Finite Group Databasebeta

← #49170273

Subgroups — #49170273

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×#22627651C211yesmaximal, characteristic, complemented
2C22×#17989789C211yesmaximal, characteristic, complemented
3#34036790C211yesmaximal, characteristic, complemented
4#34036763C211yesmaximal, characteristic, complemented
5#34036515C211yesmaximal, characteristic, complemented
6#34036378C211yesmaximal, characteristic, complemented
7#34036363C211yesmaximal, characteristic, complemented
8#34036347C211yesmaximal, characteristic, complemented
9#34036327C211yesmaximal, characteristic, complemented
10#34036278C211yesmaximal, characteristic, complemented
11#34036277C211yesmaximal, characteristic, complemented
12#34036265C211yesmaximal, characteristic, complemented
13#34036232C211yesmaximal, characteristic, complemented
14#34036197C211yesmaximal, characteristic, complemented
15#448×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic, complemented
17C22×D1172C2211yescharacteristic
18C2×D4×D293—21no—
19C2×D4×D293C2211yescharacteristic
20#22627651—24no—
21C2×#17989789—24no—
22C2×#17989789C2211yescharacteristic
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic, complemented
25C2×#17989789C2211yescharacteristic, complemented
26C2×D2344C2211yescharacteristic
27C2×D2344C2211yescharacteristic, complemented
28C2×#17989787C2211yescharacteristic, complemented
29C2×#17989785C2211yescharacteristic, complemented
30C22×#14556873C2211yescharacteristic, complemented
31#22627620C2211yescharacteristic
32#22627613—21no—
33#22627613C2211yescharacteristic, complemented
34#22627600—21no—
35#22627600C2211yescharacteristic, complemented
36#22627599C2211yescharacteristic
37#22627598—21no—
38#22627598C2211yescharacteristic, complemented
39#22627597—21no—
40#22627594—21no—
41#22627592—21no—
42Q8×Dic293C2211yescharacteristic, complemented
43#22627586C2211yescharacteristic
44C4×C22⋊D293—21no—
45C4×C22⋊D293C2211yescharacteristic
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic
49#22627546C2211yescharacteristic, complemented
50#22627545C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C22×Q8×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic, complemented
56C2×SD16×C293C2211yescharacteristic, complemented
57#121×C293C2211yescharacteristic
58#103×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60C22⋊C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989803—22no—
64#17989803—41no—
65C2×D1172—22no—
66C2×D1172—21no—
67C2×D1172—21no—
68C2×D1172—21no—
69C2×D1172—41no—
70C2×D1172C2311yescharacteristic
71C2×D1172C2311yescharacteristic
72C2×D1172C2311yescharacteristic
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293D411yescharacteristic
78D4×D293—22no—
79D4×D293—41no—
80D4×D293—44no—
81C2×C4×D293—21no—
82C2×C4×D293—21no—
83C2×C4×D293C2311yescharacteristic
84#17989789—22no—
85#17989789—22no—
86#17989789—22no—
87#17989789D414yescomplemented
88D2344—22no—
89D2344—41no—
90#17989787—22no—
91#17989785—22no—
92C2×#14556873—22no—
93C2×#14556873C2311yescharacteristic
94C2×#14556873C2311yescharacteristic
95C4⋊Dic293—21no—
96C4⋊Dic293C2311yescharacteristic
97#17989781—21no—
98#17989781D411yescharacteristic
99C22⋊Dic293D411yescharacteristic
100#17989779—21no—
101#17989779—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107C4×Dic293—21no—
108C4×Dic293C2311yescharacteristic
109C2×Q8×C293—21no—
110C2×Q8×C293—22no—
111C2×Q8×C293—41no—
112C2×Q8×C293C2311yescharacteristic
113C2×Q8×C293C2311yescharacteristic
114C2×D4×C293—21no—
115C2×D4×C293C2311yescharacteristic
116C22×C4×C293—21no—
117C22×C4×C293C2311yescharacteristic
118SD16×C293—22no—
119SD16×C293—41no—
120C2×C8×C293C2311yescharacteristic
121C2×C8×C293C2311yescharacteristic
122C4⋊C4×C293C2311yescharacteristic
123C22⋊C4×C293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—21no—
127C22×D293—42no—
128C22×D293—42no—
129C22×D293—42no—
130C22×D293—42no—
131C22×D293—81no—
132C22×D293—81no—
133C22×D293C2×D411yescharacteristic
134D1172—21no—
135D1172—21no—
136D1172—21no—
137D1172—21no—
138D1172—21no—
139D1172—42no—
140D1172—42no—
141D1172—42no—
142D1172C2×D412yes—
143C22⋊D293—22no—
144C22⋊D293—42no—
145C22⋊D293—42no—
146C22⋊D293—42no—
147C22⋊D293—81no—
148C2×Dic293—21no—
149C2×Dic293C2×D411yescharacteristic
150C2×Dic293C2×D411yescharacteristic
151C2×Dic293C2×D411yescharacteristic
152C4×D293—21no—
153C4×D293—21no—
154C4×D293—42no—
155C4×D293—42no—
156#14556873—21no—
157#14556873C2×D412yescomplemented
158C23×C293—21no—
159C23×C293C2×D411yescharacteristic
160Q8×C293—21no—
161Q8×C293—21no—
162Q8×C293—21no—
163Q8×C293—42no—
164Q8×C293C2×D412yescomplemented
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—42no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—21no—
171C2×C4×C293—21no—
172C2×C4×C293—21no—
173C2×C4×C293—42no—
174C2×C4×C293C2×D411yescharacteristic
175C2×C4×C293C2411yescharacteristic
176C8×C293—21no—
177C8×C293—21no—
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293AGL(1,ℤ/8)12yescomplemented
184C22×C293C22×D411yescharacteristic
185C2×D293—21no—
186C2×D293—22no—
187C2×D293—21no—
188C2×D293—21no—
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—22no—
201C4×C293—41no—
202C4×C293—41no—
203C4×C293C22×D411yescharacteristic
204C4×C293H(5,𝔽2)11yescharacteristic
205Dic293—21no—
206Dic293—21no—
207Dic293—21no—
208Dic293—41no—
209C2×C293—22no—
210C2×C293—81no—
211C2×C293D4211yescharacteristic
212C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C293#57711yescharacteristic
214D293—41no—
215D293—42no—
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#448D29311yescharacteristic, complemented
232#447—2931no—
233#440—2931no—
234#435—2931no—
235C2×#41—2931no—
236C22×Q8C2×D29311yescharacteristic, complemented
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—5864no—
246C2×SD16C2×D29311yescharacteristic, complemented
247C2×SD16C2×D29311yescharacteristic, complemented
248C2×D8—2931no—
249C2×D8—2931no—
250C2×D8—2931no—
251C2×M16—2931no—
252C22×C8—2931no—
253#127—2931no—
254#124—2931no—
255#124—5861no—
256C22⋊Q8—2931no—
257C22⋊Q8—5861no—
258#121—2931no—
259#121—2931no—
260#121—5861no—
261#121—5861no—
262#121C2×D29311yescharacteristic
263#120—5861no—
264#120—5861no—
265C4×Q8—2931no—
266C4×D4—2931no—
267C4×D4—2931no—
268C4×D4—5861no—
269#107—2931no—
270C4⋊C8—2931no—
271#103C2×D29311yescharacteristic, complemented
272#103C2×D29311yescharacteristic, complemented
273#102—2931no—
274#102—2931no—
275#102—2931no—
276#102—2931no—
277C22⋊C8—2931no—
278C22⋊C8C2×D29311yescharacteristic, complemented
279#97—2931no—
280C24—5861no—
281C24—5861no—
282#41—5862no—
283#41—11721no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D29311yescharacteristic
288C2×Q8C22×D29311yescharacteristic
289C2×D4—21no—
290C2×D4—2931no—
291C2×D4—2931no—
292C2×D4—2931no—
293C2×D4—2931no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5861no—
300C2×D4—5862no—
301C2×D4—5861no—
302C2×D4—5861no—
303C2×D4—11721no—
304C2×D4—11721no—
305C2×D4—11724no—
306C2×D4C22×D29311yescharacteristic
307C22×C4—21no—
308C22×C4—2931no—
309C22×C4—5861no—
310C22×C4—5861no—
311C22×C4C22×D29311yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—2934no—
315SD16—5862no—
316SD16—5862no—
317SD16—5862no—
318D8—5862no—
319D8—5862no—
320D8—11721no—
321M16—5862no—
322C2×C8—2931no—
323C2×C8—2931no—
324C2×C8—5862no—
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—2931no—
328C4⋊C4—2931no—
329C4⋊C4—5861no—
330C4⋊C4—5861no—
331C4⋊C4C22×D29311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2931no—
334C22⋊C4—2931no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C22⋊C4—5861no—
339C22⋊C4—5861no—
340C22⋊C4—5861no—
341C42—2931no—
342C42—5861no—
343C23—21no—
344C23—2931no—
345C23—23441no—
346C23—23441no—
347C23—5861no—
348C23—5861no—
349C23—5861no—
350C23—11722no—
351C23—11722no—
352C23—11722no—
353C23—11722no—
354C23D4×D29311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D29312yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—2932no—
364D4—23441no—
365D4—5862no—
366D4—5861no—
367D4—5861no—
368D4—5861no—
369D4—5861no—
370D4—5861no—
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—21no—
382C2×C4—42no—
383C2×C4—2931no—
384C2×C4—2931no—
385C2×C4—2931no—
386C2×C4—5861no—
387C2×C4—5861no—
388C2×C4—5861no—
389C2×C4—11722no—
390C2×C4—11722no—
391C2×C4D4×D29311yescharacteristic
392C2×C4C23×D29311yescharacteristic
393C8—21no—
394C8—21no—
395C8—2932no—
396C8—5861no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—23444no—
403C22—23444no—
404C22—5861no—
405C22—5862no—
406C22—5861no—
407C22—5861no—
408C22—11721no—
409C22—11721no—
410C22—11721no—
411C22—11721no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22#2262765112yescomplemented
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—5861no—
423C4—5861no—
424C4—5861no—
425C4—11721no—
426C4C2×D4×D29311yescharacteristic
427C4#2262766911yescharacteristic
428C2—22no—
429C2—81no—
430C2—23441no—
431C2—23441no—
432C2—11721no—
433C2—11722no—
434C2#3403676111yescharacteristic
435C2C2×#2262765111yescharacteristic
436C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#448×C293C2
socleC22×C293C22×D4
hypercenter#448D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C22×#17989789C2semidirect
3. #34036790C2semidirect
4. #34036763C2semidirect
5. #34036515C2semidirect
6. #34036378C2semidirect
7. #34036363C2semidirect
8. #34036347C2semidirect
9. #34036327C2semidirect
10. #34036278C2semidirect
11. #34036277C2semidirect
12. #34036265C2semidirect
13. #34036232C2semidirect
14. #34036197C2semidirect
15. #448×C293C2semidirect
16. C2×#17989803C22semidirect
24. C2×#17989789C22semidirect
25. C2×#17989789C22semidirect
27. C2×D2344C22semidirect
28. C2×#17989787C22semidirect
29. C2×#17989785C22semidirect
30. C22×#14556873C22semidirect
33. #22627613C22semidirect
35. #22627600C22semidirect
38. #22627598C22semidirect
42. Q8×Dic293C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
49. #22627546C22semidirect
50. #22627545C22semidirect
51. #22627544C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
54. C22×Q8×C293C22semidirect
55. C2×SD16×C293C22semidirect
56. C2×SD16×C293C22semidirect
58. #103×C293C22semidirect
59. #103×C293C22semidirect
60. C22⋊C8×C293C22semidirect
87. #17989789D4semidirect
157. #14556873C2×D4semidirect
164. Q8×C293C2×D4semidirect
183. C22×C293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
231. #448D293semidirect
236. C22×Q8C2×D293semidirect
246. C2×SD16C2×D293semidirect
247. C2×SD16C2×D293semidirect
271. #103C2×D293semidirect
272. #103C2×D293semidirect
278. C22⋊C8C2×D293semidirect
359. Q8D4×D293semidirect
416. C22#22627651semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C22×#179897892
#340367902
#340367632
#340365152
#340363782
#340363632
#340363472
#340363272
#340362782
#340362772
#340362652
#340362322
#340361972
#448×C2932
#3265293