Finite Group Databasebeta

← #49170261

Subgroups — #49170261

Order 37504.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627650C211yesmaximal, characteristic, complemented
2C22×#17989787C211yesmaximal, characteristic, complemented
3#34036775C211yesmaximal, characteristic, complemented
4#34036516C211yesmaximal, characteristic, complemented
5#34036380C211yesmaximal, characteristic, complemented
6#34036356C211yesmaximal, characteristic, complemented
7#34036344C211yesmaximal, characteristic, complemented
8#34036321C211yesmaximal, characteristic, complemented
9#34036280C211yesmaximal, characteristic, complemented
10#34036272C211yesmaximal, characteristic, complemented
11#34036263C211yesmaximal, characteristic, complemented
12#34036262C211yesmaximal, characteristic, complemented
13#34036239C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15D42×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic, complemented
17C22×D1172C2211yescharacteristic, complemented
18#22627650—24no—
19C2×#17989790C2211yescharacteristic, complemented
20C2×#17989790C2211yescharacteristic, complemented
21C2×#17989790C2211yescharacteristic, complemented
22C2×#17989787—24no—
23C2×#17989787C2211yescharacteristic
24C2×#17989787C2211yescharacteristic
25C2×#17989787C2211yescharacteristic, complemented
26C2×#17989787C2211yescharacteristic, complemented
27C2×#17989787C2211yescharacteristic, complemented
28C2×#17989787C2211yescharacteristic, complemented
29C2×#17989786C2211yescharacteristic, complemented
30C22×#14556873C2211yescharacteristic, complemented
31#22627612C2211yescharacteristic, complemented
32#22627609—21no—
33#22627603C2211yescharacteristic, complemented
34#22627598C2211yescharacteristic, complemented
35#22627597—21no—
36#22627592—21no—
37C4×D1172C2211yescharacteristic, complemented
38C22⋊C4×D293—21no—
39#22627555C2211yescharacteristic, complemented
40C4⋊#14556873C2211yescharacteristic, complemented
41#22627547C2211yescharacteristic, complemented
42#22627545C2211yescharacteristic, complemented
43#22627545C2211yescharacteristic, complemented
44#22627545C2211yescharacteristic, complemented
45#22627544C2211yescharacteristic, complemented
46#22627544C2211yescharacteristic, complemented
47C22⋊#14556873C2211yescharacteristic, complemented
48C22⋊#14556873C2211yescharacteristic, complemented
49#22627535C2211yescharacteristic, complemented
50C22×D4×C293—21no—
51C22×D4×C293C2211yescharacteristic
52C22×D4×C293C2211yescharacteristic, complemented
53#127×C293C2211yescharacteristic, complemented
54#121×C293—21no—
55#121×C293C2211yescharacteristic
56#121×C293C2211yescharacteristic, complemented
57#120×C293—21no—
58#120×C293—21no—
59C4×D4×C293C2211yescharacteristic
60C4×D4×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586C2311yescharacteristic
65C2×D1172—22no—
66C2×D1172—21no—
67C2×D1172—41no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293—21no—
74#17989790—22no—
75#17989790—22no—
76#17989790—41no—
77#17989787—22no—
78#17989787—22no—
79#17989787—22no—
80#17989787—22no—
81#17989787—41no—
82#17989787D414yescomplemented
83#17989786—22no—
84C2×#14556873—22no—
85C2×#14556873C2311yescharacteristic
86C2×#14556873C2311yescharacteristic
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C4⋊Dic293C2311yescharacteristic
90#17989781—21no—
91C22⋊Dic293—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96C24×C293—21no—
97C24×C293—21no—
98C2×D4×C293—22no—
99C2×D4×C293—22no—
100C2×D4×C293—21no—
101C2×D4×C293—21no—
102C2×D4×C293—21no—
103C2×D4×C293—21no—
104C2×D4×C293—21no—
105C2×D4×C293—21no—
106C2×D4×C293—41no—
107C2×D4×C293—41no—
108C2×D4×C293—44no—
109C2×D4×C293D411yescharacteristic
110C2×D4×C293C2311yescharacteristic
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114C22×C4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116C4⋊C4×C293D411yescharacteristic
117C4⋊C4×C293C2311yescharacteristic
118C22⋊C4×C293—21no—
119C22⋊C4×C293—21no—
120C22⋊C4×C293—21no—
121C22⋊C4×C293D411yescharacteristic
122C22⋊C4×C293D411yescharacteristic
123C42×C293C2311yescharacteristic
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—42no—
127C22×D293—42no—
128C22×D293—81no—
129Dic586—21no—
130Dic586—21no—
131D1172—21no—
132D1172—21no—
133D1172—21no—
134D1172—42no—
135D1172C2×D412yescomplemented
136C22⋊D293—42no—
137C22⋊D293—81no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C4×D293—42no—
141C4×D293—81no—
142#14556873—21no—
143#14556873—21no—
144#14556873—21no—
145#14556873C2×D412yescomplemented
146C23×C293—21no—
147C23×C293—21no—
148C23×C293—21no—
149C23×C293—42no—
150C23×C293—42no—
151C23×C293—42no—
152C23×C293—42no—
153C23×C293—81no—
154C23×C293—81no—
155C23×C293C2×D411yescharacteristic
156C23×C293C2×D411yescharacteristic
157D4×C293—22no—
158D4×C293—21no—
159D4×C293—21no—
160D4×C293—21no—
161D4×C293—21no—
162D4×C293—21no—
163D4×C293—42no—
164D4×C293—42no—
165D4×C293—42no—
166D4×C293—42no—
167D4×C293—42no—
168D4×C293—81no—
169D4×C293C2×D412yes—
170C2×C4×C293—21no—
171C2×C4×C293—21no—
172C2×C4×C293—21no—
173C2×C4×C293—21no—
174C2×C4×C293—21no—
175C2×C4×C293—42no—
176C2×C4×C293C2×D411yescharacteristic
177C2×C4×C293C2×D411yescharacteristic
178C2×C4×C293C2×D411yescharacteristic
179C2×C4×C293C2×D411yescharacteristic
180C2×C4×C293C2411yescharacteristic
181C22×C293—21no—
182C22×C293—21no—
183C22×C293—21no—
184C22×C293—22no—
185C22×C293—21no—
186C22×C293—21no—
187C22×C293—41no—
188C22×C293—41no—
189C22×C293—41no—
190C22×C293—41no—
191C22×C293—41no—
192C22×C293—41no—
193C22×C293—41no—
194C22×C293—41no—
195C22×C293—84no—
196C22×C293—84no—
197C22×C293AGL(1,ℤ/8)12yescomplemented
198C22×C293C22×D411yescharacteristic
199C2×D293—22no—
200C2×D293—41no—
201C2×D293—41no—
202C2×D293—41no—
203C2×D293—41no—
204C2×D293—41no—
205C2×D293—84no—
206C4×C293—21no—
207C4×C293—21no—
208C4×C293—21no—
209C4×C293—21no—
210C4×C293—41no—
211C4×C293C22×D411yescharacteristic
212C4×C293H(5,𝔽2)11yescharacteristic
213Dic293—41no—
214Dic293—41no—
215C2×C293—22no—
216C2×C293—41no—
217C2×C293—42no—
218C2×C293—81no—
219C2×C293—81no—
220C2×C293D4211yescharacteristic
221C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C293#57611yescharacteristic
223D293—42no—
224D293—81no—
225C293#326411yescharacteristic, complemented, Sylow
226#3264—2931nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2931no—
228C22×D8—2931no—
229#546—2931no—
230D42D29311yescharacteristic, complemented
231#496—2931no—
232#469—2931no—
233#465—2931no—
234#463—2931no—
235#459—2931no—
236#450—2931no—
237#449—2931no—
238#447—2931no—
239#447—2931no—
240#442—2931no—
241#435—2931no—
242C2×#41—2931no—
243C22×D4—21no—
244C22×D4—2931no—
245C22×D4C2×D29311yescharacteristic
246C22×D4C2×D29311yescharacteristic, complemented
247AGL(1,ℤ/8)—5864no—
248C2×SD16—2931no—
249C2×SD16—2931no—
250C2×SD16—2931no—
251C2×D8—2931no—
252C2×D8—2931no—
253C2×D8—2931no—
254C2×D8—2931no—
255C2×D8—2931no—
256C2×D8—2931no—
257C2×D8—5864no—
258C2×M16—2931no—
259C22×C8—2931no—
260#127C2×D29311yescharacteristic, complemented
261#124—2931no—
262#123—5861no—
263C22⋊Q8—2931no—
264#121—21no—
265#121—2931no—
266#121—5861no—
267#121C2×D29311yescharacteristic
268#121C2×D29311yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—5861no—
272C4×D4—2931no—
273C4×D4C2×D29311yescharacteristic
274C4×D4C2×D29311yescharacteristic, complemented
275C2×C22⋊C4—5861no—
276#106—2931no—
277C4⋊C8—2931no—
278#103—2931no—
279#102—2931no—
280#102—2931no—
281#102—2931no—
282#102—2931no—
283#102—2931no—
284C22⋊C8—2931no—
285C22⋊C8—2931no—
286#97—2931no—
287C24—21no—
288C24—21no—
289C24—5861no—
290#41—5862no—
291#41—11721no—
292C2×Q8—2931no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—2931no—
305C2×D4—2931no—
306C2×D4—5862no—
307C2×D4—5861no—
308C2×D4—5861no—
309C2×D4—5861no—
310C2×D4—11721no—
311C2×D4C22⋊D29311yescharacteristic
312C2×D4C22×D29311yescharacteristic
313C2×D4C22×D29311yescharacteristic
314C2×D4C22×D29311yescharacteristic
315C22×C4—21no—
316C22×C4—5861no—
317C22×C4—5861no—
318C22×C4C22×D29311yescharacteristic
319C22×C4C22×D29311yescharacteristic
320SD16—5862no—
321SD16—5862no—
322SD16—11721no—
323D8—2934no—
324D8—5862no—
325D8—5862no—
326D8—5862no—
327D8—5862no—
328D8—11721no—
329M16—5862no—
330C2×C8—2931no—
331C2×C8—2931no—
332C2×C8—2931no—
333C2×C8—2931no—
334C2×C8—5862no—
335C4⋊C4—2931no—
336C4⋊C4—5861no—
337C4⋊C4C22⋊D29311yescharacteristic
338C4⋊C4C22×D29311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—5861no—
343C22⋊C4—5861no—
344C22⋊C4—5861no—
345C22⋊C4—5861no—
346C22⋊C4—5861no—
347C22⋊C4C22⋊D29311yescharacteristic
348C22⋊C4C22⋊D29311yescharacteristic
349C42C22×D29311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—23441no—
360C23—5861no—
361C23—5861no—
362C23—11722no—
363C23—11722no—
364C23C2×C22⋊D29311yescharacteristic
365C23C2×C22⋊D29311yescharacteristic
366Q8—5861no—
367Q8—5861no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—2932no—
381D4—23441no—
382D4—5861no—
383D4—5861no—
384D4—5861no—
385D4—11722no—
386D4—11722no—
387D4D4×D29312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—23441no—
395C2×C4—5861no—
396C2×C4—5861no—
397C2×C4—11722no—
398C2×C4C2×C22⋊D29311yescharacteristic
399C2×C4C2×C22⋊D29311yescharacteristic
400C2×C4C2×C22⋊D29311yescharacteristic
401C2×C4C2×C22⋊D29311yescharacteristic
402C2×C4C23×D29311yescharacteristic
403C8—2932no—
404C8—5861no—
405C8—5861no—
406C8—5861no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—23444no—
424C22—5862no—
425C22—11721no—
426C22—11721no—
427C22—11721no—
428C22—11721no—
429C22—11721no—
430C22#2262765012yescomplemented
431C22C22×C22⋊D29311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—11721no—
438C4—11721no—
439C4C2×D4×D29311yescharacteristic
440C4#2262766911yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—23441no—
447C2—11722no—
448C2D4×C22⋊D29311yescharacteristic
449C2C2×#2262765011yescharacteristic
450C2#3403693211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingD42×C293C2
socleC22×C293C22×D4
hypercenterD42D293
hypocenterC293#3264
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#22627650C2semidirect
2. C22×#17989787C2semidirect
3. #34036775C2semidirect
4. #34036516C2semidirect
5. #34036380C2semidirect
6. #34036356C2semidirect
7. #34036344C2semidirect
8. #34036321C2semidirect
9. #34036280C2semidirect
10. #34036272C2semidirect
11. #34036263C2semidirect
12. #34036262C2semidirect
13. #34036239C2semidirect
14. #34036201C2semidirect
15. D42×C293C2semidirect
16. C2×#17989802C22semidirect
17. C22×D1172C22semidirect
19. C2×#17989790C22semidirect
20. C2×#17989790C22semidirect
21. C2×#17989790C22semidirect
25. C2×#17989787C22semidirect
26. C2×#17989787C22semidirect
27. C2×#17989787C22semidirect
28. C2×#17989787C22semidirect
29. C2×#17989786C22semidirect
30. C22×#14556873C22semidirect
31. #22627612C22semidirect
33. #22627603C22semidirect
34. #22627598C22semidirect
37. C4×D1172C22semidirect
39. #22627555C22semidirect
40. C4⋊#14556873C22semidirect
41. #22627547C22semidirect
42. #22627545C22semidirect
43. #22627545C22semidirect
44. #22627545C22semidirect
45. #22627544C22semidirect
46. #22627544C22semidirect
47. C22⋊#14556873C22semidirect
48. C22⋊#14556873C22semidirect
49. #22627535C22semidirect
52. C22×D4×C293C22semidirect
53. #127×C293C22semidirect
56. #121×C293C22semidirect
60. C4×D4×C293C22semidirect
82. #17989787D4semidirect
135. D1172C2×D4semidirect
145. #14556873C2×D4semidirect
197. C22×C293AGL(1,ℤ/8)semidirect
225. C293#3264semidirect
230. D42D293semidirect
246. C22×D4C2×D293semidirect
260. #127C2×D293semidirect
268. #121C2×D293semidirect
274. C4×D4C2×D293semidirect
430. C22#22627650semidirect

Maximal subgroups

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

subgroupindex
C2×#226276502
C22×#179897872
#340367752
#340365162
#340363802
#340363562
#340363442
#340363212
#340362802
#340362722
#340362632
#340362622
#340362392
#340362012
D42×C2932
#3264293