Finite Group Databasebeta

← #18003886

Subgroups — #18003886

Order 4736.

700 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924584C211yesmaximal, characteristic
2C2×#11924568C211yesmaximal, characteristic, complemented
3C2×Q16×D37C211yesmaximal, characteristic, complemented
4#14558106C211yesmaximal, characteristic, complemented
5#14557698C211yesmaximal, characteristic, complemented
6#14557697C211yesmaximal, characteristic, complemented
7#14557694C211yesmaximal, characteristic, complemented
8#14557693C211yesmaximal, characteristic, complemented
9#14557690C211yesmaximal, characteristic, complemented
10#14557689C211yesmaximal, characteristic, complemented
11#14557687C211yesmaximal, characteristic, complemented
12#14557540C211yesmaximal, characteristic, complemented
13#14557528C211yesmaximal, characteristic, complemented
14#14557498C211yesmaximal, characteristic, complemented
15C37×#451C211yesmaximal, characteristic, complemented
16#11924584—24no—
17#11924584—24no—
18C2×#10596966—21no—
19C2×#10596966C2211yescharacteristic
20C2×#10596966C2211yescharacteristic
21C2×#10596962—21no—
22C2×#10596962—21no—
23C2×#10596962C2211yescharacteristic
24C2×#10596962C2211yescharacteristic
25C2×Q8×D37—21no—
26C2×Q8×D37C2211yescharacteristic
27C2×Q8×D37C2211yescharacteristic
28C2×Q8×D37C2211yescharacteristic
29#11924568—24no—
30C2×Dic148C2211yescharacteristic
31C2×#10596942C2211yescharacteristic
32C2×#10596942C2211yescharacteristic
33C2×#10596942C2211yescharacteristic
34Q16×D37—24no—
35C2×#10596930C2211yescharacteristic
36C2×#10596941C2211yescharacteristic
37C2×#10596929C2211yescharacteristic
38C2×C8×D37C2211yescharacteristic
39#11924535—21no—
40#11924522—21no—
41#11924521—21no—
42#11924518—21no—
43#11924516C2211yescharacteristic
44#11924515C2211yescharacteristic
45#11924512C2211yescharacteristic
46#11924494C2211yescharacteristic
47#11924461C2211yescharacteristic
48#11924460C2211yescharacteristic
49#11924460C2211yescharacteristic
50#11924459C2211yescharacteristic
51#11924458C2211yescharacteristic
52#11924456C2211yescharacteristic
53C22⋊#64436C2211yescharacteristic
54#11924448C2211yescharacteristic
55#11924448C2211yescharacteristic
56C22×Q8×C37C2211yescharacteristic
57C2×Q16×C37C2211yescharacteristic
58C2×Q16×C37C2211yescharacteristic
59C22⋊Q8×C37C2211yescharacteristic
60#103×C37C2211yescharacteristic
61#103×C37C2211yescharacteristic
62C22⋊C8×C37C2211yescharacteristic
63C2×Q8×C37—22no—
64C2×Q8×C37—22no—
65C2×Q8×C37—21no—
66C2×Q8×C37C2311yescharacteristic
67C2×Q8×C37C2311yescharacteristic
68C2×Q8×C37C2311yescharacteristic
69C22×C4×C37—21no—
70C22×C4×C37C2311yescharacteristic
71C2×C22⋊D37—21no—
72C2×C22⋊D37D411yescharacteristic
73C2×C22⋊D37D411yescharacteristic
74#10596966—28no—
75#10596966—22no—
76#10596966—22no—
77#10596966—22no—
78#10596966D414yes—
79Q8×D37—28no—
80Q8×D37—22no—
81Q8×D37—22no—
82Q8×D37—22no—
83Q8×D37—22no—
84Q8×D37—22no—
85Q8×D37D414yes—
86#10596962—28no—
87#10596962—28no—
88#10596962—22no—
89#10596962—22no—
90#10596962—22no—
91#10596962—22no—
92C2×D148—21no—
93C2×D148—21no—
94C2×D148D411yescharacteristic
95C2×D148C2311yescharacteristic
96C2×C4×D37—21no—
97C2×C4×D37—21no—
98C2×C4×D37—21no—
99C2×C4×D37—21no—
100C2×C4×D37—21no—
101C2×C4×D37—21no—
102C2×C4×D37C2311yescharacteristic
103C2×C4×D37C2311yescharacteristic
104C2×Dic74—21no—
105C2×Dic74—21no—
106C2×Dic74—21no—
107C2×Dic74D411yescharacteristic
108C2×Dic74C2311yescharacteristic
109C2×Dic74C2311yescharacteristic
110Q16×C37—22no—
111Q16×C37—22no—
112C2×C8×C37C2311yescharacteristic
113C2×C8×C37C2311yescharacteristic
114C4⋊C4×C37—21no—
115C4⋊C4×C37C2311yescharacteristic
116C22⋊C4×C37—21no—
117C22⋊Dic37—21no—
118#10596942—22no—
119#10596942—22no—
120#10596942—22no—
121#10596941—22no—
122#10596938—21no—
123#10596938—21no—
124#10596938—21no—
125C4⋊Dic37—21no—
126C4⋊Dic37C2311yescharacteristic
127#10596936—21no—
128#10596936—21no—
129C4×Dic37—21no—
130C2×#64436C2311yescharacteristic
131C2×#64436C2311yescharacteristic
132Dic148—22no—
133#10596930—22no—
134#10596929—22no—
135C8×D37—22no—
136C23×C37C2×D411yescharacteristic
137C22×D37—21no—
138C22×D37C2×D411yescharacteristic
139C22×D37C2×D411yescharacteristic
140Q8×C37—24no—
141Q8×C37—21no—
142Q8×C37—21no—
143Q8×C37—21no—
144Q8×C37—21no—
145Q8×C37C2×D412yes—
146Q8×C37C2×D412yes—
147C2×C4×C37—24no—
148C2×C4×C37—21no—
149C2×C4×C37—21no—
150C2×C4×C37—21no—
151C2×C4×C37—21no—
152C2×C4×C37—21no—
153C2×C4×C37—21no—
154C2×C4×C37C2×D411yescharacteristic
155C2×C4×C37C2411yescharacteristic
156C22⋊D37—24no—
157C22⋊D37—22no—
158C22⋊D37—22no—
159C2×Dic37—21no—
160C2×Dic37—21no—
161C2×Dic37—21no—
162C2×Dic37C2×D411yescharacteristic
163C2×Dic37C2×D411yescharacteristic
164D148—24no—
165D148—24no—
166D148—22no—
167D148—21no—
168D148C2×D412yes—
169C4×D37—24no—
170C4×D37—24no—
171C4×D37—24no—
172C4×D37—24no—
173C4×D37—21no—
174C4×D37—21no—
175C4×D37—24no—
176C4×D37—42no—
177C4×D37C2×D412yes—
178C4×D37C2×D412yes—
179Dic74—24no—
180Dic74—21no—
181Dic74—24no—
182Dic74—22no—
183Dic74—21no—
184Dic74—21no—
185Dic74—42no—
186Dic74C2×D412yes—
187C8×C37—21no—
188C8×C37—21no—
189#64436—21no—
190#64436—21no—
191C22×C37—21no—
192C22×C37—21no—
193C22×C37—21no—
194C22×C37C22×D411yescharacteristic
195C2×D37—21no—
196C2×D37—21no—
197C2×D37—21no—
198C2×D37—22no—
199C2×D37—21no—
200C2×D37—21no—
201C2×D37—41no—
202C2×D37—41no—
203C2×D37#12012yes—
204C4×C37—22no—
205C4×C37—22no—
206C4×C37—21no—
207C4×C37—41no—
208C4×C37—41no—
209C4×C37C22×D411yescharacteristic
210C4×C37C22×D411yescharacteristic
211Dic37—21no—
212Dic37—22no—
213Dic37—41no—
214Dic37—41no—
215Dic37#12012yes—
216#3011—371nomaximal, Sylow
217C2×C37—41no—
218C2×C37C2×#12011yescharacteristic
219C2×C37#57811yescharacteristic
220C2×C37#57811yescharacteristic
221D37—22no—
222D37—41no—
223D37—42no—
224C2×2−1+4—371no—
225C2×#137—371no—
226C22×Q16—371no—
227#536—371no—
228#452—371no—
229#452—371no—
230#452—371no—
231#452—371no—
232#451—371no—
233#451—371no—
234#451—371no—
235#451D3711yescharacteristic, complemented
236#419—371no—
237#416—371no—
238#408—371no—
239C37#301111yescharacteristic, complemented, Sylow
2402−1+4—744no—
2412−1+4—744no—
242C2×#41—371no—
243C2×#41—371no—
244C2×#41—371no—
245C2×#41—371no—
246C2×#41—741no—
247C2×#41—741no—
248C2×#41—741no—
249C22×Q8—371no—
250C22×Q8—371no—
251C22×Q8—371no—
252C22×Q8—741no—
253C22×Q8C2×D3711yescharacteristic
254#137—744no—
255C2×Q16—371no—
256C2×Q16—371no—
257C2×Q16—371no—
258C2×Q16—371no—
259C2×Q16—744no—
260C2×Q16C2×D3711yescharacteristic
261C2×Q16C2×D3711yescharacteristic
262C2×SD16—371no—
263C2×SD16—371no—
264C2×M16—371no—
265C22×C8—371no—
266C4⋊Q8—741no—
267#124—741no—
268#123—741no—
269#123—741no—
270C22⋊Q8—371no—
271C22⋊Q8—371no—
272C22⋊Q8—371no—
273C22⋊Q8C2×D3711yescharacteristic
274#117—371no—
275#103—371no—
276#103—371no—
277#103—371no—
278#103—371no—
279#103C2×D3711yescharacteristic
280#103C2×D3711yescharacteristic
281#102—371no—
282#102—371no—
283C22⋊C8—371no—
284C22⋊C8—371no—
285C22⋊C8—371no—
286C22⋊C8C2×D3711yescharacteristic
287#41—374no—
288#41—742no—
289#41—742no—
290#41—742no—
291#41—742no—
292#41—748no—
293#41—748no—
294#41—742no—
295#41—742no—
296#41—748no—
297#41—742no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—371no—
302C2×Q8—374no—
303C2×Q8—371no—
304C2×Q8—371no—
305C2×Q8—741no—
306C2×Q8—741no—
307C2×Q8—741no—
308C2×Q8—742no—
309C2×Q8—742no—
310C2×Q8—742no—
311C2×Q8—742no—
312C2×Q8—748no—
313C2×Q8—742no—
314C2×Q8C22×D3711yescharacteristic
315C2×Q8C22×D3711yescharacteristic
316C2×Q8C22×D3711yescharacteristic
317C2×D4—371no—
318C2×D4—371no—
319C2×D4—371no—
320C2×D4—371no—
321C2×D4—741no—
322C2×D4—741no—
323C2×D4—741no—
324C22×C4—21no—
325C22×C4—371no—
326C22×C4—371no—
327C22×C4—741no—
328C22×C4—741no—
329C22×C4—741no—
330C22×C4—741no—
331C22×C4—741no—
332C22×C4—741no—
333C22×C4C22×D3711yescharacteristic
334Q16—22no—
335Q16—22no—
336Q16—742no—
337Q16—742no—
338Q16—742no—
339Q16—742no—
340SD16—742no—
341SD16—742no—
342M16—742no—
343C2×C8—371no—
344C2×C8—371no—
345C2×C8—742no—
346C2×C8C22×D3711yescharacteristic
347C2×C8C22×D3711yescharacteristic
348C4⋊C4—21no—
349C4⋊C4—371no—
350C4⋊C4—741no—
351C4⋊C4—741no—
352C4⋊C4—741no—
353C4⋊C4C22×D3711yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—741no—
356C22⋊C4—741no—
357C22⋊C4—741no—
358C22⋊C4—741no—
359C42—741no—
360C23—371no—
361C23—371no—
362C23—741no—
363C23D4×D3711yescharacteristic
364Q8—24no—
365Q8—21no—
366Q8—21no—
367Q8—21no—
368Q8—21no—
369Q8—372no—
370Q8—744no—
371Q8—742no—
372Q8—744no—
373Q8—741no—
374Q8—741no—
375Q8—741no—
376Q8—1482no—
377Q8D4×D3712yes—
378Q8D4×D3712yes—
379D4—372no—
380D4—742no—
381D4—744no—
382D4—744no—
383D4—742no—
384D4—742no—
385D4—744no—
386D4—741no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—371no—
395C2×C4—372no—
396C2×C4—372no—
397C2×C4—371no—
398C2×C4—744no—
399C2×C4—744no—
400C2×C4—744no—
401C2×C4—744no—
402C2×C4—744no—
403C2×C4—741no—
404C2×C4—741no—
405C2×C4—741no—
406C2×C4—741no—
407C2×C4—741no—
408C2×C4—1482no—
409C2×C4D4×D3711yescharacteristic
410C2×C4C23×D3711yescharacteristic
411C8—21no—
412C8—21no—
413C8—741no—
414C8—741no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—372no—
419C22—741no—
420C22—741no—
421C22—742no—
422C22—741no—
423C22—741no—
424C22—741no—
425C22—1481no—
426C22—1481no—
427C22C2×D4×D3711yescharacteristic
428C4—22no—
429C4—22no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—372no—
434C4—741no—
435C4—742no—
436C4—1481no—
437C4—1481no—
438C4C2×D4×D3711yescharacteristic
439C4C2×D4×D3711yescharacteristic
440C2—41no—
441C2—742no—
442C2—1481no—
443C2—1482no—
444C2#120×D3711yescharacteristic
445C2#1455834611yescharacteristic
446C2#1455834911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#451C2
socleC22×C37C22×D4
prime radicalC2×#11924584C2
hypercenter#451D37
hypocenterC37#3011
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
2. C2×#11924568C2semidirect
3. C2×Q16×D37C2semidirect
4. #14558106C2semidirect
5. #14557698C2semidirect
6. #14557697C2semidirect
7. #14557694C2semidirect
8. #14557693C2semidirect
9. #14557690C2semidirect
10. #14557689C2semidirect
11. #14557687C2semidirect
12. #14557540C2semidirect
13. #14557528C2semidirect
14. #14557498C2semidirect
15. C37×#451C2semidirect
235. #451D37semidirect
239. C37#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#119245842
C2×#119245682
C2×Q16×D372
#145581062
#145576982
#145576972
#145576942
#145576932
#145576902
#145576892
#145576872
#145575402
#145575282
#145574982
C37×#4512
#301137