Finite Group Databasebeta

← #49170285

Subgroups — #49170285

Order 37504.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627652C211yesmaximal, characteristic, complemented
2C22×#17989791C211yesmaximal, characteristic, complemented
3#34036788C211yesmaximal, characteristic, complemented
4D4×D1172C211yesmaximal, characteristic, complemented
5#34036518C211yesmaximal, characteristic, complemented
6#34036375C211yesmaximal, characteristic, complemented
7#34036365C211yesmaximal, characteristic, complemented
8#34036345C211yesmaximal, characteristic, complemented
9#34036330C211yesmaximal, characteristic, complemented
10#34036282C211yesmaximal, characteristic, complemented
11#34036276C211yesmaximal, characteristic, complemented
12#34036270C211yesmaximal, characteristic, complemented
13#34036265C211yesmaximal, characteristic, complemented
14#34036231C211yesmaximal, characteristic, complemented
15#435×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic, complemented
17C22×Dic586C2211yescharacteristic, complemented
18C22×D1172C2211yescharacteristic
19C22×D1172C2211yescharacteristic
20C2×D4×D293—21no—
21#22627652—24no—
22C2×#17989791—24no—
23C2×#17989791C2211yescharacteristic
24C2×#17989791C2211yescharacteristic
25C2×#17989791C2211yescharacteristic, complemented
26C2×#17989791C2211yescharacteristic, complemented
27C2×#17989791C2211yescharacteristic, complemented
28C2×#17989791C2211yescharacteristic, complemented
29C2×D2344C2211yescharacteristic
30C2×D2344C2211yescharacteristic, complemented
31C2×D2344C2211yescharacteristic, complemented
32#22627621C2211yescharacteristic
33#22627612C2211yescharacteristic, complemented
34#22627610—21no—
35#22627603C2211yescharacteristic, complemented
36#22627602—21no—
37#22627599—21no—
38#22627598C2211yescharacteristic
39#22627598C2211yescharacteristic
40#22627598C2211yescharacteristic, complemented
41#22627596—21no—
42#22627594—21no—
43#22627594—21no—
44C4×Dic586C2211yescharacteristic, complemented
45C4×D1172C2211yescharacteristic
46#22627584—21no—
47#22627558C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627549C2211yescharacteristic, complemented
50#22627546C2211yescharacteristic
51#22627546C2211yescharacteristic, complemented
52#22627546C2211yescharacteristic, complemented
53#22627546C2211yescharacteristic, complemented
54C2×M16×C293C2211yescharacteristic, complemented
55C22×C8×C293C2211yescharacteristic, complemented
56C4×D4×C293C2211yescharacteristic
57C4⋊C8×C293C2211yescharacteristic, complemented
58C22⋊C8×C293C2211yescharacteristic, complemented
59C22⋊C8×C293C2211yescharacteristic, complemented
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989802—22no—
64#17989802—41no—
65C2×Dic586—22no—
66C2×Dic586—21no—
67C2×Dic586—41no—
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70C2×D1172—22no—
71C2×D1172—22no—
72C2×D1172—21no—
73C2×D1172—21no—
74C2×D1172—21no—
75C2×D1172—21no—
76C2×D1172—41no—
77C2×D1172—41no—
78C2×D1172C2311yescharacteristic
79C2×D1172C2311yescharacteristic
80C2×D1172C2311yescharacteristic
81C2×D1172C2311yescharacteristic
82C2×C22⋊D293—21no—
83C2×C22⋊D293—21no—
84C2×C22⋊D293—21no—
85C2×C22⋊D293—21no—
86D4×D293—44no—
87C22×Dic293—21no—
88C2×C4×D293—21no—
89C2×C4×D293—21no—
90#17989791—22no—
91#17989791—22no—
92#17989791—22no—
93#17989791—22no—
94#17989791—41no—
95#17989791D414yescomplemented
96D2344—22no—
97D2344—22no—
98D2344—41no—
99C4⋊Dic293C2311yescharacteristic
100C4⋊Dic293C2311yescharacteristic
101#17989781—21no—
102#17989781—21no—
103C22⋊Dic293—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779—21no—
109#17989779—21no—
110C4×Dic293—21no—
111C2×D4×C293D411yescharacteristic
112C22×C4×C293C2311yescharacteristic
113C22×C4×C293C2311yescharacteristic
114M16×C293—22no—
115C2×C8×C293—22no—
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293D411yescharacteristic
121C22⋊C4×C293D411yescharacteristic
122C22⋊C4×C293D411yescharacteristic
123C42×C293C2311yescharacteristic
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—21no—
127C22×D293—21no—
128C22×D293—42no—
129C22×D293—42no—
130C22×D293—42no—
131C22×D293—42no—
132C22×D293—81no—
133C22×D293—81no—
134Dic586—21no—
135Dic586—21no—
136Dic586—21no—
137Dic586—42no—
138Dic586C2×D412yescomplemented
139D1172—21no—
140D1172—21no—
141D1172—21no—
142D1172—21no—
143D1172—21no—
144D1172—21no—
145D1172—21no—
146D1172—42no—
147D1172—42no—
148D1172—42no—
149D1172—81no—
150D1172C2×D412yes—
151C22⋊D293—42no—
152C22⋊D293—42no—
153C22⋊D293—42no—
154C22⋊D293—42no—
155C2×Dic293—21no—
156C2×Dic293—21no—
157C2×Dic293—21no—
158C2×Dic293—21no—
159C2×Dic293—42no—
160C4×D293—42no—
161C4×D293—42no—
162C23×C293C2×D411yescharacteristic
163C23×C293C2×D411yescharacteristic
164D4×C293—22no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293C2×D411yescharacteristic
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2×D411yescharacteristic
173C2×C4×C293C2411yescharacteristic
174C8×C293—21no—
175C8×C293—21no—
176C8×C293—21no—
177C8×C293C2×D412yescomplemented
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—21no—
181C22×C293—21no—
182C22×C293—21no—
183C22×C293AGL(1,ℤ/8)12yescomplemented
184C22×C293C22×D411yescharacteristic
185C2×D293—22no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—41no—
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—84no—
198C2×D293—84no—
199C4×C293—21no—
200C4×C293—21no—
201C4×C293—21no—
202C4×C293—21no—
203C4×C293C22×D411yescharacteristic
204C4×C293H(5,𝔽2)11yescharacteristic
205Dic293—22no—
206Dic293—41no—
207Dic293—41no—
208Dic293—41no—
209C2×C293—22no—
210C2×C293—41no—
211C2×C293D4211yescharacteristic
212C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C293#57711yescharacteristic
214D293—42no—
215D293—81no—
216D293—81no—
217D293—81no—
218C293#326511yescharacteristic, complemented, Sylow
219#3265—2931nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2931no—
221C22×SD16—2931no—
222#548—2931no—
223D42—2931no—
224#496—2931no—
225#468—2931no—
226#466—2931no—
227#463—2931no—
228#460—2931no—
229#450—2931no—
230#449—2931no—
231#448—2931no—
232#447—2931no—
233#440—2931no—
234#435D29311yescharacteristic, complemented
235C2×#41—2931no—
236C22×Q8—2931no—
237C22×D4—2931no—
238C22×D4—2931no—
239C22×D4—5861no—
240AGL(1,ℤ/8)—5864no—
241C2×SD16—2931no—
242C2×SD16—2931no—
243C2×SD16—2931no—
244C2×SD16—2931no—
245C2×SD16—2931no—
246C2×SD16—2931no—
247C2×SD16—5864no—
248C2×D8—2931no—
249C2×D8—2931no—
250C2×D8—2931no—
251C2×M16C2×D29311yescharacteristic, complemented
252C22×C8C2×D29311yescharacteristic, complemented
253#127—2931no—
254#124—2931no—
255#124—5861no—
256C22⋊Q8—2931no—
257C22⋊Q8—5861no—
258#121—2931no—
259#121—2931no—
260#121—2931no—
261#121—5861no—
262#121—5861no—
263#120—5861no—
264#120—5861no—
265C4×Q8—2931no—
266C4×D4—2931no—
267C4×D4—5861no—
268C4×D4C2×D29311yescharacteristic
269#107—2931no—
270C4⋊C8C2×D29311yescharacteristic, complemented
271#103—2931no—
272#103—2931no—
273#102—2931no—
274#102—2931no—
275#102—2931no—
276#102—2931no—
277C22⋊C8C2×D29311yescharacteristic, complemented
278C22⋊C8C2×D29311yescharacteristic, complemented
279#97C2×D29311yescharacteristic, complemented
280C24—5861no—
281C24—5861no—
282#41—5862no—
283#41—11721no—
284C2×Q8—2931no—
285C2×Q8—2931no—
286C2×Q8—5862no—
287C2×Q8—5861no—
288C2×Q8—11721no—
289C2×D4—2931no—
290C2×D4—2931no—
291C2×D4—2931no—
292C2×D4—2931no—
293C2×D4—5862no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5861no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—5861no—
303C2×D4—11721no—
304C2×D4—11721no—
305C2×D4—11724no—
306C2×D4D117211yescharacteristic
307C22×C4—5861no—
308C22×C4—5861no—
309C22×C4—5861no—
310C22×C4C22×D29311yescharacteristic
311C22×C4C22×D29311yescharacteristic
312SD16—2934no—
313SD16—5862no—
314SD16—5862no—
315SD16—5862no—
316SD16—5862no—
317SD16—11721no—
318D8—5862no—
319D8—5862no—
320D8—11721no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D29311yescharacteristic
324C2×C8C22×D29311yescharacteristic
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—2931no—
328C4⋊C4—2931no—
329C4⋊C4—5861no—
330C4⋊C4—5861no—
331C4⋊C4D117211yescharacteristic
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C22⋊C4—5861no—
339C22⋊C4D117211yescharacteristic
340C22⋊C4D117211yescharacteristic
341C42—5861no—
342C42C22×D29311yescharacteristic
343C23—23441no—
344C23—23441no—
345C23—5861no—
346C23—5861no—
347C23—5861no—
348C23—5861no—
349C23—11722no—
350C23—11722no—
351C23—11722no—
352C23—11722no—
353C23C2×D117211yescharacteristic
354C23C2×D117211yescharacteristic
355Q8—2932no—
356Q8—5861no—
357Q8—5861no—
358Q8—5861no—
359Q8—11722no—
360D4—22no—
361D4—2932no—
362D4—23441no—
363D4—5861no—
364D4—5861no—
365D4—5861no—
366D4—5861no—
367D4—5861no—
368D4—5861no—
369D4—5861no—
370D4—11722no—
371D4—11722no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4—11722no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—5861no—
385C2×C4—11722no—
386C2×C4—11722no—
387C2×C4—11722no—
388C2×C4C2×D117211yescharacteristic
389C2×C4C2×D117211yescharacteristic
390C2×C4C2×D117211yescharacteristic
391C2×C4C2×D117211yescharacteristic
392C2×C4C23×D29311yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D29312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—23444no—
403C22—23444no—
404C22—5862no—
405C22—11721no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22—11721no—
410C22—11721no—
411C22—11721no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22#2262765212yescomplemented
417C22C22×D117211yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—5862no—
423C4—11721no—
424C4—11721no—
425C4—11721no—
426C4C2×D4×D29311yescharacteristic
427C4#2262767011yescharacteristic
428C2—22no—
429C2—41no—
430C2—23441no—
431C2—23441no—
432C2—23441no—
433C2—11722no—
434C2D4×D117211yescharacteristic
435C2C2×#2262765211yescharacteristic
436C2#3403694111yescharacteristic
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#435×C293C2
socleC22×C293C22×D4
hypercenter#435D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. C22×#17989791C2semidirect
3. #34036788C2semidirect
4. D4×D1172C2semidirect
5. #34036518C2semidirect
6. #34036375C2semidirect
7. #34036365C2semidirect
8. #34036345C2semidirect
9. #34036330C2semidirect
10. #34036282C2semidirect
11. #34036276C2semidirect
12. #34036270C2semidirect
13. #34036265C2semidirect
14. #34036231C2semidirect
15. #435×C293C2semidirect
16. C2×#17989802C22semidirect
17. C22×Dic586C22semidirect
25. C2×#17989791C22semidirect
26. C2×#17989791C22semidirect
27. C2×#17989791C22semidirect
28. C2×#17989791C22semidirect
30. C2×D2344C22semidirect
31. C2×D2344C22semidirect
33. #22627612C22semidirect
35. #22627603C22semidirect
40. #22627598C22semidirect
44. C4×Dic586C22semidirect
47. #22627558C22semidirect
48. #22627549C22semidirect
49. #22627549C22semidirect
51. #22627546C22semidirect
52. #22627546C22semidirect
53. #22627546C22semidirect
54. C2×M16×C293C22semidirect
55. C22×C8×C293C22semidirect
57. C4⋊C8×C293C22semidirect
58. C22⋊C8×C293C22semidirect
59. C22⋊C8×C293C22semidirect
60. #97×C293C22semidirect
95. #17989791D4semidirect
138. Dic586C2×D4semidirect
177. C8×C293C2×D4semidirect
183. C22×C293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
234. #435D293semidirect
251. C2×M16C2×D293semidirect
252. C22×C8C2×D293semidirect
270. C4⋊C8C2×D293semidirect
277. C22⋊C8C2×D293semidirect
278. C22⋊C8C2×D293semidirect
279. #97C2×D293semidirect
396. C8D4×D293semidirect
416. C22#22627652semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
C22×#179897912
#340367882
D4×D11722
#340365182
#340363752
#340363652
#340363452
#340363302
#340362822
#340362762
#340362702
#340362652
#340362312
#435×C2932
#3265293