Finite Group Databasebeta

← #34056261

Subgroups — #34056261

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
2Q8×C22⋊D149C211yesmaximal, characteristic, complemented
3#23750286C211yesmaximal, characteristic
4#23750018C211yesmaximal, characteristic, complemented
5#23749880C211yesmaximal, characteristic, complemented
6#23749854C211yesmaximal, characteristic, complemented
7#23749849C211yesmaximal, characteristic, complemented
8#23749833C211yesmaximal, characteristic, complemented
9#23749786C211yesmaximal, characteristic, complemented
10#23749781C211yesmaximal, characteristic, complemented
11#23749779C211yesmaximal, characteristic, complemented
12#23749764C211yesmaximal, characteristic, complemented
13#23749732C211yesmaximal, characteristic, complemented
14#23749696C211yesmaximal, characteristic, complemented
15C22×Q16×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18#18011943—24no—
19C2×#14558726C2211yescharacteristic
20C2×#14558726C2211yescharacteristic
21C2×#14558725C2211yescharacteristic
22C2×#14558723C2211yescharacteristic
23C2×#14558723C2211yescharacteristic
24C2×#14558719C2211yescharacteristic
25#18011909—21no—
26#18011909C2211yescharacteristic
27#18011900—21no—
28#18011900C2211yescharacteristic
29#18011890—21no—
30#18011890C2211yescharacteristic
31#18011887—21no—
32#18011887—21no—
33#18011887C2211yescharacteristic
34#18011887C2211yescharacteristic
35#18011885—21no—
36#18011885C2211yescharacteristic
37Q8×Dic149C2211yescharacteristic
38Q8×Dic149C2211yescharacteristic
39C4×C22⋊D149—21no—
40C4×C22⋊D149—21no—
41C4×C22⋊D149C2211yescharacteristic
42#18011843C2211yescharacteristic
43#18011840C2211yescharacteristic
44#18011836C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011835C2211yescharacteristic
48#18011835C2211yescharacteristic
49#18011833C2211yescharacteristic
50C22⋊#11924746C2211yescharacteristic
51#18011823C2211yescharacteristic
52#18011821C2211yescharacteristic
53C22×Q8×C149C2211yescharacteristic
54C22×Q8×C149C2211yescharacteristic
55C2×Q16×C149—24no—
56C2×Q16×C149C2211yescharacteristic
57C2×Q16×C149C2211yescharacteristic
58C2×Q16×C149C2211yescharacteristic
59C2×Q16×C149C2211yescharacteristic
60C22×C8×C149C2211yescharacteristic
61#14558737—22no—
62#14558737—41no—
63C2×Dic298—21no—
64C2×Dic298C2311yescharacteristic
65Q8×D149—22no—
66Q8×D149—41no—
67C2×D596—21no—
68C2×D596C2311yescharacteristic
69C2×C22⋊D149—21no—
70C2×C22⋊D149D411yescharacteristic
71C2×C4×D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149C2311yescharacteristic
74#14558726—22no—
75#14558726—41no—
76#14558725—22no—
77#14558723—22no—
78#14558723—41no—
79#14558719—22no—
80C2×#11924746C2311yescharacteristic
81C2×#11924746C2311yescharacteristic
82C4⋊Dic149—21no—
83C4⋊Dic149—21no—
84C4⋊Dic149C2311yescharacteristic
85C4⋊Dic149C2311yescharacteristic
86#14558715—21no—
87#14558715—21no—
88#14558715—21no—
89#14558715—21no—
90#14558715—21no—
91#14558715D411yescharacteristic
92C22⋊Dic149—21no—
93C22⋊Dic149D411yescharacteristic
94#14558713—21no—
95#14558713—21no—
96#14558713—21no—
97#14558713—21no—
98#14558713—21no—
99#14558713D411yescharacteristic
100C4×Dic149—21no—
101C4×Dic149—21no—
102C4×Dic149C2311yescharacteristic
103C2×Q8×C149—21no—
104C2×Q8×C149—21no—
105C2×Q8×C149—22no—
106C2×Q8×C149—22no—
107C2×Q8×C149—41no—
108C2×Q8×C149—41no—
109C2×Q8×C149C2311yescharacteristic
110C2×Q8×C149C2311yescharacteristic
111C2×Q8×C149C2311yescharacteristic
112C2×Q8×C149C2311yescharacteristic
113C22×C4×C149—21no—
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116Q16×C149—22no—
117Q16×C149—22no—
118Q16×C149—22no—
119Q16×C149D414yescomplemented
120C2×C8×C149—22no—
121C2×C8×C149C2311yescharacteristic
122C2×C8×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149C2×D411yescharacteristic
125Dic298—21no—
126Dic298—21no—
127Dic298—42no—
128D596—21no—
129D596—21no—
130D596—42no—
131C22⋊D149—22no—
132C22⋊D149—42no—
133C2×Dic149—21no—
134C2×Dic149—21no—
135C2×Dic149—21no—
136C2×Dic149C2×D411yescharacteristic
137C2×Dic149C2×D411yescharacteristic
138C2×Dic149C2×D411yescharacteristic
139C4×D149—21no—
140C4×D149—21no—
141C4×D149—42no—
142C4×D149—42no—
143#11924746—21no—
144#11924746—21no—
145C23×C149C2×D411yescharacteristic
146Q8×C149—21no—
147Q8×C149—21no—
148Q8×C149—21no—
149Q8×C149—21no—
150Q8×C149—21no—
151Q8×C149—21no—
152Q8×C149—42no—
153Q8×C149—42no—
154Q8×C149C2×D412yes—
155Q8×C149C2×D412yes—
156C2×C4×C149—21no—
157C2×C4×C149—21no—
158C2×C4×C149—21no—
159C2×C4×C149—21no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149—42no—
163C2×C4×C149—42no—
164C2×C4×C149C2×D411yescharacteristic
165C2×C4×C149C2411yescharacteristic
166C8×C149—21no—
167C8×C149C2×D412yes—
168C22×C149—21no—
169C22×C149—21no—
170C22×C149#13712yescomplemented
171C22×C149C22×D411yescharacteristic
172C2×D149—21no—
173C2×D149—21no—
174C2×D149—21no—
175C2×D149—41no—
176C2×D149—41no—
177C2×D149—41no—
178C4×C149—21no—
179C4×C149—22no—
180C4×C149—22no—
181C4×C149—41no—
182C4×C149—41no—
183C4×C149C22×D411yescharacteristic
184C4×C149H(5,𝔽2)11yescharacteristic
185Dic149—21no—
186Dic149—21no—
187Dic149—21no—
188Dic149—41no—
189Dic149—41no—
190Dic149—41no—
191C2×C149—22no—
192C2×C149D4211yescharacteristic
193C2×C149C2×#13711yescharacteristic
194C2×C149#57811yescharacteristic
195D149—41no—
196D149—81no—
197C149#327011yescharacteristic, complemented, Sylow
198#3270—1491nomaximal, Sylow
199C2×#137—1491no—
200C22×Q16D14911yescharacteristic, complemented
201D4×Q8—1491no—
202#548—1491no—
203#497—1491no—
204#470—1491no—
205#465—1491no—
206#464—1491no—
207#462—1491no—
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×Q8C2×D14911yescharacteristic
217C22×Q8C2×D14911yescharacteristic
218#137—2984no—
219C2×Q16—24no—
220C2×Q16—1491no—
221C2×Q16—1491no—
222C2×Q16C2×D14911yescharacteristic
223C2×Q16C2×D14911yescharacteristic
224C2×Q16C2×D14911yescharacteristic
225C2×Q16C2×D14911yescharacteristic
226C2×SD16—1491no—
227C2×SD16—1491no—
228C2×SD16—1491no—
229C2×M16—1491no—
230C22×C8C2×D14911yescharacteristic
231C4⋊Q8—2981no—
232C4⋊Q8—1491no—
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×Q8—1491no—
245C4×D4—2981no—
246C4×D4—2981no—
247C4×D4—1491no—
248#106—1491no—
249C4⋊C8—1491no—
250#103—1491no—
251#103—1491no—
252#103—1491no—
253#103—1491no—
254#103—1491no—
255#102—1491no—
256C22⋊C8—1491no—
257C22⋊C8—1491no—
258#97—1491no—
259#41—2982no—
260#41—5961no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—2981no—
268C2×Q8—2982no—
269C2×Q8—5961no—
270C2×Q8—1491no—
271C2×Q8C22×D14911yescharacteristic
272C2×Q8C22×D14911yescharacteristic
273C2×Q8C22×D14911yescharacteristic
274C2×Q8C22×D14911yescharacteristic
275C2×D4—2981no—
276C2×D4—2981no—
277C2×D4—1491no—
278C2×D4—1491no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2981no—
282C22×C4—2981no—
283C22×C4—1491no—
284C22×C4C22×D14911yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—2982no—
289Q16—5961no—
290Q16C22⋊D14914yescomplemented
291SD16—2982no—
292SD16—2982no—
293SD16—5961no—
294M16—2982no—
295C2×C8—22no—
296C2×C8—1491no—
297C2×C8—1491no—
298C2×C8C22×D14911yescharacteristic
299C2×C8C22×D14911yescharacteristic
300C4⋊C4—2981no—
301C4⋊C4—2981no—
302C4⋊C4—2981no—
303C4⋊C4—2981no—
304C4⋊C4—2981no—
305C4⋊C4—2981no—
306C4⋊C4—2981no—
307C4⋊C4—1491no—
308C4⋊C4—1491no—
309C4⋊C4—1491no—
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—2981no—
319C42—2981no—
320C42—1491no—
321C23—2981no—
322C23—1491no—
323C23D4×D14911yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—2981no—
333Q8—2981no—
334Q8—5962no—
335Q8C2×C22⋊D14912yes—
336Q8C2×C22⋊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—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—2981no—
351C2×C4—2981no—
352C2×C4—2981no—
353C2×C4—2981no—
354C2×C4—2981no—
355C2×C4—5962no—
356C2×C4—5962no—
357C2×C4—1491no—
358C2×C4—1491no—
359C2×C4—1491no—
360C2×C4D4×D14911yescharacteristic
361C2×C4C23×D14911yescharacteristic
362C8—21no—
363C8—2981no—
364C8—2981no—
365C8C2×C22⋊D14912yes—
366C22—21no—
367C22—21no—
368C22—2981no—
369C22—2981no—
370C22—2981no—
371C22—5961no—
372C22—5961no—
373C22—5961no—
374C22#1801194312yescomplemented
375C22C2×D4×D14911yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—2981no—
382C4—2981no—
383C4—2981no—
384C4—5961no—
385C4—5961no—
386C4—5961no—
387C4C22×C22⋊D14911yescharacteristic
388C4#1801195611yescharacteristic
389C2—22no—
390C2—5961no—
391C2—11921no—
392C2D4×C22⋊D14911yescharacteristic
393C2C2×#1801194311yescharacteristic
394C2#2375043811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC22×Q16×C149C2
socleC22×C149C22×D4
prime radical#23750286C2
hypercenterC22×Q16D149
hypocenterC149#3270
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#18011943C2semidirect
2. Q8×C22⋊D149C2semidirect
4. #23750018C2semidirect
5. #23749880C2semidirect
6. #23749854C2semidirect
7. #23749849C2semidirect
8. #23749833C2semidirect
9. #23749786C2semidirect
10. #23749781C2semidirect
11. #23749779C2semidirect
12. #23749764C2semidirect
13. #23749732C2semidirect
14. #23749696C2semidirect
15. C22×Q16×C149C2semidirect
119. Q16×C149D4semidirect
170. C22×C149#137semidirect
197. C149#3270semidirect
200. C22×Q16D149semidirect
290. Q16C22⋊D149semidirect
374. C22#18011943semidirect

Maximal subgroups

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

subgroupindex
C2×#180119432
Q8×C22⋊D1492
#237502862
#237500182
#237498802
#237498542
#237498492
#237498332
#237497862
#237497812
#237497792
#237497642
#237497322
#237496962
C22×Q16×C1492
#3270149