Finite Group Databasebeta

← #18006107

Subgroups — #18006107

Order 4736.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924567C211yesmaximal, characteristic, complemented
2C22×#10596941C211yesmaximal, characteristic, complemented
3#14558179C211yesmaximal, characteristic
4#14557925C211yesmaximal, characteristic, complemented
5#14557781C211yesmaximal, characteristic, complemented
6#14557772C211yesmaximal, characteristic, complemented
7#14557755C211yesmaximal, characteristic, complemented
8#14557735C211yesmaximal, characteristic, complemented
9#14557693C211yesmaximal, characteristic, complemented
10#14557689C211yesmaximal, characteristic, complemented
11#14557683C211yesmaximal, characteristic, complemented
12#14557672C211yesmaximal, characteristic, complemented
13#14557631C211yesmaximal, characteristic, complemented
14#14557605C211yesmaximal, characteristic, complemented
15D4×Q8×C37C211yesmaximal, characteristic, complemented
16C2×#10596962C2211yescharacteristic
17C22×D148C2211yescharacteristic
18#11924567—24no—
19C2×#10596942C2211yescharacteristic, complemented
20C2×#10596942C2211yescharacteristic, complemented
21C2×#10596942C2211yescharacteristic, complemented
22C2×#10596941—24no—
23C2×#10596941C2211yescharacteristic
24C2×#10596941C2211yescharacteristic
25C2×#10596941C2211yescharacteristic
26C2×#10596941C2211yescharacteristic
27C2×#10596941C2211yescharacteristic
28C2×#10596941C2211yescharacteristic
29C2×#10596934C2211yescharacteristic, complemented
30C22×#64436C2211yescharacteristic
31#11924524C2211yescharacteristic
32#11924521—21no—
33#11924515C2211yescharacteristic
34#11924510C2211yescharacteristic
35#11924509—21no—
36#11924504—21no—
37C4×D148C2211yescharacteristic
38C22⋊C4×D37—21no—
39#11924469C2211yescharacteristic, complemented
40C4⋊#64436C2211yescharacteristic, complemented
41#11924460C2211yescharacteristic, complemented
42#11924460C2211yescharacteristic, complemented
43#11924460C2211yescharacteristic, complemented
44#11924459C2211yescharacteristic, complemented
45#11924456C2211yescharacteristic
46#11924456C2211yescharacteristic
47C22⋊#64436C2211yescharacteristic
48C22⋊#64436C2211yescharacteristic, complemented
49#11924447C2211yescharacteristic, complemented
50C22×Q8×C37C2211yescharacteristic
51C22×Q8×C37C2211yescharacteristic, complemented
52C4⋊Q8×C37—21no—
53C4⋊Q8×C37C2211yescharacteristic, complemented
54C22⋊Q8×C37—21no—
55C22⋊Q8×C37—21no—
56C22⋊Q8×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic, complemented
58C4×Q8×C37C2211yescharacteristic, complemented
59C4×D4×C37—21no—
60C4×D4×C37C2211yescharacteristic
61C23×D37—21no—
62C2×Q8×C37—21no—
63C2×Q8×C37—21no—
64C2×Q8×C37—22no—
65C2×Q8×C37—22no—
66C2×Q8×C37—41no—
67C2×Q8×C37—41no—
68C2×Q8×C37C2311yescharacteristic
69C2×Q8×C37C2311yescharacteristic
70C2×Q8×C37C2311yescharacteristic
71C2×D4×C37D411yescharacteristic
72C22×C4×C37—21no—
73C22×C4×C37—21no—
74C22×C4×C37C2311yescharacteristic
75C22×C4×C37C2311yescharacteristic
76C2×C22⋊D37—21no—
77C2×C22⋊D37—21no—
78#10596962—22no—
79#10596962—41no—
80C2×D148—22no—
81C2×D148—21no—
82C2×D148—41no—
83C2×D148C2311yescharacteristic
84C2×D148C2311yescharacteristic
85C2×C4×D37—21no—
86C2×C4×D37—21no—
87C2×Dic74C2311yescharacteristic
88C4⋊C4×C37—21no—
89C4⋊C4×C37—21no—
90C4⋊C4×C37—21no—
91C4⋊C4×C37—21no—
92C4⋊C4×C37—21no—
93C4⋊C4×C37D411yescharacteristic
94C4⋊C4×C37C2311yescharacteristic
95C22⋊C4×C37—21no—
96C22⋊C4×C37—21no—
97C22⋊C4×C37D411yescharacteristic
98C22⋊C4×C37D411yescharacteristic
99C42×C37—21no—
100C42×C37C2311yescharacteristic
101C22⋊Dic37—21no—
102#10596942—22no—
103#10596942—22no—
104#10596942—41no—
105#10596941—22no—
106#10596941—22no—
107#10596941—22no—
108#10596941—22no—
109#10596941—41no—
110#10596941D414yescomplemented
111#10596938—21no—
112#10596938—21no—
113#10596938—21no—
114#10596938—21no—
115C4⋊Dic37C2311yescharacteristic
116#10596936—21no—
117#10596934—22no—
118C2×#64436—22no—
119C2×#64436C2311yescharacteristic
120C2×#64436C2311yescharacteristic
121C2×#64436C2311yescharacteristic
122C2×#64436C2311yescharacteristic
123C23×C37C2×D411yescharacteristic
124C23×C37C2×D411yescharacteristic
125C22×D37—21no—
126C22×D37—21no—
127C22×D37—42no—
128C22×D37—42no—
129C22×D37—81no—
130Q8×C37—21no—
131Q8×C37—21no—
132Q8×C37—21no—
133Q8×C37—21no—
134Q8×C37—21no—
135Q8×C37—42no—
136Q8×C37—42no—
137Q8×C37C2×D412yes—
138D4×C37—22no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37—21no—
142C2×C4×C37—21no—
143C2×C4×C37—21no—
144C2×C4×C37—21no—
145C2×C4×C37—21no—
146C2×C4×C37—21no—
147C2×C4×C37—42no—
148C2×C4×C37—42no—
149C2×C4×C37C2×D411yescharacteristic
150C2×C4×C37C2×D411yescharacteristic
151C2×C4×C37C2×D411yescharacteristic
152C2×C4×C37C2×D411yescharacteristic
153C2×C4×C37C2411yescharacteristic
154C22⋊D37—42no—
155C22⋊D37—81no—
156C2×Dic37—21no—
157C2×Dic37—21no—
158D148—21no—
159D148—21no—
160D148—21no—
161D148—42no—
162D148C2×D412yes—
163C4×D37—42no—
164C4×D37—81no—
165Dic74—21no—
166Dic74—21no—
167#64436—21no—
168#64436—21no—
169#64436—21no—
170#64436C2×D412yes—
171C22×C37—21no—
172C22×C37—21no—
173C22×C37—21no—
174C22×C37—21no—
175C22×C37—21no—
176C22×C37#13712yescomplemented
177C22×C37C22×D411yescharacteristic
178C2×D37—22no—
179C2×D37—41no—
180C2×D37—41no—
181C2×D37—41no—
182C2×D37—41no—
183C2×D37—41no—
184C2×D37—84no—
185C4×C37—21no—
186C4×C37—21no—
187C4×C37—22no—
188C4×C37—21no—
189C4×C37—21no—
190C4×C37—41no—
191C4×C37—41no—
192C4×C37—41no—
193C4×C37C22×D411yescharacteristic
194C4×C37H(5,𝔽2)11yescharacteristic
195Dic37—41no—
196Dic37—41no—
197#3266—371nomaximal, Sylow
198C2×C37—22no—
199C2×C37—41no—
200C2×C37D4211yescharacteristic
201C2×C37C2×#13711yescharacteristic
202C2×C37#57711yescharacteristic
203D37—42no—
204D37—81no—
205C2×#137—371no—
206C22×SD16—371no—
207D4×Q8D3711yescharacteristic, complemented
208#546—371no—
209#497—371no—
210#468—371no—
211#467—371no—
212#464—371no—
213#461—371no—
214#452—371no—
215#451—371no—
216#450—371no—
217#448—371no—
218#440—371no—
219#435—371no—
220C37#326611yescharacteristic, complemented, Sylow
221C2×#41—371no—
222C22×Q8C2×D3711yescharacteristic
223C22×Q8C2×D3711yescharacteristic, complemented
224C22×D4—371no—
225#137—744no—
226C2×Q16—371no—
227C2×Q16—371no—
228C2×Q16—371no—
229C2×SD16—371no—
230C2×SD16—371no—
231C2×SD16—371no—
232C2×SD16—371no—
233C2×SD16—371no—
234C2×SD16—371no—
235C2×SD16—744no—
236C2×M16—371no—
237C22×C8—371no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D3711yescharacteristic, complemented
240#124—371no—
241#123—741no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—371no—
245C22⋊Q8C2×D3711yescharacteristic
246C22⋊Q8C2×D3711yescharacteristic, complemented
247#121—371no—
248#121—741no—
249#120—741no—
250C4×Q8C2×D3711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—371no—
253C4×D4C2×D3711yescharacteristic
254C2×C22⋊C4—741no—
255#107—371no—
256C4⋊C8—371no—
257#103—371no—
258#103—371no—
259#103—371no—
260#103—371no—
261#102—371no—
262#102—371no—
263C22⋊C8—371no—
264C22⋊C8—371no—
265#97—371no—
266C24—741no—
267#41—742no—
268#41—1481no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—371no—
276C2×Q8C22×D3711yescharacteristic
277C2×Q8C22×D3711yescharacteristic
278C2×Q8C22×D3711yescharacteristic
279C2×D4—371no—
280C2×D4—371no—
281C2×D4—742no—
282C2×D4—741no—
283C2×D4—741no—
284C2×D4—741no—
285C2×D4—1481no—
286C2×D4C22⋊D3711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—741no—
290C22×C4—741no—
291C22×C4C22×D3711yescharacteristic
292C22×C4C22×D3711yescharacteristic
293Q16—742no—
294Q16—742no—
295Q16—1481no—
296SD16—374no—
297SD16—742no—
298SD16—742no—
299SD16—742no—
300SD16—742no—
301SD16—1481no—
302M16—742no—
303C2×C8—371no—
304C2×C8—371no—
305C2×C8—371no—
306C2×C8—371no—
307C2×C8—742no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—371no—
314C4⋊C4—741no—
315C4⋊C4C22⋊D3711yescharacteristic
316C4⋊C4C22×D3711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—741no—
320C22⋊C4—741no—
321C22⋊C4—741no—
322C22⋊C4—741no—
323C22⋊C4—741no—
324C22⋊C4C22⋊D3711yescharacteristic
325C22⋊C4C22⋊D3711yescharacteristic
326C42—21no—
327C42C22×D3711yescharacteristic
328C23—2961no—
329C23—741no—
330C23—741no—
331C23—1482no—
332C23—1482no—
333C23C2×C22⋊D3711yescharacteristic
334C23C2×C22⋊D3711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—741no—
343Q8—741no—
344Q8D4×D3712yes—
345D4—22no—
346D4—372no—
347D4—2961no—
348D4—741no—
349D4—741no—
350D4—741no—
351D4—1482no—
352D4—1482no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—2961no—
364C2×C4—741no—
365C2×C4—741no—
366C2×C4—1482no—
367C2×C4C2×C22⋊D3711yescharacteristic
368C2×C4C2×C22⋊D3711yescharacteristic
369C2×C4C2×C22⋊D3711yescharacteristic
370C2×C4C2×C22⋊D3711yescharacteristic
371C2×C4C23×D3711yescharacteristic
372C8—372no—
373C8—741no—
374C8—741no—
375C8—741no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—2964no—
382C22—742no—
383C22—1481no—
384C22—1481no—
385C22—1481no—
386C22—1481no—
387C22—1481no—
388C22#1192456712yescomplemented
389C22C22×C22⋊D3711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—1481no—
399C4—1481no—
400C4C2×D4×D3711yescharacteristic
401C4#1192458111yescharacteristic
402C2—22no—
403C2—41no—
404C2—2961no—
405C2—1482no—
406C2D4×C22⋊D3711yescharacteristic
407C2C2×#1192456711yescharacteristic
408C2#1455834111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingD4×Q8×C37C2
socleC22×C37C22×D4
prime radical#14558179C2
hypercenterD4×Q8D37
hypocenterC37#3266
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11924567C2semidirect
2. C22×#10596941C2semidirect
4. #14557925C2semidirect
5. #14557781C2semidirect
6. #14557772C2semidirect
7. #14557755C2semidirect
8. #14557735C2semidirect
9. #14557693C2semidirect
10. #14557689C2semidirect
11. #14557683C2semidirect
12. #14557672C2semidirect
13. #14557631C2semidirect
14. #14557605C2semidirect
15. D4×Q8×C37C2semidirect
19. C2×#10596942C22semidirect
20. C2×#10596942C22semidirect
21. C2×#10596942C22semidirect
29. C2×#10596934C22semidirect
39. #11924469C22semidirect
40. C4⋊#64436C22semidirect
41. #11924460C22semidirect
42. #11924460C22semidirect
43. #11924460C22semidirect
44. #11924459C22semidirect
48. C22⋊#64436C22semidirect
49. #11924447C22semidirect
51. C22×Q8×C37C22semidirect
53. C4⋊Q8×C37C22semidirect
57. C22⋊Q8×C37C22semidirect
58. C4×Q8×C37C22semidirect
110. #10596941D4semidirect
176. C22×C37#137semidirect
207. D4×Q8D37semidirect
220. C37#3266semidirect
223. C22×Q8C2×D37semidirect
239. C4⋊Q8C2×D37semidirect
246. C22⋊Q8C2×D37semidirect
250. C4×Q8C2×D37semidirect
388. C22#11924567semidirect

Maximal subgroups

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

subgroupindex
C2×#119245672
C22×#105969412
#145581792
#145579252
#145577812
#145577722
#145577552
#145577352
#145576932
#145576892
#145576832
#145576722
#145576312
#145576052
D4×Q8×C372
#326637