311yescharacteristic
90C2×#11924746C2311yescharacteristic
91C8×D149—22no—
92C4⋊Dic149C2311yescharacteristic
93#14558715—21no—
94#14558715—21no—
95#14558715D411yescharacteristic
96C22⋊Dic149—21no—
97C22⋊Dic149D411yescharacteristic
98#14558713—21no—
99#14558713—21no—
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×Q8×C149C2311yescharacteristic
112C2×D4×C149—21no—
113C2×D4×C149C2311yescharacteristic
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116Q16×C149—41no—
117SD16×C149—22no—
118C2×C8×C149C2311yescharacteristic
119C2×C8×C149C2311yescharacteristic
120C4⋊C4×C149—21no—
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—
130Dic298—42no—
131Dic298C2×D412yes—
132D596—21no—
133D596—21no—
134D596—21no—
135D596—21no—
136D596—42no—
137C22⋊D149—22no—
138C22⋊D149—42no—
139C22⋊D149—42no—
140C22⋊D149—81no—
141C2×Dic149—21no—
142C2×Dic149—21no—
143C2×Dic149—81no—
144C2×Dic149C2×D411yescharacteristic
145C2×Dic149C2×D411yescharacteristic
146C2×Dic149C2×D411yescharacteristic
147C4×D149—21no—
148C4×D149—21no—
149C4×D149—42no—
150C4×D149—42no—
151#11924746—21no—
152#11924746C2×D412yes—
153C23×C149—21no—
154C23×C149C2×D411yescharacteristic
155Q8×C149—21no—
156Q8×C149—21no—
157Q8×C149—21no—
158Q8×C149—21no—
159D4×C149—21no—
160D4×C149—42no—
161D4×C149C2×D412yes—
162C2×C4×C149—21no—
163C2×C4×C149—21no—
164C2×C4×C149—21no—
165C2×C4×C149—21no—
166C2×C4×C149—21no—
167C2×C4×C149—42no—
168C2×C4×C149C2×D411yescharacteristic
169C2×C4×C149C2411yescharacteristic
170C8×C149—21no—
171C8×C149—21no—
172C22×C149—21no—
173C22×C149—22no—
174C22×C149—21no—
175C22×C149—21no—
176C22×C149—41no—
177C22×C149—41no—
178C22×C149C22×D411yescharacteristic
179C2×D149—21no—
180C2×D149—21no—
181C2×D149—41no—
182C2×D149—41no—
183C2×D149—41no—
184C2×D149—41no—
185C2×D149—41no—
186C2×D149—41no—
187C2×D149—84no—
188C2×D149#13512yescomplemented
189C4×C149—21no—
190C4×C149—41no—
191C4×C149—41no—
192C4×C149—41no—
193C4×C149C22×D411yescharacteristic
194C4×C149H(5,𝔽2)11yescharacteristic
195Dic149—21no—
196Dic149—22no—
197Dic149—21no—
198Dic149—21no—
199Dic149—41no—
200C2×C149—42no—
201C2×C149—41no—
202C2×C149D4211yescharacteristic
203C2×C149C2×#13511yescharacteristic
204C2×C149#57711yescharacteristic
205D149—22no—
206D149—81no—
207D149—81no—
208C149#328211yescharacteristic, complemented, Sylow
209#3282—1491nomaximal, Sylow
210C2×#135—1491no—
211C22×SD16—1491no—
212#548—1491no—
213#546—1491no—
214#495—1491no—
215#465—1491no—
216#465—1491no—
217#464—1491no—
218#463—1491no—
219#452D14911yescharacteristic, complemented
220#451—1491no—
221#449—1491no—
222#447—1491no—
223C4×SD16—1491no—
224C8×D4—1491no—
225C2×#41—1491no—
226C2×#41C2×D14911yescharacteristic
227C22×Q8—1491no—
228C22×D4—1491no—
229#135—2984no—
230C2×Q16—1491no—
231C2×Q16C2×D14911yescharacteristic, complemented
232C2×SD16—2984no—
233C2×SD16—1491no—
234C2×SD16—1491no—
235C2×SD16—1491no—
236C2×SD16—1491no—
237C2×SD16C2×D14911yescharacteristic
238C2×D8—1491no—
239C2×D8—1491no—
240C22×C8—1491no—
241C22×C8—1491no—
242#124—2981no—
243#124—1491no—
244#124—1491no—
245#123—2981no—
246C22⋊Q8—2981no—
247C22⋊Q8—1491no—
248C22⋊Q8C2×D14911yescharacteristic, complemented
249#121—2981no—
250#121—2981no—
251#121—1491no—
252#121—1491no—
253#120—2981no—
254C4×Q8—1491no—
255C4×D4—2981no—
256C4×D4—1491no—
257C4×D4—1491no—
258C2×C22⋊C4—2981no—
259#106—1491no—
260C4⋊C8—1491no—
261#103—1491no—
262#103—1491no—
263#103C2×D14911yescharacteristic, complemented
264#102—1491no—
265#102—1491no—
266#102C2×D14911yescharacteristic
267C22⋊C8—1491no—
268C22⋊C8C2×D14911yescharacteristic, complemented
269C4×C8—1491no—
270C24—2981no—
271#41—22no—
272#41—41no—
273#41—2982no—
274#41—5961no—
275C2×Q8—2981no—
276C2×Q8—2982no—
277C2×Q8—5961no—
278C2×Q8—1491no—
279C2×Q8C22×D14911yescharacteristic
280C2×Q8C22×D14911yescharacteristic
281C2×D4—21no—
282C2×D4—2981no—
283C2×D4—2982no—
284C2×D4—2981no—
285C2×D4—2981no—
286C2×D4—2981no—
287C2×D4—5961no—
288C2×D4—1491no—
289C2×D4—1491no—
290C2×D4—1491no—
291C2×D4C22×D14911yescharacteristic
292C22×C4—21no—
293C22×C4—2981no—
294C22×C4—2981no—
295C22×C4—2981no—
296C22×C4—1491no—
297C22×C4C22×D14911yescharacteristic
298Q16—41no—
299Q16—2982no—
300SD16—22no—
301SD16—2982no—
302SD16—2982no—
303SD16—2982no—
304SD16—1494no—
305D8—2982no—
306D8—5961no—
307C2×C8—2982no—
308C2×C8—2982no—
309C2×C8—1491no—
310C2×C8—1491no—
311C2×C8C22×D14911yescharacteristic
312C2×C8C22×D14911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2981no—
315C4⋊C4—2981no—
316C4⋊C4—1491no—
317C4⋊C4—1491no—
318C4⋊C4C22×D14911yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—2981no—
321C22⋊C4—2981no—
322C22⋊C4—2981no—
323C22⋊C4—2981no—
324C22⋊C4—2981no—
325C22⋊C4—2981no—
326C22⋊C4—2981no—
327C22⋊C4—2981no—
328C22⋊C4—1491no—
329C22⋊C4—1491no—
330C42—2981no—
331C42—1491no—
332C23—21no—
333C23—2981no—
334C23—2981no—
335C23—5962no—
336C23—5962no—
337C23—1491no—
338C23—11921no—
339C23D4×D14911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2981no—
345Q8—5962no—
346Q8—1492no—
347D4—21no—
348D4—42no—
349D4—2982no—
350D4—2981no—
351D4—2981no—
352D4—2981no—
353D4—2981no—
354D4—5962no—
355D4—5962no—
356D4—5962no—
357D4—11921no—
358D4D4×D14912yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—2981no—
366C2×C4—2981no—
367C2×C4—2981no—
368C2×C4—2981no—
369C2×C4—5962no—
370C2×C4—5962no—
371C2×C4—1491no—
372C2×C4—1491no—
373C2×C4—1491no—
374C2×C4—11921no—
375C2×C4D4×D14911yescharacteristic
376C2×C4C23×D14911yescharacteristic
377C8—21no—
378C8—21no—
379C8—2981no—
380C8—1492no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—2981no—
388C22—2981no—
389C22—5961no—
390C22—5961no—
391C22—5961no—
392C22—5961no—
393C22—5961no—
394C22—5961no—
395C22—1492no—
396C22—11924no—
397C22C2×D4×D14911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—2981no—
403C4—2981no—
404C4—2981no—
405C4—2982no—
406C4—5961no—
407C4C2×D4×D14911yescharacteristic
408C4#1801195611yescharacteristic
409C2—41no—
410C2—42no—
411C2—2982no—
412C2—11921no—
413C2—11921no—
414C2#2375025711yescharacteristic
415C2#135×D14911yescharacteristic
416C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#452×C149C2
socleC22×C149C22×D4
hypercenter#452D149
hypocenterC149#3282
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#18011931C2semidirect
2. C2×SD16×D149C2semidirect
3. #23750285C2semidirect
4. #23750263C2semidirect
5. #23750010C2semidirect
6. #23749857C2semidirect
7. #23749856C2semidirect
8. #23749850C2semidirect
9. #23749842C2semidirect
10. #23749779C2semidirect
11. #23749772C2semidirect
12. #23749760C2semidirect
13. #23749716C2semidirect
14. #23749688C2semidirect
15. #452×C149C2semidirect
16. C2×#14558736C22semidirect
20. C2×#14558726C22semidirect
24. C2×#14558723C22semidirect
28. C22×#11924746C22semidirect
31. #18011900C22semidirect
41. #18011876C22semidirect
46. #18011842C22semidirect
47. #18011840C22semidirect
48. #18011836C22semidirect
49. #18011835C22semidirect
50. #18011833C22semidirect
53. C8×Dic149C22semidirect
55. C2×Q16×C149C22semidirect
57. C22⋊Q8×C149C22semidirect
58. #103×C149C22semidirect
60. C22⋊C8×C149C22semidirect
83. #14558724D4semidirect
188. C2×D149#135semidirect
208. C149#3282semidirect
219. #452D149semidirect
231. C2×Q16C2×D149semidirect
248. C22⋊Q8C2×D149semidirect
263. #103C2×D149semidirect
268. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119312
C2×SD16×D1492
#237502852
#237502632
#237500102
#237498572
#237498562
#237498502
#237498422
#237497792
#237497722
#237497602
#237497162
#237496882
#452×C1492
#3282149