Finite Group Databasebeta

← #34056331

Subgroups — #34056331

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×#18011940C211yesmaximal, characteristic, complemented
2Q8×C22⋊D149C211yesmaximal, characteristic, complemented
3#23750274C211yesmaximal, characteristic
4#23750003C211yesmaximal, characteristic, complemented
5#23749883C211yesmaximal, characteristic, complemented
6#23749880C211yesmaximal, characteristic, complemented
7#23749833C211yesmaximal, characteristic, complemented
8#23749828C211yesmaximal, characteristic, complemented
9#23749787C211yesmaximal, characteristic, complemented
10#23749784C211yesmaximal, characteristic, complemented
11#23749781C211yesmaximal, characteristic, complemented
12#23749779C211yesmaximal, characteristic, complemented
13SD16×Dic149C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15C2×#137×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18#18011940—24no—
19C2×Dic596C2211yescharacteristic
20C2×#14558726C2211yescharacteristic
21C2×#14558726C2211yescharacteristic
22C2×#14558726C2211yescharacteristic
23C2×#14558724C2211yescharacteristic
24C2×#14558719C2211yescharacteristic
25#18011909—21no—
26#18011909C2211yescharacteristic
27#18011909C2211yescharacteristic
28#18011894—21no—
29#18011892—21no—
30#18011890—21no—
31#18011890C2211yescharacteristic
32#18011890C2211yescharacteristic
33#18011887—21no—
34#18011887C2211yescharacteristic
35#18011887C2211yescharacteristic
36#18011883—21no—
37Q8×Dic149C2211yescharacteristic
38D4×Dic149C2211yescharacteristic
39C4×C22⋊D149—21no—
40C4×C22⋊D149C2211yescharacteristic
41C2×#14558715—21no—
42#18011843C2211yescharacteristic
43#18011840C2211yescharacteristic
44#18011836C2211yescharacteristic
45#18011836C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011835C2211yescharacteristic
48#18011835C2211yescharacteristic
49#18011832C2211yescharacteristic
50C22⋊#11924746C2211yescharacteristic
51#18011823C2211yescharacteristic
52C8×Dic149C2211yescharacteristic
53C2×#41×C149C2211yescharacteristic
54C22×Q8×C149C2211yescharacteristic
55#137×C149—24no—
56C2×Q16×C149C2211yescharacteristic
57C2×Q16×C149C2211yescharacteristic
58C2×SD16×C149C2211yescharacteristic
59C2×SD16×C149C2211yescharacteristic
60C2×M16×C149C2211yescharacteristic
61#14558735—22no—
62#14558735—41no—
63C2×Dic298—21no—
64C2×Dic298C2311yescharacteristic
65C2×Dic298C2311yescharacteristic
66Q8×D149—22no—
67Q8×D149—41no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149D411yescharacteristic
70C22×Dic149—21no—
71C22×Dic149—21no—
72C2×C4×D149—21no—
73C2×C4×D149C2311yescharacteristic
74Dic596—22no—
75#14558726—22no—
76#14558726—41no—
77#14558726—41no—
78#14558724—22no—
79#14558719—22no—
80C2×#11924746C2311yescharacteristic
81C2×#11924746C2311yescharacteristic
82C4⋊Dic149—21no—
83C4⋊Dic149C2311yescharacteristic
84C4⋊Dic149C2311yescharacteristic
85#14558715—21no—
86#14558715—21no—
87#14558715—21no—
88#14558715—21no—
89#14558715—21no—
90#14558715—21no—
91#14558715—21no—
92#14558715—21no—
93#14558715D411yescharacteristic
94C22⋊Dic149—21no—
95C22⋊Dic149—21no—
96C22⋊Dic149—21no—
97C22⋊Dic149D411yescharacteristic
98#14558713—21no—
99#14558713—21no—
100#14558713D411yescharacteristic
101C4×Dic149—21no—
102C4×Dic149C2311yescharacteristic
103#41×C149—22no—
104#41×C149—41no—
105C2×Q8×C149—21no—
106C2×Q8×C149—22no—
107C2×Q8×C149—41no—
108C2×Q8×C149C2311yescharacteristic
109C2×Q8×C149C2311yescharacteristic
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149C2311yescharacteristic
113C22×C4×C149—21no—
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116Q16×C149—22no—
117Q16×C149—22no—
118SD16×C149—22no—
119SD16×C149D414yescomplemented
120M16×C149—22no—
121C2×C8×C149C2311yescharacteristic
122C2×C8×C149C2311yescharacteristic
123C22×D149C2×D411yescharacteristic
124Dic298—21no—
125Dic298—21no—
126Dic298—21no—
127Dic298—21no—
128Dic298—42no—
129C22⋊D149—22no—
130C22⋊D149—42no—
131C2×Dic149—21no—
132C2×Dic149—21no—
133C2×Dic149—21no—
134C2×Dic149—21no—
135C2×Dic149—42no—
136C2×Dic149—42no—
137C2×Dic149C2×D411yescharacteristic
138C2×Dic149C2×D411yescharacteristic
139C2×Dic149C2×D411yescharacteristic
140C4×D149—21no—
141C4×D149—21no—
142C4×D149—42no—
143#11924746—21no—
144#11924746—21no—
145C23×C149—21no—
146C23×C149C2×D411yescharacteristic
147Q8×C149—21no—
148Q8×C149—21no—
149Q8×C149—21no—
150Q8×C149—21no—
151Q8×C149—21no—
152Q8×C149—42no—
153Q8×C149C2×D412yes—
154D4×C149—21no—
155D4×C149—42no—
156D4×C149C2×D412yes—
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—22no—
170C22×C149—21no—
171C22×C149—21no—
172C22×C149—41no—
173C22×C149—41no—
174C22×C149C22×D411yescharacteristic
175C2×D149—21no—
176C2×D149—21no—
177C2×D149—21no—
178C4×C149—21no—
179C4×C149—22no—
180C4×C149—41no—
181C4×C149—41no—
182C4×C149C22×D411yescharacteristic
183C4×C149H(5,𝔽2)11yescharacteristic
184Dic149—21no—
185Dic149—21no—
186Dic149—41no—
187Dic149—41no—
188Dic149—41no—
189Dic149—41no—
190Dic149#13712yescomplemented
191C2×C149—41no—
192C2×C149—42no—
193C2×C149D4211yescharacteristic
194C2×C149C2×#13711yescharacteristic
195C2×C149#57811yescharacteristic
196D149—41no—
197C149#327611yescharacteristic, complemented, Sylow
198#3276—1491nomaximal, Sylow
199C2×#137—1491no—
200C2×#137D14911yescharacteristic, complemented
201D4×Q8—1491no—
202#547—1491no—
203#494—1491no—
204#470—1491no—
205#470—1491no—
206#462—1491no—
207#461—1491no—
208#452—1491no—
209#452—1491no—
210#451—1491no—
211#451—1491no—
212C4×SD16—1491no—
213#436—1491no—
214C2×#41—1491no—
215C2×#41C2×D14911yescharacteristic
216C22×Q8—1491no—
217C22×Q8C2×D14911yescharacteristic
218#137—24no—
219#137—2984no—
220C2×Q16—1491no—
221C2×Q16—1491no—
222C2×Q16—1491no—
223C2×Q16—1491no—
224C2×Q16C2×D14911yescharacteristic
225C2×Q16C2×D14911yescharacteristic
226C2×SD16—1491no—
227C2×SD16C2×D14911yescharacteristic
228C2×SD16C2×D14911yescharacteristic
229C2×M16—1491no—
230C2×M16C2×D14911yescharacteristic
231C4⋊Q8—2981no—
232C4⋊Q8—1491no—
233C4⋊Q8—1491no—
234#123—2981no—
235#123—2981no—
236C22⋊Q8—2981no—
237C22⋊Q8—2981no—
238C22⋊Q8—1491no—
239C22⋊Q8—1491no—
240C22⋊Q8—1491no—
241C22⋊Q8—1491no—
242#121—2981no—
243C4×Q8—1491no—
244C4×D4—2981no—
245C4×D4—1491no—
246C4×D4—1491no—
247C2×C4⋊C4—2981no—
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—
258C4×C8—1491no—
259#41—22no—
260#41—41no—
261#41—2982no—
262#41—5961no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—2981no—
267C2×Q8—2982no—
268C2×Q8—5961no—
269C2×Q8—1491no—
270C2×Q8—1491no—
271C2×Q8C22×D14911yescharacteristic
272C2×Q8C22×D14911yescharacteristic
273C2×Q8C22×D14911yescharacteristic
274C2×D4—21no—
275C2×D4—2981no—
276C2×D4—1491no—
277C2×D4C22×D14911yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—2981no—
281C22×C4—2981no—
282C22×C4—2981no—
283C22×C4—1491no—
284C22×C4C22×D14911yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—2982no—
288Q16—2982no—
289Q16—5961no—
290Q16—5961no—
291SD16—22no—
292SD16—2982no—
293SD16C22⋊D14914yescomplemented
294M16—22no—
295M16—2982no—
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—2981no—
308C4⋊C4—2981no—
309C4⋊C4—1491no—
310C4⋊C4—1491no—
311C4⋊C4—1491no—
312C22⋊C4—2981no—
313C22⋊C4—2981no—
314C22⋊C4—2981no—
315C22⋊C4—2981no—
316C22⋊C4—2981no—
317C22⋊C4—1491no—
318C22⋊C4—1491no—
319C42—2981no—
320C42—1491no—
321C23—21no—
322C23—1491no—
323C23D4×D14911yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—2981no—
331Q8—2981no—
332Q8—2981no—
333Q8—2981no—
334Q8—5962no—
335Q8C2×C22⋊D14912yes—
336D4—21no—
337D4—42no—
338D4—2982no—
339D4—5962no—
340D4C2×C22⋊D14912yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—2981no—
349C2×C4—2981no—
350C2×C4—2981no—
351C2×C4—2981no—
352C2×C4—2981no—
353C2×C4—2981no—
354C2×C4—5962no—
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—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—2981no—
373C22—2981no—
374C22—2981no—
375C22C2×D4×D14911yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—2981no—
381C4—2981no—
382C4—5961no—
383C4—5961no—
384C4—5961no—
385C4—5961no—
386C4—1492no—
387C4C22×C22⋊D14911yescharacteristic
388C4#1801195611yescharacteristic
389C2—41no—
390C2—42no—
391C2—5961no—
392C2D4×C22⋊D14911yescharacteristic
393C2#137×D14911yescharacteristic
394C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×#137×C149C2
socleC22×C149C22×D4
prime radical#23750274C2
hypercenterC2×#137D149
hypocenterC149#3276
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#18011940C2semidirect
2. Q8×C22⋊D149C2semidirect
4. #23750003C2semidirect
5. #23749883C2semidirect
6. #23749880C2semidirect
7. #23749833C2semidirect
8. #23749828C2semidirect
9. #23749787C2semidirect
10. #23749784C2semidirect
11. #23749781C2semidirect
12. #23749779C2semidirect
13. SD16×Dic149C2semidirect
14. #23749701C2semidirect
15. C2×#137×C149C2semidirect
119. SD16×C149D4semidirect
190. Dic149#137semidirect
197. C149#3276semidirect
200. C2×#137D149semidirect
293. SD16C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119402
Q8×C22⋊D1492
#237502742
#237500032
#237498832
#237498802
#237498332
#237498282
#237497872
#237497842
#237497812
#237497792
SD16×Dic1492
#237497012
C2×#137×C1492
#3276149