Finite Group Databasebeta

← #34047687

Subgroups — #34047687

Order 19072.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic149C211yesmaximal, characteristic
2C2×#18011870C211yesmaximal, characteristic, complemented
3#23749524C211yesmaximal, characteristic, complemented
4#23749518C211yesmaximal, characteristic, complemented
5#23749518C211yesmaximal, characteristic, complemented
6#23749511C211yesmaximal, characteristic, complemented
7#23749511C211yesmaximal, characteristic, complemented
8#23749505C211yesmaximal, characteristic
9#23749500C211yesmaximal, characteristic, complemented
10#23749500C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic149C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic149C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic149C211yesmaximal, characteristic, complemented
14C2×#18011818C211yesmaximal, characteristic, complemented
15C2×#123×C149C211yesmaximal, characteristic
16C23×Dic149C2211yescharacteristic
17C23×Dic149C2211yescharacteristic
18C23×Dic149C2211yescharacteristic
19D4×Dic149—24no—
20#18011870—24no—
21C2×C4⋊Dic149C2211yescharacteristic
22C2×C4⋊Dic149C2211yescharacteristic
23C2×C4⋊Dic149C2211yescharacteristic
24C2×C4⋊Dic149C2211yescharacteristic
25C2×C4⋊Dic149C2211yescharacteristic
26C2×C22⋊Dic149C2211yescharacteristic
27C2×C22⋊Dic149C2211yescharacteristic
28C2×C22⋊Dic149C2211yescharacteristic
29C2×C22⋊Dic149C2211yescharacteristic
30C2×C22⋊Dic149C2211yescharacteristic
31C2×C4×Dic149C2211yescharacteristic
32C2×C4×Dic149C2211yescharacteristic
33C2×C4×Dic149C2211yescharacteristic
34C2×C4×Dic149C2211yescharacteristic
35C2×C4×Dic149C2211yescharacteristic
36#18011818C2211yescharacteristic
37#18011818C2211yescharacteristic
38#18011818C2211yescharacteristic
39#18011818C2211yescharacteristic
40#18011818C2211yescharacteristic
41#18011818C2211yescharacteristic
42#18011818C2211yescharacteristic
43#18011818C2211yescharacteristic
44#18011818C2211yescharacteristic
45#18011818C2211yescharacteristic
46C22×D4×C149C2211yescharacteristic
47C23×C4×C149C2211yescharacteristic
48#123×C149C418yescomplemented
49C2×C4⋊C4×C149C2211yescharacteristic
50C2×C4⋊C4×C149C2211yescharacteristic
51C2×C22⋊C4×C149C2211yescharacteristic
52C2×C22⋊C4×C149C2211yescharacteristic
53C2×C22⋊C4×C149C2211yescharacteristic
54C22×Dic149—22no—
55C22×Dic149—22no—
56C22×Dic149—22no—
57C22×Dic149—22no—
58C22×Dic149—22no—
59C22×Dic149—22no—
60C22×Dic149—41no—
61C22×Dic149—41no—
62C22×Dic149—41no—
63C22×Dic149C2311yescharacteristic
64C22×Dic149C2311yescharacteristic
65C22×Dic149C2311yescharacteristic
66C22×Dic149C2311yescharacteristic
67C22×Dic149C2311yescharacteristic
68C22×Dic149C2311yescharacteristic
69C22×Dic149C2311yescharacteristic
70C22×Dic149C2311yescharacteristic
71C4⋊Dic149—22no—
72C4⋊Dic149—22no—
73C4⋊Dic149—22no—
74C4⋊Dic149—22no—
75C4⋊Dic149—41no—
76C22⋊Dic149—22no—
77C22⋊Dic149—22no—
78C22⋊Dic149—22no—
79C22⋊Dic149—41no—
80C22⋊Dic149D414yes—
81C4×Dic149—22no—
82C4×Dic149—22no—
83C4×Dic149—22no—
84C4×Dic149—22no—
85C4×Dic149—41no—
86C24×C149C2311yescharacteristic
87C24×C149C2311yescharacteristic
88C2×D4×C149—22no—
89C2×D4×C149—22no—
90C2×D4×C149C2×C414yes—
91C22×C4×C149—22no—
92C22×C4×C149—22no—
93C22×C4×C149C2×C414yescomplemented
94C22×C4×C149C2311yescharacteristic
95C22×C4×C149C2311yescharacteristic
96C22×C4×C149C2311yescharacteristic
97C22×C4×C149C2311yescharacteristic
98C22×C4×C149C2311yescharacteristic
99C4⋊C4×C149C2×C414yescomplemented
100C4⋊C4×C149C2×C414yescomplemented
101C22⋊C4×C149C2×C414yescomplemented
102C22⋊C4×C149C2×C414yescomplemented
103C22⋊C4×C149C2×C414yescomplemented
104C2×Dic149—21no—
105C2×Dic149—21no—
106C2×Dic149—21no—
107C2×Dic149—21no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C2×Dic149—21no—
111C2×Dic149—21no—
112C2×Dic149—21no—
113C2×Dic149—21no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149—21no—
118C2×Dic149—21no—
119C2×Dic149—21no—
120C2×Dic149—21no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—21no—
124C2×Dic149—44no—
125C2×Dic149—44no—
126C2×Dic149—44no—
127C2×Dic149C2×D412yes—
128C2×Dic149C2×D412yes—
129C2×Dic149#4112yes—
130C2×Dic149#4112yes—
131C23×C149—21no—
132C23×C149—21no—
133C23×C149—21no—
134C23×C149—21no—
135C23×C149—21no—
136C23×C149—21no—
137C23×C149—21no—
138C23×C149—21no—
139C23×C149—21no—
140C23×C149—21no—
141C23×C149C22×C412yes—
142C23×C149C22×C412yes—
143C23×C149C2×D412yes—
144C23×C149#4112yes—
145C23×C149C2411yescharacteristic
146D4×C149—28no—
147C2×C4×C149—28no—
148C2×C4×C149—21no—
149C2×C4×C149—21no—
150C2×C4×C149—21no—
151C2×C4×C149—21no—
152C2×C4×C149—21no—
153C2×C4×C149—21no—
154C2×C4×C149—21no—
155C2×C4×C149—21no—
156C2×C4×C149—21no—
157C2×C4×C149—21no—
158C2×C4×C149C22×C412yes—
159C2×C4×C149C22×C412yes—
160C2×C4×C149C22×C412yes—
161C2×C4×C149C22×C412yes—
162C2×C4×C149C22×C412yes—
163C22×C149—22no—
164C22×C149—22no—
165C22×C149—22no—
166C22×C149—22no—
167C22×C149—22no—
168C22×C149—22no—
169C22×C149—22no—
170C22×C149—22no—
171C22×C149—22no—
172C22×C149—41no—
173C22×C149—41no—
174C22×C149—41no—
175C22×C149—41no—
176C22×C149C4×D414yes—
177C22×C149C23×C411yescharacteristic
178C22×C149C22×D411yescharacteristic
179C22×C149C2×#4111yescharacteristic
180C22×C149C2×#4111yescharacteristic
181C22×C149C2×#4111yescharacteristic
182C22×C149H(5,𝔽2)11yescharacteristic
183C22×C1492−1+411yescharacteristic
184C4×C149—22no—
185C4×C149—22no—
186C4×C149—22no—
187C4×C149—22no—
188C4×C149—22no—
189Dic149—22no—
190Dic149—22no—
191Dic149—22no—
192Dic149—22no—
193Dic149—41no—
194Dic149—41no—
195Dic149—41no—
196Dic149—41no—
197C2×C149—24no—
198C2×C149—42no—
199C2×C149C2×C4×D411yescharacteristic
200C2×C149C4×#4111yescharacteristic
201C2×C149#52011yescharacteristic
202C2×C149#54611yescharacteristic
203C2×C149#54711yescharacteristic
204C2×C149#55111yescharacteristic
205C2×C149#55211yescharacteristic
206C149#235911yescharacteristic, complemented, Sylow
207#2359—1491nomaximal, Sylow
208C2×#123D14911yescharacteristic
209C2×C4×D4—1491no—
210C2×#117—1491no—
211#389—1491no—
212#388—1491no—
213#388—1491no—
214#387—1491no—
215#387—1491no—
216#386—1491no—
217#385—1491no—
218#385—1491no—
219C4×C4⋊C4—1491no—
220C4×C22⋊C4—1491no—
221C4×C22⋊C4—1491no—
222C2×#95—1491no—
223C22×D4C2×D14911yescharacteristic
224C23×C4—1491no—
225C23×C4—1491no—
226C23×C4—1491no—
227C23×C4C2×D14911yescharacteristic
228#123Dic14918yescomplemented
229C4×D4—2984no—
230#117—2984no—
231C2×C4⋊C4—1491no—
232C2×C4⋊C4—1491no—
233C2×C4⋊C4—1491no—
234C2×C4⋊C4—1491no—
235C2×C4⋊C4—1491no—
236C2×C4⋊C4C2×D14911yescharacteristic
237C2×C4⋊C4C2×D14911yescharacteristic
238C2×C22⋊C4—1491no—
239C2×C22⋊C4—1491no—
240C2×C22⋊C4—1491no—
241C2×C22⋊C4—1491no—
242C2×C22⋊C4—1491no—
243C2×C22⋊C4C2×D14911yescharacteristic
244C2×C22⋊C4C2×D14911yescharacteristic
245C2×C22⋊C4C2×D14911yescharacteristic
246C2×C42—1491no—
247C2×C42—1491no—
248C2×C42—1491no—
249C2×C42—1491no—
250C2×C42—1491no—
251#95—1491no—
252#95—1491no—
253#95—1491no—
254#95—1491no—
255#95—1491no—
256#95—1491no—
257#95—1491no—
258#95—1491no—
259#95—1491no—
260#95—1491no—
261C24C22×D14911yescharacteristic
262C24C22×D14911yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C2×Dic14914yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—2982no—
269C22×C4—2982no—
270C22×C4—2982no—
271C22×C4—2982no—
272C22×C4—2982no—
273C22×C4—2982no—
274C22×C4—5961no—
275C22×C4—5961no—
276C22×C4—5961no—
277C22×C4—1491no—
278C22×C4—1491no—
279C22×C4—1491no—
280C22×C4—1491no—
281C22×C4—1491no—
282C22×C4—1491no—
283C22×C4—1491no—
284C22×C4—1491no—
285C22×C4C2×Dic14914yescomplemented
286C22×C4C22×D14911yescharacteristic
287C22×C4C22×D14911yescharacteristic
288C22×C4C22×D14911yescharacteristic
289C22×C4C22×D14911yescharacteristic
290C22×C4C22×D14911yescharacteristic
291C4⋊C4—2982no—
292C4⋊C4—2982no—
293C4⋊C4—2982no—
294C4⋊C4—2982no—
295C4⋊C4—5961no—
296C4⋊C4C2×Dic14914yescomplemented
297C4⋊C4C2×Dic14914yescomplemented
298C22⋊C4—2982no—
299C22⋊C4—2982no—
300C22⋊C4—2982no—
301C22⋊C4—5961no—
302C22⋊C4—1494no—
303C22⋊C4C2×Dic14914yescomplemented
304C22⋊C4C2×Dic14914yescomplemented
305C22⋊C4C2×Dic14914yescomplemented
306C42—2982no—
307C42—2982no—
308C42—2982no—
309C42—2982no—
310C42—5961no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C22×Dic14912yes—
322C23C22×Dic14912yes—
323C23D4×D14912yes—
324C23#1455873512yes—
325C23C23×D14911yescharacteristic
326D4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—28no—
338C2×C4—2981no—
339C2×C4—2981no—
340C2×C4—2981no—
341C2×C4—2981no—
342C2×C4—2981no—
343C2×C4—2981no—
344C2×C4—2981no—
345C2×C4—2981no—
346C2×C4—2981no—
347C2×C4—2981no—
348C2×C4—2981no—
349C2×C4—2981no—
350C2×C4—2981no—
351C2×C4—2981no—
352C2×C4—2981no—
353C2×C4—2981no—
354C2×C4—2981no—
355C2×C4—2981no—
356C2×C4—2981no—
357C2×C4—2981no—
358C2×C4—5964no—
359C2×C4—5964no—
360C2×C4—5964no—
361C2×C4—1492no—
362C2×C4—1492no—
363C2×C4—1492no—
364C2×C4—1492no—
365C2×C4C22×Dic14912yes—
366C2×C4C22×Dic14912yes—
367C2×C4C22×Dic14912yes—
368C2×C4C22×Dic14912yes—
369C2×C4C22×Dic14912yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22D4×Dic14914yes—
384C22C23×Dic14911yescharacteristic
385C22C2×D4×D14911yescharacteristic
386C22#41×D14911yescharacteristic
387C22#41×D14911yescharacteristic
388C22C2×#1455873511yescharacteristic
389C22#1801195711yescharacteristic
390C22#1801195811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—2982no—
397C4—2982no—
398C4—2982no—
399C4—2982no—
400C4—5961no—
401C4—5961no—
402C4—5961no—
403C4—5961no—
404C2—24no—
405C2—42no—
406C2C2×D4×Dic14911yescharacteristic
407C2#41×Dic14911yescharacteristic
408C2#2375010811yescharacteristic
409C2#2375026711yescharacteristic
410C2#2375027311yescharacteristic
411C2#2375029811yescharacteristic
412C2#2375031211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
FittingC2×#123×C149C2
socle, cosocleC23×C149C24
prime radicalC22×D4×C149C22
hypercenterC2×#123D149
hypocenterC149#2359
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#18011870C2semidirect
3. #23749524C2semidirect
4. #23749518C2semidirect
5. #23749518C2semidirect
6. #23749511C2semidirect
7. #23749511C2semidirect
9. #23749500C2semidirect
10. #23749500C2semidirect
11. C4⋊C4×Dic149C2semidirect
12. C22⋊C4×Dic149C2semidirect
13. C22⋊C4×Dic149C2semidirect
14. C2×#18011818C2semidirect
48. #123×C149C4semidirect
93. C22×C4×C149C2×C4semidirect
99. C4⋊C4×C149C2×C4semidirect
100. C4⋊C4×C149C2×C4semidirect
101. C22⋊C4×C149C2×C4semidirect
102. C22⋊C4×C149C2×C4semidirect
103. C22⋊C4×C149C2×C4semidirect
206. C149#2359semidirect
228. #123Dic149semidirect
285. C22×C4C2×Dic149semidirect
296. C4⋊C4C2×Dic149semidirect
297. C4⋊C4C2×Dic149semidirect
303. C22⋊C4C2×Dic149semidirect
304. C22⋊C4C2×Dic149semidirect
305. C22⋊C4C2×Dic149semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1492
C2×#180118702
#237495242
#237495182
#237495112
#237495052
#237495002
C4⋊C4×Dic1492
C22⋊C4×Dic1492
C2×#180118182
C2×#123×C1492
#2359149