Finite Group Databasebeta

← #34056267

Subgroups — #34056267

Order 19072.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011943C211yesmaximal, characteristic, complemented
2C2×Q16×D149C211yesmaximal, characteristic, complemented
3#23750290C211yesmaximal, characteristic, complemented
4#23750287C211yesmaximal, characteristic
5#23750017C211yesmaximal, characteristic, complemented
6#23749882C211yesmaximal, characteristic, complemented
7#23749857C211yesmaximal, characteristic, complemented
8#23749848C211yesmaximal, characteristic, complemented
9#23749786C211yesmaximal, characteristic, complemented
10#23749782C211yesmaximal, characteristic, complemented
11#23749779C211yesmaximal, characteristic, complemented
12#23749765C211yesmaximal, characteristic, complemented
13#23749730C211yesmaximal, characteristic, complemented
14#23749695C211yesmaximal, characteristic, complemented
15#462×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×Q8×D149C2211yescharacteristic
19C2×Q8×D149C2211yescharacteristic
20#18011943—24no—
21C2×Dic596C2211yescharacteristic
22C2×#14558726C2211yescharacteristic
23C2×#14558726C2211yescharacteristic
24C2×#14558726C2211yescharacteristic
25Q16×D149—24no—
26C2×#14558725C2211yescharacteristic
27C2×#14558723C2211yescharacteristic
28C2×#14558723C2211yescharacteristic
29C2×#14558719C2211yescharacteristic
30C2×C8×D149C2211yescharacteristic
31#18011910—21no—
32#18011899—21no—
33#18011899C2211yescharacteristic
34#18011891—21no—
35#18011891C2211yescharacteristic
36#18011889—21no—
37#18011889—21no—
38#18011887C2211yescharacteristic
39#18011887C2211yescharacteristic
40#18011886—21no—
41#18011886C2211yescharacteristic
42C4×Dic298C2211yescharacteristic
43C4×D596—21no—
44C4×D596C2211yescharacteristic
45#18011873—21no—
46#18011842C2211yescharacteristic
47#18011836C2211yescharacteristic
48#18011835C2211yescharacteristic
49#18011834C2211yescharacteristic
50#18011833C2211yescharacteristic
51#18011823C2211yescharacteristic
52#18011823C2211yescharacteristic
53#18011822C2211yescharacteristic
54C2×Q16×C149C2211yescharacteristic
55C2×Q16×C149C2211yescharacteristic
56C4⋊Q8×C149C2211yescharacteristic
57C4×Q8×C149C2211yescharacteristic
58C4⋊C8×C149C2211yescharacteristic
59#103×C149C2211yescharacteristic
60#103×C149C2211yescharacteristic
61#14558737—22no—
62#14558737—41no—
63C2×Dic298—21no—
64C2×Dic298—21no—
65C2×Dic298—21no—
66C2×Dic298C2311yescharacteristic
67C2×Dic298C2311yescharacteristic
68Q8×D149—22no—
69Q8×D149—22no—
70Q8×D149—22no—
71Q8×D149—41no—
72Q8×D149—41no—
73Q8×D149—41no—
74C2×D596—21no—
75C2×D596—21no—
76C2×D596D411yescharacteristic
77C2×D596C2311yescharacteristic
78C2×C4×D149—21no—
79C2×C4×D149—21no—
80C2×C4×D149—21no—
81C2×C4×D149—21no—
82C2×C4×D149C2311yescharacteristic
83C2×C4×D149C2311yescharacteristic
84Dic596—22no—
85#14558726—22no—
86#14558726—41no—
87#14558726D414yescomplemented
88#14558725—22no—
89#14558723—22no—
90#14558723—41no—
91#14558719—22no—
92C2×#11924746C2311yescharacteristic
93C2×#11924746C2311yescharacteristic
94C8×D149—22no—
95C4⋊Dic149—21no—
96C4⋊Dic149D411yescharacteristic
97C4⋊Dic149C2311yescharacteristic
98#14558715—21no—
99#14558715—21no—
100#14558715—21no—
101#14558713—21no—
102#14558713—21no—
103#14558713—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713—21no—
107#14558713D411yescharacteristic
108#14558713D411yescharacteristic
109C4×Dic149—21no—
110C2×Q8×C149C2311yescharacteristic
111C2×Q8×C149C2311yescharacteristic
112C2×Q8×C149C2311yescharacteristic
113Q16×C149—22no—
114Q16×C149—22no—
115C2×C8×C149C2311yescharacteristic
116C2×C8×C149C2311yescharacteristic
117C4⋊C4×C149—21no—
118C4⋊C4×C149—21no—
119C4⋊C4×C149—21no—
120C4⋊C4×C149C2311yescharacteristic
121C42×C149—21no—
122C42×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149C2×D411yescharacteristic
125C22×D149C2×D411yescharacteristic
126Dic298—21no—
127Dic298—21no—
128Dic298—21no—
129Dic298—42no—
130Dic298—42no—
131Dic298—42no—
132Dic298C2×D412yes—
133D596—22no—
134D596—21no—
135D596—21no—
136D596—42no—
137D596—42no—
138C2×Dic149—21no—
139C2×Dic149—21no—
140C2×Dic149—21no—
141C2×Dic149C2×D411yescharacteristic
142C2×Dic149C2×D411yescharacteristic
143C4×D149—21no—
144C4×D149—21no—
145C4×D149—21no—
146C4×D149—21no—
147C4×D149—42no—
148C4×D149—42no—
149C4×D149—42no—
150C4×D149—42no—
151#11924746—21no—
152#11924746C2×D412yes—
153Q8×C149—21no—
154Q8×C149—21no—
155Q8×C149—21no—
156Q8×C149—21no—
157Q8×C149—21no—
158Q8×C149C2×D412yes—
159C2×C4×C149—21no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149—21no—
163C2×C4×C149C2×D411yescharacteristic
164C2×C4×C149C2×D411yescharacteristic
165C2×C4×C149C2411yescharacteristic
166C8×C149—21no—
167C8×C149—21no—
168C22×C149C22×D411yescharacteristic
169C2×D149—21no—
170C2×D149—21no—
171C2×D149—21no—
172C2×D149—21no—
173C2×D149—21no—
174C2×D149—41no—
175C2×D149—41no—
176C2×D149—41no—
177C2×D149#13712yescomplemented
178C4×C149—22no—
179C4×C149—21no—
180C4×C149—21no—
181C4×C149—41no—
182C4×C149—41no—
183C4×C149—41no—
184C4×C149C22×D411yescharacteristic
185C4×C149H(5,𝔽2)11yescharacteristic
186Dic149—21no—
187Dic149—21no—
188Dic149—22no—
189Dic149—41no—
190Dic149—41no—
191C2×C149D4211yescharacteristic
192C2×C149C2×#13711yescharacteristic
193C2×C149#57811yescharacteristic
194D149—22no—
195D149—41no—
196D149—81no—
197C149#327011yescharacteristic, complemented, Sylow
198#3270—1491nomaximal, Sylow
199C2×#137—1491no—
200C22×Q16—1491no—
201D4×Q8—1491no—
202#548—1491no—
203#497—1491no—
204#470—1491no—
205#465—1491no—
206#464—1491no—
207#462D14911yescharacteristic, complemented
208#452—1491no—
209#451—1491no—
210#451—1491no—
211#448—1491no—
212#441—1491no—
213#435—1491no—
214C2×#41—1491no—
215C22×Q8—1491no—
216C22×Q8—1491no—
217C22×Q8—1491no—
218#137—2984no—
219C2×Q16—2984no—
220C2×Q16—1491no—
221C2×Q16—1491no—
222C2×Q16—1491no—
223C2×Q16—1491no—
224C2×Q16C2×D14911yescharacteristic
225C2×Q16C2×D14911yescharacteristic
226C2×SD16—1491no—
227C2×SD16—1491no—
228C2×SD16—1491no—
229C2×M16—1491no—
230C22×C8—1491no—
231C4⋊Q8—2981no—
232C4⋊Q8C2×D14911yescharacteristic
233#124—2981no—
234#124—1491no—
235C22⋊Q8—2981no—
236C22⋊Q8—2981no—
237C22⋊Q8—2981no—
238C22⋊Q8—1491no—
239C22⋊Q8—1491no—
240C22⋊Q8—1491no—
241#121—2981no—
242#121—1491no—
243C4×Q8—1491no—
244C4×Q8C2×D14911yescharacteristic
245C4×D4—2981no—
246C4×D4—2981no—
247C4×D4—1491no—
248#106—1491no—
249C4⋊C8C2×D14911yescharacteristic
250#103—1491no—
251#103—1491no—
252#103—1491no—
253#103C2×D14911yescharacteristic
254#103C2×D14911yescharacteristic
255#102—1491no—
256C22⋊C8—1491no—
257C22⋊C8—1491no—
258#97—1491no—
259#41—2982no—
260#41—5961no—
261C2×Q8—2981no—
262C2×Q8—2981no—
263C2×Q8—2981no—
264C2×Q8—2982no—
265C2×Q8—2982no—
266C2×Q8—2982no—
267C2×Q8—5961no—
268C2×Q8—5961no—
269C2×Q8—5961no—
270C2×Q8—1491no—
271C2×Q8—1491no—
272C2×Q8C22×D14911yescharacteristic
273C2×Q8C22×D14911yescharacteristic
274C2×Q8C22×D14911yescharacteristic
275C2×D4—2981no—
276C2×D4—2981no—
277C2×D4—1491no—
278C2×D4—1491no—
279C22×C4—2981no—
280C22×C4—2981no—
281C22×C4—2981no—
282C22×C4—2981no—
283C22×C4—1491no—
284C22×C4—1491no—
285Q16—22no—
286Q16—22no—
287Q16—2982no—
288Q16—2982no—
289Q16—5961no—
290Q16—1494no—
291SD16—2982no—
292SD16—2982no—
293SD16—5961no—
294M16—2982no—
295C2×C8—2982no—
296C2×C8—1491no—
297C2×C8—1491no—
298C2×C8C22×D14911yescharacteristic
299C2×C8C22×D14911yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—2981no—
304C4⋊C4—2981no—
305C4⋊C4—2981no—
306C4⋊C4—2981no—
307C4⋊C4—1491no—
308C4⋊C4—1491no—
309C4⋊C4C22×D14911yescharacteristic
310C22⋊C4—2981no—
311C22⋊C4—2981no—
312C22⋊C4—2981no—
313C22⋊C4—2981no—
314C22⋊C4—2981no—
315C22⋊C4—2981no—
316C22⋊C4—1491no—
317C22⋊C4—1491no—
318C42—21no—
319C42—2981no—
320C42C22×D14911yescharacteristic
321C23—2981no—
322C23—1491no—
323C23—1491no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—2981no—
330Q8—2981no—
331Q8—2981no—
332Q8—5962no—
333Q8—5962no—
334Q8—5962no—
335Q8—1492no—
336Q8D4×D14912yes—
337D4—2982no—
338D4—2981no—
339D4—2981no—
340D4—5962no—
341D4—5962no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—2981no—
347C2×C4—2981no—
348C2×C4—2981no—
349C2×C4—2981no—
350C2×C4—2981no—
351C2×C4—2981no—
352C2×C4—2981no—
353C2×C4—5962no—
354C2×C4—5962no—
355C2×C4—5962no—
356C2×C4—5962no—
357C2×C4—1491no—
358C2×C4—1491no—
359C2×C4D4×D14911yescharacteristic
360C2×C4D4×D14911yescharacteristic
361C2×C4C23×D14911yescharacteristic
362C8—21no—
363C8—21no—
364C8—2981no—
365C8—1492no—
366C22—2981no—
367C22—2981no—
368C22—2981no—
369C22—2981no—
370C22—2981no—
371C22—5961no—
372C22—5961no—
373C22—5961no—
374C22—1492no—
375C22C2×D4×D14911yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2981no—
383C4—2981no—
384C4—2982no—
385C4—5961no—
386C4—5961no—
387C4C2×D4×D14911yescharacteristic
388C4#1801195711yescharacteristic
389C2—2982no—
390C2—5961no—
391C2—11921no—
392C2#2375025911yescharacteristic
393C2#137×D14911yescharacteristic
394C2#2375043811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#462×C149C2
socleC22×C149C22×D4
prime radical#23750287C2
hypercenter#462D149
hypocenterC149#3270
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#18011943C2semidirect
2. C2×Q16×D149C2semidirect
3. #23750290C2semidirect
5. #23750017C2semidirect
6. #23749882C2semidirect
7. #23749857C2semidirect
8. #23749848C2semidirect
9. #23749786C2semidirect
10. #23749782C2semidirect
11. #23749779C2semidirect
12. #23749765C2semidirect
13. #23749730C2semidirect
14. #23749695C2semidirect
15. #462×C149C2semidirect
87. #14558726D4semidirect
177. C2×D149#137semidirect
197. C149#3270semidirect
207. #462D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119432
C2×Q16×D1492
#237502902
#237502872
#237500172
#237498822
#237498572
#237498482
#237497862
#237497822
#237497792
#237497652
#237497302
#237496952
#462×C1492
#3270149