Finite Group Databasebeta

← #34055492

Subgroups — #34055492

Order 19072.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D149C211yesmaximal, characteristic, complemented
2#121×D149C211yesmaximal, characteristic
3#23749962C211yesmaximal, characteristic, complemented
4#23749959C211yesmaximal, characteristic, complemented
5#23749940C211yesmaximal, characteristic, complemented
6#23749937C211yesmaximal, characteristic, complemented
7#23749873C211yesmaximal, characteristic, complemented
8#23749871C211yesmaximal, characteristic, complemented
9#23749869C211yesmaximal, characteristic, complemented
10#23749777C211yesmaximal, characteristic, complemented
11#23749774C211yesmaximal, characteristic, complemented
12#23749765C211yesmaximal, characteristic, complemented
13#23749762C211yesmaximal, characteristic, complemented
14#23749588C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic, complemented
17C2×D4×D149—21no—
18C2×D4×D149C2211yescharacteristic
19C22×C4×D149C2211yescharacteristic
20C2×#14558725C2211yescharacteristic, complemented
21C2×#14558724C2211yescharacteristic, complemented
22C2×#14558723C2211yescharacteristic, complemented
23C2×#14558720C2211yescharacteristic, complemented
24#18011891C2211yescharacteristic
25#18011890C2211yescharacteristic, complemented
26#18011889—21no—
27#18011888—21no—
28#18011887C2211yescharacteristic
29#18011886C2211yescharacteristic
30#18011885C2211yescharacteristic
31#18011884—21no—
32#18011883—21no—
33#18011882C2211yescharacteristic
34C4⋊C4×D149—21no—
35C4⋊C4×D149C2211yescharacteristic
36C4⋊C4×D149C2211yescharacteristic, complemented
37C22⋊C4×D149—21no—
38C22⋊C4×D149—21no—
39#18011845C2211yescharacteristic, complemented
40#18011844C2211yescharacteristic, complemented
41#18011844C2211yescharacteristic, complemented
42#18011836C2211yescharacteristic, complemented
43#18011835C2211yescharacteristic, complemented
44#18011834C2211yescharacteristic, complemented
45#18011833C2211yescharacteristic, complemented
46#18011832C2211yescharacteristic, complemented
47#18011831C2211yescharacteristic, complemented
48#18011823C2211yescharacteristic
49#18011823C2211yescharacteristic
50#18011823C2211yescharacteristic, complemented
51#18011823C2211yescharacteristic, complemented
52C2×SD16×C149C2211yescharacteristic, complemented
53C2×M16×C149C2211yescharacteristic, complemented
54C22⋊Q8×C149C2211yescharacteristic, complemented
55#121×C149C2211yescharacteristic
56#107×C149C2211yescharacteristic, complemented
57#103×C149C2211yescharacteristic, complemented
58#102×C149C2211yescharacteristic, complemented
59C23×D149—21no—
60C23×D149D411yescharacteristic
61C2×Dic298—21no—
62C2×Dic298C2311yescharacteristic
63Q8×D149—41no—
64Q8×D149—41no—
65C2×D596—21no—
66C2×D596C2311yescharacteristic
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149—21no—
70D4×D149—41no—
71D4×D149—41no—
72D4×D149—44no—
73C22×Dic149D411yescharacteristic
74C2×C4×D149—21no—
75C2×C4×D149—21no—
76C2×C4×D149—21no—
77C2×C4×D149—41no—
78C2×C4×D149—41no—
79C2×C4×D149D411yescharacteristic
80C2×C4×D149D411yescharacteristic
81C2×C4×D149C2311yescharacteristic
82C2×C4×D149C2311yescharacteristic
83#14558725—41no—
84#14558724—41no—
85#14558723—41no—
86#14558720—41no—
87C2×#11924746C2311yescharacteristic
88C2×#11924746C2311yescharacteristic
89C4⋊Dic149—21no—
90C4⋊Dic149C2311yescharacteristic
91C4⋊Dic149C2311yescharacteristic
92#14558715—21no—
93#14558715—21no—
94#14558715—21no—
95#14558715—21no—
96C22⋊Dic149—21no—
97C22⋊Dic149—21no—
98#14558713—21no—
99#14558713—21no—
100#14558713—21no—
101#14558713—21no—
102C2×Q8×C149C2311yescharacteristic
103C2×D4×C149—21no—
104C2×D4×C149C2311yescharacteristic
105C22×C4×C149C2311yescharacteristic
106SD16×C149—41no—
107M16×C149—41no—
108C2×C8×C149C2311yescharacteristic
109C2×C8×C149C2311yescharacteristic
110C4⋊C4×C149—21no—
111C4⋊C4×C149C2311yescharacteristic
112C4⋊C4×C149C2311yescharacteristic
113C22⋊C4×C149—21no—
114C22⋊C4×C149—21no—
115C22×D149—22no—
116C22×D149—22no—
117C22×D149—21no—
118C22×D149—41no—
119C22×D149—42no—
120C22×D149—42no—
121C22×D149—42no—
122C22×D149C2×D411yescharacteristic
123C22×D149C2×D411yescharacteristic
124Dic298—21no—
125Dic298—21no—
126Dic298—42no—
127D596—21no—
128D596—21no—
129D596—42no—
130C22⋊D149—42no—
131C22⋊D149—42no—
132C22⋊D149—42no—
133C2×Dic149—21no—
134C2×Dic149—21no—
135C2×Dic149—21no—
136C2×Dic149—41no—
137C2×Dic149C2×D411yescharacteristic
138C2×Dic149C2×D411yescharacteristic
139C4×D149—21no—
140C4×D149—21no—
141C4×D149—21no—
142C4×D149—21no—
143C4×D149—41no—
144C4×D149—41no—
145C4×D149—42no—
146C4×D149—42no—
147C4×D149—42no—
148#11924746—21no—
149#11924746—21no—
150C23×C149—21no—
151C23×C149C2×D411yescharacteristic
152Q8×C149—21no—
153Q8×C149—21no—
154D4×C149—21no—
155D4×C149—21no—
156D4×C149—42no—
157C2×C4×C149—21no—
158C2×C4×C149—21no—
159C2×C4×C149—21no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149C2×D411yescharacteristic
163C2×C4×C149C2411yescharacteristic
164C8×C149—21no—
165C8×C149—21no—
166C22×C149—21no—
167C22×C149—21no—
168C22×C149—21no—
169C22×C149—41no—
170C22×C149—41no—
171C22×C149—41no—
172C22×C149C22×D411yescharacteristic
173C2×D149—21no—
174C2×D149—21no—
175C2×D149—21no—
176C2×D149—21no—
177C2×D149—44no—
178C2×D149—41no—
179C2×D149—41no—
180C2×D149—41no—
181C2×D149—84no—
182C2×D149AGL(1,ℤ/8)12yes—
183C2×D149#13712yes—
184C4×C149—21no—
185C4×C149—41no—
186C4×C149—41no—
187C4×C149—41no—
188C4×C149H(5,𝔽2)11yescharacteristic
189C4×C149H(5,𝔽2)11yescharacteristic
190Dic149—21no—
191Dic149—21no—
192Dic149—41no—
193Dic149—41no—
194Dic149—41no—
195C2×C149—41no—
196C2×C149—81no—
197C2×C149#53411yescharacteristic
198C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C149C2×#13711yescharacteristic
200D149—22no—
201D149—22no—
202D149—81no—
203C149#319311yescharacteristic, complemented, Sylow
204#3193—1491nomaximal, Sylow
205C2×C22⋊Q8—1491no—
206C2×#121—1491no—
207#484—1491no—
208#484—1491no—
209#480—1491no—
210#480—1491no—
211#468—1491no—
212#468—1491no—
213#468—1491no—
214#468D14911yescharacteristic, complemented
215#450—1491no—
216#450—1491no—
217#448—1491no—
218#448—1491no—
219#407—1491no—
220C22×Q8—1491no—
221C22×D4—2981no—
222C22×D4—1491no—
223C23×C4—1491no—
224C2×SD16—1491no—
225C2×SD16—1491no—
226C2×SD16—1491no—
227C2×SD16C2×D14911yescharacteristic, complemented
228C2×M16—1491no—
229C2×M16C2×D14911yescharacteristic, complemented
230C22⋊Q8—2981no—
231C22⋊Q8—2981no—
232C22⋊Q8—1491no—
233C22⋊Q8—1491no—
234C22⋊Q8—1491no—
235C22⋊Q8C2×D14911yescharacteristic, complemented
236#121—2981no—
237#121—2981no—
238#121—1491no—
239#121—1491no—
240#121—1491no—
241#121C2×D14911yescharacteristic
242C2×C4⋊C4—2981no—
243C2×C4⋊C4—1491no—
244C2×C4⋊C4—1491no—
245C2×C22⋊C4—2981no—
246C2×C22⋊C4—2981no—
247#107—1491no—
248#107—1491no—
249#107—1491no—
250#107C2×D14911yescharacteristic, complemented
251#103—1491no—
252#103—1491no—
253#103—1491no—
254#103C2×D14911yescharacteristic, complemented
255#102—1491no—
256#102—1491no—
257#102—1491no—
258#102C2×D14911yescharacteristic, complemented
259C22⋊C8—1491no—
260C22⋊C8—1491no—
261C22⋊C8—1491no—
262C22⋊C8—1491no—
263C24—2981no—
264C24—1491no—
265C2×Q8—2981no—
266C2×Q8—5961no—
267C2×Q8—5961no—
268C2×Q8—1491no—
269C2×Q8C22×D14911yescharacteristic
270C2×D4—21no—
271C2×D4—2981no—
272C2×D4—2981no—
273C2×D4—2981no—
274C2×D4—2981no—
275C2×D4—5961no—
276C2×D4—5961no—
277C2×D4—5964no—
278C2×D4—1491no—
279C2×D4C22×D14911yescharacteristic
280C22×C4—2981no—
281C22×C4—2981no—
282C22×C4—2981no—
283C22×C4—5961no—
284C22×C4—5961no—
285C22×C4—1491no—
286C22×C4—1491no—
287C22×C4—1491no—
288C22×C4—1491no—
289C22×C4—1491no—
290C22×C4C22×D14911yescharacteristic
291SD16—41no—
292SD16—5961no—
293SD16—5961no—
294SD16—5961no—
295M16—41no—
296M16—5961no—
297C2×C8—1491no—
298C2×C8—1491no—
299C2×C8C22×D14911yescharacteristic
300C2×C8C22×D14911yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—2981no—
303C4⋊C4—2981no—
304C4⋊C4—2981no—
305C4⋊C4—2981no—
306C4⋊C4—2981no—
307C4⋊C4—1491no—
308C4⋊C4—1491no—
309C4⋊C4C22×D14911yescharacteristic
310C4⋊C4C22×D14911yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—2981no—
314C22⋊C4—2981no—
315C22⋊C4—2981no—
316C22⋊C4—2981no—
317C22⋊C4—2981no—
318C22⋊C4—2981no—
319C23—21no—
320C23—2982no—
321C23—2982no—
322C23—2981no—
323C23—5961no—
324C23—5962no—
325C23—5962no—
326C23—5962no—
327C23—1491no—
328C23—1491no—
329C23D4×D14911yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—2981no—
333Q8—2981no—
334Q8—5962no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—2981no—
339D4—2981no—
340D4—5962no—
341D4—5962no—
342D4—5962no—
343D4—5962no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—2981no—
350C2×C4—2981no—
351C2×C4—2981no—
352C2×C4—2981no—
353C2×C4—2981no—
354C2×C4—2981no—
355C2×C4—2981no—
356C2×C4—5961no—
357C2×C4—5961no—
358C2×C4—5962no—
359C2×C4—5962no—
360C2×C4—5962no—
361C2×C4—5961no—
362C2×C4—1491no—
363C2×C4—1491no—
364C2×C4D4×D14911yescharacteristic
365C2×C4C23×D14911yescharacteristic
366C8—21no—
367C8—21no—
368C8—2981no—
369C8—2981no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—2981no—
377C22—2981no—
378C22—2981no—
379C22—2981no—
380C22—5964no—
381C22—5961no—
382C22—5961no—
383C22—5961no—
384C22—1492no—
385C22—1492no—
386C22—11924no—
387C22C2×D4×D14911yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2981no—
393C4—2981no—
394C4—5961no—
395C4—5961no—
396C4—5961no—
397C4#1801195611yescharacteristic
398C4#1801195711yescharacteristic
399C2—41no—
400C2—81no—
401C2—2982no—
402C2—2982no—
403C2—11921no—
404C2#2375018311yescharacteristic
405C2AGL(1,ℤ/8)×D14911yescharacteristic
406C2#137×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
prime radical#121×D149C2
hypercenter#468D149
hypocenterC149#3193
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D149C2semidirect
3. #23749962C2semidirect
4. #23749959C2semidirect
5. #23749940C2semidirect
6. #23749937C2semidirect
7. #23749873C2semidirect
8. #23749871C2semidirect
9. #23749869C2semidirect
10. #23749777C2semidirect
11. #23749774C2semidirect
12. #23749765C2semidirect
13. #23749762C2semidirect
14. #23749588C2semidirect
15. #468×C149C2semidirect
16. C2×Q8×D149C22semidirect
20. C2×#14558725C22semidirect
21. C2×#14558724C22semidirect
22. C2×#14558723C22semidirect
23. C2×#14558720C22semidirect
25. #18011890C22semidirect
36. C4⋊C4×D149C22semidirect
39. #18011845C22semidirect
40. #18011844C22semidirect
41. #18011844C22semidirect
42. #18011836C22semidirect
43. #18011835C22semidirect
44. #18011834C22semidirect
45. #18011833C22semidirect
46. #18011832C22semidirect
47. #18011831C22semidirect
50. #18011823C22semidirect
51. #18011823C22semidirect
52. C2×SD16×C149C22semidirect
53. C2×M16×C149C22semidirect
54. C22⋊Q8×C149C22semidirect
56. #107×C149C22semidirect
57. #103×C149C22semidirect
58. #102×C149C22semidirect
203. C149#3193semidirect
214. #468D149semidirect
227. C2×SD16C2×D149semidirect
229. C2×M16C2×D149semidirect
235. C22⋊Q8C2×D149semidirect
250. #107C2×D149semidirect
254. #103C2×D149semidirect
258. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1492
#121×D1492
#237499622
#237499592
#237499402
#237499372
#237498732
#237498712
#237498692
#237497772
#237497742
#237497652
#237497622
#237495882
#468×C1492
#3193149