Finite Group Databasebeta

← #18006341

Subgroups — #18006341

Order 4736.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10596942C211yesmaximal, characteristic, complemented
2C2×Q16×D37C211yesmaximal, characteristic, complemented
3Q8×C22⋊D37C211yesmaximal, characteristic
4#14558196C211yesmaximal, characteristic
5#14557911C211yesmaximal, characteristic, complemented
6#14557773C211yesmaximal, characteristic, complemented
7#14557772C211yesmaximal, characteristic, complemented
8#14557742C211yesmaximal, characteristic, complemented
9#14557741C211yesmaximal, characteristic, complemented
10#14557691C211yesmaximal, characteristic, complemented
11#14557690C211yesmaximal, characteristic, complemented
12#14557687C211yesmaximal, characteristic, complemented
13#14557625C211yesmaximal, characteristic, complemented
14#14557596C211yesmaximal, characteristic
15C37×#451C211yesmaximal, characteristic, complemented
16C22×Dic74C2211yescharacteristic
17C2×Q8×D37C2211yescharacteristic
18C2×Q8×D37C2211yescharacteristic
19C2×Dic148C2211yescharacteristic
20C2×Dic148C2211yescharacteristic
21C2×#10596942—24no—
22C2×#10596942C2211yescharacteristic
23C2×#10596942C2211yescharacteristic
24C2×#10596942C2211yescharacteristic
25C2×#10596942C2211yescharacteristic
26C2×#10596942C2211yescharacteristic
27Q16×D37—24no—
28C22×#64436C2211yescharacteristic
29C2×C8×D37C2211yescharacteristic
30#11924535—21no—
31#11924534—21no—
32#11924534C2211yescharacteristic
33#11924534C2211yescharacteristic
34#11924516C2211yescharacteristic
35#11924515—21no—
36#11924515C2211yescharacteristic
37#11924514—21no—
38#11924513—21no—
39#11924512—21no—
40#11924512C2211yescharacteristic
41Q8×Dic37C2211yescharacteristic
42#11924501C2211yescharacteristic
43C4×C22⋊D37—21no—
44C4×C22⋊D37C2211yescharacteristic
45#11924496—21no—
46#11924469C2211yescharacteristic
47#11924465C2211yescharacteristic
48#11924461C2211yescharacteristic
49#11924461C2211yescharacteristic
50#11924460C2211yescharacteristic
51#11924459C2211yescharacteristic
52#11924448C2211yescharacteristic
53C8×Dic37C2211yescharacteristic
54C22×Q8×C37C2211yescharacteristic
55C2×Q16×C37C2211yescharacteristic
56C2×Q16×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic
58#103×C37C2211yescharacteristic
59#103×C37C2211yescharacteristic
60C22⋊C8×C37C2211yescharacteristic
61C2×Q8×C37—21no—
62C2×Q8×C37—22no—
63C2×Q8×C37—41no—
64C2×Q8×C37C2311yescharacteristic
65C2×Q8×C37C2311yescharacteristic
66C2×Q8×C37C2311yescharacteristic
67C22×C4×C37—21no—
68C22×C4×C37C2311yescharacteristic
69C2×C22⋊D37D411yescharacteristic
70C22×Dic37—21no—
71Q8×D37—22no—
72Q8×D37—22no—
73Q8×D37—41no—
74Q8×D37—41no—
75C2×C4×D37—21no—
76C2×C4×D37—21no—
77C2×C4×D37C2311yescharacteristic
78C2×Dic74—22no—
79C2×Dic74—21no—
80C2×Dic74—21no—
81C2×Dic74—21no—
82C2×Dic74—41no—
83C2×Dic74C2311yescharacteristic
84C2×Dic74C2311yescharacteristic
85C2×Dic74C2311yescharacteristic
86Q16×C37—22no—
87Q16×C37—41no—
88C2×C8×C37C2311yescharacteristic
89C2×C8×C37C2311yescharacteristic
90C4⋊C4×C37—21no—
91C4⋊C4×C37C2311yescharacteristic
92C22⋊C4×C37—21no—
93C22⋊Dic37—21no—
94C22⋊Dic37D411yescharacteristic
95#10596942—22no—
96#10596942—22no—
97#10596942—22no—
98#10596942—22no—
99#10596942D414yescomplemented
100#10596938—21no—
101#10596938—21no—
102#10596938D411yescharacteristic
103C4⋊Dic37—21no—
104C4⋊Dic37—21no—
105C4⋊Dic37C2311yescharacteristic
106#10596936—21no—
107#10596936—21no—
108#10596936—21no—
109#10596936—21no—
110#10596936—21no—
111#10596936—21no—
112#10596936—21no—
113#10596936D411yescharacteristic
114C4×Dic37—21no—
115C4×Dic37—21no—
116C4×Dic37C2311yescharacteristic
117C2×#64436—22no—
118C2×#64436C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120Dic148—22no—
121Dic148—41no—
122C8×D37—22no—
123C23×C37C2×D411yescharacteristic
124C22×D37C2×D411yescharacteristic
125Q8×C37—21no—
126Q8×C37—21no—
127Q8×C37—21no—
128Q8×C37—21no—
129Q8×C37—21no—
130Q8×C37—42no—
131Q8×C37C2×D412yes—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—42no—
139C2×C4×C37C2×D411yescharacteristic
140C2×C4×C37C2411yescharacteristic
141C22⋊D37Q1614yescomplemented
142C2×Dic37—21no—
143C2×Dic37—21no—
144C2×Dic37—21no—
145C2×Dic37—21no—
146C2×Dic37—42no—
147C2×Dic37C2×D411yescharacteristic
148C2×Dic37C2×D411yescharacteristic
149C2×Dic37C2×D411yescharacteristic
150C4×D37—21no—
151C4×D37—21no—
152C4×D37—42no—
153C4×D37—42no—
154Dic74—21no—
155Dic74—21no—
156Dic74—21no—
157Dic74—21no—
158Dic74—21no—
159Dic74—42no—
160Dic74—42no—
161Dic74—42no—
162Dic74C2×D412yes—
163C8×C37—21no—
164C8×C37—21no—
165#64436—21no—
166#64436C2×D412yes—
167C22×C37—21no—
168C22×C37—21no—
169C22×C37C2×Q1612yescomplemented
170C22×C37C22×D411yescharacteristic
171C2×D37—21no—
172C2×D37—21no—
173C2×D37C2×Q1612yescomplemented
174C4×C37—21no—
175C4×C37—22no—
176C4×C37—41no—
177C4×C37—41no—
178C4×C37—41no—
179C4×C37C22×D411yescharacteristic
180C4×C37H(5,𝔽2)11yescharacteristic
181Dic37—22no—
182Dic37—21no—
183Dic37—21no—
184Dic37—41no—
185Dic37—41no—
186Dic37—41no—
187Dic37C2×Q1612yescomplemented
188D4×Q16—371nomaximal, Sylow
189C2×C37—22no—
190C2×C37D4211yescharacteristic
191C2×C37C22×Q1611yescharacteristic
192C2×C37#57811yescharacteristic
193D37—22no—
194C22×Q16—371no—
195C22×Q16—371no—
196D4×Q8—371no—
197D4×Q8—371no—
198#494—371no—
199#467—371no—
200#467—371no—
201#462—371no—
202#462—371no—
203#451—371no—
204#451—371no—
205#451—371no—
206#451D3711yescharacteristic, complemented
207C4×Q16—371no—
208C8×D4—371no—
209C37D4×Q1611yescharacteristic, complemented, Sylow
210C22×Q8—371no—
211C22×Q8—371no—
212C22×Q8—371no—
213C22×Q8C2×D3711yescharacteristic
214C2×Q16—371no—
215C2×Q16—371no—
216C2×Q16—371no—
217C2×Q16—371no—
218C2×Q16—371no—
219C2×Q16—371no—
220C2×Q16—371no—
221C2×Q16—744no—
222C2×Q16—744no—
223C2×Q16C2×D3711yescharacteristic
224C2×Q16C2×D3711yescharacteristic
225C22×C8—371no—
226C22×C8—371no—
227C4⋊Q8—371no—
228C4⋊Q8—371no—
229C4⋊Q8—741no—
230C4⋊Q8—741no—
231C22⋊Q8—371no—
232C22⋊Q8—371no—
233C22⋊Q8—371no—
234C22⋊Q8—741no—
235C22⋊Q8—741no—
236C22⋊Q8—741no—
237C22⋊Q8—741no—
238C22⋊Q8C2×D3711yescharacteristic
239C4×Q8—371no—
240C4×Q8—371no—
241C4×D4—371no—
242C4×D4—741no—
243C4×D4—741no—
244#107—371no—
245C4⋊C8—371no—
246#103—371no—
247#103—371no—
248#103—371no—
249#103—371no—
250#103C2×D3711yescharacteristic
251#103C2×D3711yescharacteristic
252C22⋊C8—371no—
253C22⋊C8C2×D3711yescharacteristic
254C4×C8—371no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—371no—
259C2×Q8—371no—
260C2×Q8—371no—
261C2×Q8—741no—
262C2×Q8—741no—
263C2×Q8—742no—
264C2×Q8—741no—
265C2×Q8—742no—
266C2×Q8—742no—
267C2×Q8—1481no—
268C2×Q8—1481no—
269C2×Q8—1481no—
270C2×Q8C22×D3711yescharacteristic
271C2×Q8C22×D3711yescharacteristic
272C2×Q8C22×D3711yescharacteristic
273C2×D4—371no—
274C22×C4—21no—
275C22×C4—371no—
276C22×C4—741no—
277C22×C4—741no—
278C22×C4—741no—
279C22×C4C22×D3711yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—374no—
283Q16—742no—
284Q16—742no—
285Q16—742no—
286Q16—742no—
287Q16—742no—
288Q16—1481no—
289C2×C8—371no—
290C2×C8—371no—
291C2×C8—742no—
292C2×C8—742no—
293C2×C8C22×D3711yescharacteristic
294C2×C8C22×D3711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—371no—
297C4⋊C4—371no—
298C4⋊C4—741no—
299C4⋊C4—741no—
300C4⋊C4—741no—
301C4⋊C4—741no—
302C4⋊C4—741no—
303C4⋊C4—741no—
304C4⋊C4—741no—
305C4⋊C4—741no—
306C4⋊C4—741no—
307C4⋊C4C22×D3711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—371no—
310C22⋊C4—371no—
311C22⋊C4—741no—
312C22⋊C4—741no—
313C22⋊C4—741no—
314C42—371no—
315C42—741no—
316C42—741no—
317C23—371no—
318C23D4×D3711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—372no—
326Q8—741no—
327Q8—741no—
328Q8—741no—
329Q8—741no—
330Q8—741no—
331Q8—1482no—
332Q8—1482no—
333Q8—1482no—
334Q8D4×D3712yes—
335D4—374no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—371no—
344C2×C4—371no—
345C2×C4—371no—
346C2×C4—741no—
347C2×C4—741no—
348C2×C4—741no—
349C2×C4—741no—
350C2×C4—741no—
351C2×C4—741no—
352C2×C4—1482no—
353C2×C4—1482no—
354C2×C4—1482no—
355C2×C4D4×D3711yescharacteristic
356C2×C4C23×D3711yescharacteristic
357C8—21no—
358C8—21no—
359C8—372no—
360C8—741no—
361C22—21no—
362C22—21no—
363C22—372no—
364C22—741no—
365C22—741no—
366C22Q16×D3712yescomplemented
367C22C2×D4×D3711yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—372no—
374C4—741no—
375C4—741no—
376C4—742no—
377C4—1481no—
378C4—1481no—
379C4—1481no—
380C4C2×D4×D3711yescharacteristic
381C4#1192458111yescharacteristic
382C2—22no—
383C2—742no—
384C2#1455816511yescharacteristic
385C2C2×Q16×D3711yescharacteristic
386C2#1455834911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#451C2
socleC22×C37C22×D4
prime radicalC2×C22⋊D37D4
hypercenter#451D37
hypocenterC37D4×Q16
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#10596942C2semidirect
2. C2×Q16×D37C2semidirect
5. #14557911C2semidirect
6. #14557773C2semidirect
7. #14557772C2semidirect
8. #14557742C2semidirect
9. #14557741C2semidirect
10. #14557691C2semidirect
11. #14557690C2semidirect
12. #14557687C2semidirect
13. #14557625C2semidirect
15. C37×#451C2semidirect
99. #10596942D4semidirect
141. C22⋊D37Q16semidirect
169. C22×C37C2×Q16semidirect
173. C2×D37C2×Q16semidirect
187. Dic37C2×Q16semidirect
206. #451D37semidirect
209. C37D4×Q16semidirect
366. C22Q16×D37semidirect

Maximal subgroups

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

subgroupindex
C22×#105969422
C2×Q16×D372
Q8×C22⋊D372
#145581962
#145579112
#145577732
#145577722
#145577422
#145577412
#145576912
#145576902
#145576872
#145576252
#145575962
C37×#4512
D4×Q1637