Finite Group Databasebeta

← #34056917

Subgroups — #34056917

Order 19072.

448 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750346C211yesmaximal, characteristic, complemented
2#23750259C211yesmaximal, characteristic
3#23749970C211yesmaximal, characteristic, complemented
4#23749952C211yesmaximal, characteristic, complemented
5#23749946C211yesmaximal, characteristic, complemented
6#23749904C211yesmaximal, characteristic, complemented
7#23749874C211yesmaximal, characteristic, complemented
8#23749823C211yesmaximal, characteristic, complemented
9#23749775C211yesmaximal, characteristic, complemented
10#23749774C211yesmaximal, characteristic, complemented
11#23749710C211yesmaximal, characteristic, complemented
12#23749701C211yesmaximal, characteristic, complemented
13#23749631C211yesmaximal, characteristic, complemented
14#23749628C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C22×D596C2211yescharacteristic
17C2×D4×D149—21no—
18C2×D4×D149C2211yescharacteristic
19C2×#14558723C2211yescharacteristic
20C2×#14558723C2211yescharacteristic, complemented
21C2×#14558719C2211yescharacteristic, complemented
22#18011910C2211yescharacteristic, complemented
23#18011907C2211yescharacteristic
24#18011906—21no—
25#18011904—21no—
26#18011897—21no—
27#18011887C2211yescharacteristic, complemented
28#18011886C2211yescharacteristic
29#18011884—21no—
30#18011881—21no—
31#18011879—21no—
32Q8×Dic149C2211yescharacteristic, complemented
33#18011876—21no—
34#18011873C2211yescharacteristic
35C4×C22⋊D149C2211yescharacteristic
36#18011870C2211yescharacteristic, complemented
37#18011869C2211yescharacteristic, complemented
38#18011844C2211yescharacteristic, complemented
39#18011842C2211yescharacteristic, complemented
40#18011840C2211yescharacteristic, complemented
41#18011833C2211yescharacteristic
42#18011833C2211yescharacteristic
43#18011833C2211yescharacteristic, complemented
44#18011833C2211yescharacteristic, complemented
45#18011832C2211yescharacteristic, complemented
46#18011832C2211yescharacteristic, complemented
47#18011831C2211yescharacteristic
48#18011831C2211yescharacteristic, complemented
49C22⋊#11924746C2211yescharacteristic, complemented
50#18011823C2211yescharacteristic, complemented
51C8×Dic149C2211yescharacteristic, 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×D149—21no—
61C2×Dic298—21no—
62C2×D596—22no—
63C2×D596—21no—
64C2×D596—21no—
65C2×D596—41no—
66C2×D596C2311yescharacteristic
67C2×D596C2311yescharacteristic
68C2×C22⋊D149—21no—
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C2×C22⋊D149D411yescharacteristic
72D4×D149—22no—
73D4×D149—41no—
74D4×D149—44no—
75C2×C4×D149—21no—
76C2×C4×D149C2311yescharacteristic
77#14558723—22no—
78#14558723—41no—
79#14558719—41no—
80C2×#11924746C2311yescharacteristic
81C2×#11924746C2311yescharacteristic
82C4⋊Dic149—21no—
83C4⋊Dic149C2311yescharacteristic
84C4⋊Dic149C2311yescharacteristic
85#14558715—21no—
86#14558715—21no—
87#14558715D411yescharacteristic
88C22⋊Dic149—21no—
89C22⋊Dic149D411yescharacteristic
90#14558713—21no—
91#14558713—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713D411yescharacteristic
95C4×Dic149—21no—
96C4×Dic149—21no—
97C4×Dic149—21no—
98C4×Dic149C2311yescharacteristic
99C2×Q8×C149C2311yescharacteristic
100C2×D4×C149—21no—
101C2×D4×C149C2311yescharacteristic
102C22×C4×C149C2311yescharacteristic
103SD16×C149—41no—
104M16×C149—41no—
105C2×C8×C149C2311yescharacteristic
106C2×C8×C149C2311yescharacteristic
107C4⋊C4×C149—21no—
108C4⋊C4×C149C2311yescharacteristic
109C4⋊C4×C149C2311yescharacteristic
110C22⋊C4×C149—21no—
111C22⋊C4×C149—21no—
112C22×D149—21no—
113C22×D149—21no—
114C22×D149—42no—
115C22×D149—42no—
116C22×D149—42no—
117C22×D149—42no—
118C22×D149—81no—
119C22×D149—81no—
120C22×D149C2×D411yescharacteristic
121Dic298—42no—
122D596—21no—
123D596—21no—
124D596—21no—
125D596—42no—
126D596—42no—
127D596#4112yes—
128C22⋊D149—22no—
129C22⋊D149—42no—
130C22⋊D149—42no—
131C22⋊D149—81no—
132C2×Dic149—21no—
133C2×Dic149—21no—
134C2×Dic149C2×D411yescharacteristic
135C2×Dic149C2×D411yescharacteristic
136C2×Dic149C2×D411yescharacteristic
137C4×D149—21no—
138C4×D149—21no—
139C4×D149—42no—
140#11924746—21no—
141#11924746—21no—
142C23×C149—21no—
143C23×C149C2×D411yescharacteristic
144Q8×C149—21no—
145Q8×C149—21no—
146D4×C149—21no—
147D4×C149—21no—
148D4×C149—42no—
149C2×C4×C149—21no—
150C2×C4×C149—21no—
151C2×C4×C149—21no—
152C2×C4×C149—21no—
153C2×C4×C149—21no—
154C2×C4×C149C2×D411yescharacteristic
155C2×C4×C149C2411yescharacteristic
156C8×C149—21no—
157C8×C149—21no—
158C22×C149—21no—
159C22×C149—21no—
160C22×C149—21no—
161C22×C149—41no—
162C22×C149—41no—
163C22×C149—41no—
164C22×C149C22×D411yescharacteristic
165C2×D149—21no—
166C2×D149—22no—
167C2×D149—21no—
168C2×D149—21no—
169C2×D149—41no—
170C2×D149—41no—
171C2×D149—41no—
172C2×D149—41no—
173C2×D149—41no—
174C2×D149—84no—
175C2×D149—84no—
176C4×C149—21no—
177C4×C149—41no—
178C4×C149—41no—
179C4×C149—41no—
180C4×C149C2×#4111yescharacteristic
181C4×C149H(5,𝔽2)11yescharacteristic
182Dic149—21no—
183Dic149—21no—
184Dic149—41no—
185Dic149—41no—
186Dic149AGL(1,ℤ/8)12yes—
187C2×C149—41no—
188C2×C149—81no—
189C2×C149#54611yescharacteristic
190C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C149#57711yescharacteristic
192D149—41no—
193D149—42no—
194D149—81no—
195C149#332411yescharacteristic, complemented, Sylow
196#3324—1491nomaximal, Sylow
197#556—1491no—
198D42—1491no—
199#486—1491no—
200#482—1491no—
201#482—1491no—
202#474—1491no—
203#468—1491no—
204#468D14911yescharacteristic, complemented
205#460—1491no—
206#450—1491no—
207#450—1491no—
208C4×SD16—1491no—
209#436—1491no—
210#418—1491no—
211#418—1491no—
212C22×D4—2981no—
213C22×D4—1491no—
214C22×D4—1491no—
215C2×SD16—1491no—
216C2×SD16—1491no—
217C2×SD16C2×D14911yescharacteristic, complemented
218C2×M16—1491no—
219C2×M16C2×D14911yescharacteristic, complemented
220C4⋊Q8—1491no—
221#127—1491no—
222#126—2981no—
223#126—2981no—
224#124—2981no—
225C22⋊Q8—1491no—
226C22⋊Q8C2×D14911yescharacteristic, complemented
227#121—2981no—
228#121—1491no—
229#121C2×D14911yescharacteristic
230#120—2981no—
231#120—2981no—
232C4×Q8—2981no—
233C4×Q8—1491no—
234C4×D4—1491no—
235C4×D4—1491no—
236#117—1491no—
237#117—1491no—
238#107—1491no—
239#107C2×D14911yescharacteristic, complemented
240#106—1491no—
241C4⋊C8—1491no—
242#103C2×D14911yescharacteristic, complemented
243#102—1491no—
244#102—1491no—
245#102—1491no—
246#102—1491no—
247#102—1491no—
248#102—1491no—
249#102—1491no—
250#102—1491no—
251#102C2×D14911yescharacteristic, complemented
252C22⋊C8—1491no—
253C22⋊C8—1491no—
254C4×C8—1491no—
255C24—2981no—
256C24—2981no—
257C2×Q8—2981no—
258C2×Q8C22×D14911yescharacteristic
259C2×D4—21no—
260C2×D4—2982no—
261C2×D4—2981no—
262C2×D4—2982no—
263C2×D4—2981no—
264C2×D4—2981no—
265C2×D4—2981no—
266C2×D4—2981no—
267C2×D4—5961no—
268C2×D4—5961no—
269C2×D4—5964no—
270C2×D4—1491no—
271C2×D4—1491no—
272C2×D4—1491no—
273C2×D4C22×D14911yescharacteristic
274C22×C4—2981no—
275C22×C4—1491no—
276C22×C4C22×D14911yescharacteristic
277SD16—41no—
278SD16—2982no—
279SD16—5961no—
280M16—41no—
281M16—5961no—
282C2×C8—1491no—
283C2×C8—1491no—
284C2×C8C22×D14911yescharacteristic
285C2×C8C22×D14911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2981no—
288C4⋊C4—2981no—
289C4⋊C4—2981no—
290C4⋊C4—1491no—
291C4⋊C4—1491no—
292C4⋊C4—1491no—
293C4⋊C4C22×D14911yescharacteristic
294C4⋊C4C22×D14911yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—2981no—
298C22⋊C4—2981no—
299C22⋊C4—2981no—
300C22⋊C4—2981no—
301C22⋊C4—2981no—
302C22⋊C4—1491no—
303C22⋊C4—1491no—
304C42—2981no—
305C42—2981no—
306C42—2981no—
307C42—1491no—
308C23—21no—
309C23—2981no—
310C23—2981no—
311C23—5962no—
312C23—5962no—
313C23—5962no—
314C23—5962no—
315C23—1491no—
316C23—11921no—
317C23—11921no—
318C23D4×D14911yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—5962no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2982no—
326D4—2981no—
327D4—2981no—
328D4—2981no—
329D4—5962no—
330D4—5962no—
331D4—5962no—
332D4—5962no—
333D4—1492no—
334D4—11921no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—2981no—
341C2×C4—2981no—
342C2×C4—2981no—
343C2×C4—2981no—
344C2×C4—5962no—
345C2×C4—1491no—
346C2×C4—1491no—
347C2×C4—1491no—
348C2×C4D4×D14911yescharacteristic
349C2×C4C23×D14911yescharacteristic
350C8—21no—
351C8—21no—
352C8—2981no—
353C8—2981no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2981no—
361C22—2981no—
362C22—2981no—
363C22—2982no—
364C22—5961no—
365C22—5961no—
366C22—5961no—
367C22—5961no—
368C22—5961no—
369C22—11924no—
370C22—11924no—
371C22C2×D4×D14911yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—2981no—
377C4—2981no—
378C4—5961no—
379C4—5961no—
380C4—1492no—
381C4#41×D14911yescharacteristic
382C4#1801195611yescharacteristic
383C2—41no—
384C2—81no—
385C2—5961no—
386C2—5962no—
387C2—11921no—
388C2#2375026411yescharacteristic
389C2AGL(1,ℤ/8)×D14911yescharacteristic
390C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
prime radical#23750259C2
hypercenter#468D149
hypocenterC149#3324
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. #23750346C2semidirect
3. #23749970C2semidirect
4. #23749952C2semidirect
5. #23749946C2semidirect
6. #23749904C2semidirect
7. #23749874C2semidirect
8. #23749823C2semidirect
9. #23749775C2semidirect
10. #23749774C2semidirect
11. #23749710C2semidirect
12. #23749701C2semidirect
13. #23749631C2semidirect
14. #23749628C2semidirect
15. #468×C149C2semidirect
20. C2×#14558723C22semidirect
21. C2×#14558719C22semidirect
22. #18011910C22semidirect
27. #18011887C22semidirect
32. Q8×Dic149C22semidirect
36. #18011870C22semidirect
37. #18011869C22semidirect
38. #18011844C22semidirect
39. #18011842C22semidirect
40. #18011840C22semidirect
43. #18011833C22semidirect
44. #18011833C22semidirect
45. #18011832C22semidirect
46. #18011832C22semidirect
48. #18011831C22semidirect
49. C22⋊#11924746C22semidirect
50. #18011823C22semidirect
51. C8×Dic149C22semidirect
52. C2×SD16×C149C22semidirect
53. C2×M16×C149C22semidirect
54. C22⋊Q8×C149C22semidirect
56. #107×C149C22semidirect
57. #103×C149C22semidirect
58. #102×C149C22semidirect
195. C149#3324semidirect
204. #468D149semidirect
217. C2×SD16C2×D149semidirect
219. C2×M16C2×D149semidirect
226. C22⋊Q8C2×D149semidirect
239. #107C2×D149semidirect
242. #103C2×D149semidirect
251. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503462
#237502592
#237499702
#237499522
#237499462
#237499042
#237498742
#237498232
#237497752
#237497742
#237497102
#237497012
#237496312
#237496282
#468×C1492
#3324149