Finite Group Databasebeta

← #49170389

Subgroups — #49170389

Order 37504.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627649C211yesmaximal, characteristic, complemented
3#34036770C211yesmaximal, characteristic, complemented
4#34036767C211yesmaximal, characteristic, complemented
5#34036514C211yesmaximal, characteristic, complemented
6#34036378C211yesmaximal, characteristic, complemented
7#34036377C211yesmaximal, characteristic, complemented
8#34036347C211yesmaximal, characteristic, complemented
9#34036346C211yesmaximal, characteristic, complemented
10#34036290C211yesmaximal, characteristic, complemented
11#34036265C211yesmaximal, characteristic, complemented
12#34036264C211yesmaximal, characteristic, complemented
13#34036209C211yesmaximal, characteristic, complemented
14#34036204C211yesmaximal, characteristic, complemented
15#452×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C22×D1172C2211yescharacteristic, complemented
18C2×D4×D293C2211yescharacteristic, complemented
19#22627651—24no—
20#22627649—24no—
21C2×#17989789C2211yescharacteristic, complemented
22C2×#17989789C2211yescharacteristic, complemented
23C2×#17989789C2211yescharacteristic, complemented
24C2×D2344C2211yescharacteristic
25C2×D2344C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989786C2211yescharacteristic, complemented
29C2×#17989785C2211yescharacteristic, complemented
30#22627613C2211yescharacteristic, complemented
31#22627613C2211yescharacteristic, complemented
32#22627609—21no—
33#22627607—21no—
34#22627600C2211yescharacteristic, complemented
35#22627599—21no—
36#22627599C2211yescharacteristic
37#22627598C2211yescharacteristic
38#22627596—21no—
39#22627594—21no—
40#22627592—21no—
41D4×Dic293C2211yescharacteristic, complemented
42#22627586C2211yescharacteristic, complemented
43C4×C22⋊D293C2211yescharacteristic
44C2×#17989779—21no—
45C22⋊C4×D293—21no—
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627548C2211yescharacteristic, complemented
49#22627546C2211yescharacteristic, complemented
50#22627546C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53C8×Dic293C2211yescharacteristic, complemented
54C2×#41×C293C2211yescharacteristic
55C2×Q16×C293C2211yescharacteristic, complemented
56C2×SD16×C293C2211yescharacteristic, complemented
57C22⋊Q8×C293C2211yescharacteristic, complemented
58#103×C293C2211yescharacteristic, complemented
59#102×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⋊D293D411yescharacteristic
76D4×D293—22no—
77D4×D293—41no—
78C22×Dic293—21no—
79C2×C4×D293—21no—
80C2×C4×D293—21no—
81C2×C4×D293C2311yescharacteristic
82#17989789—22no—
83#17989789—22no—
84#17989789—22no—
85D2344—22no—
86D2344—41no—
87#17989787—22no—
88#17989787D414yescomplemented
89#17989786—22no—
90#17989785—22no—
91C2×#14556873C2311yescharacteristic
92C2×#14556873C2311yescharacteristic
93C4⋊Dic293C2311yescharacteristic
94#17989781—21no—
95#17989781D411yescharacteristic
96C22⋊Dic293—21no—
97C22⋊Dic293D411yescharacteristic
98#17989779—21no—
99#17989779—21no—
100#17989779—21no—
101#17989779—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107C4×Dic293C2311yescharacteristic
108#41×C293—22no—
109#41×C293—41no—
110C2×Q8×C293C2311yescharacteristic
111C2×Q8×C293C2311yescharacteristic
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C22×C4×C293—21no—
115C22×C4×C293C2311yescharacteristic
116Q16×C293—41no—
117SD16×C293—22no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293—21no—
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—42no—
127C22×D293—42no—
128C22×D293—42no—
129C22×D293—42no—
130C22×D293—81no—
131C22×D293—81no—
132C22×D293C2×D411yescharacteristic
133D1172—21no—
134D1172—21no—
135D1172—21no—
136D1172—21no—
137D1172—21no—
138D1172—42no—
139D1172—42no—
140D1172—81no—
141D1172C2×D412yes—
142C22⋊D293—41no—
143C22⋊D293—42no—
144C22⋊D293—81no—
145C2×Dic293—21no—
146C2×Dic293—81no—
147C2×Dic293C2×D411yescharacteristic
148C2×Dic293C2×D411yescharacteristic
149C2×Dic293C2×D411yescharacteristic
150C4×D293—21no—
151C4×D293—21no—
152C4×D293—42no—
153C4×D293—81no—
154#14556873—21no—
155#14556873C2×D412yes—
156C23×C293—21no—
157C23×C293C2×D411yescharacteristic
158Q8×C293—21no—
159Q8×C293—21no—
160Q8×C293—21no—
161Q8×C293—21no—
162D4×C293—21no—
163D4×C293—42no—
164D4×C293C2×D412yes—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—42no—
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293—21no—
175C22×C293—21no—
176C22×C293—22no—
177C22×C293—21no—
178C22×C293—21no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293C22×D411yescharacteristic
182C2×D293—21no—
183C2×D293—22no—
184C2×D293—21no—
185C2×D293—21no—
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—84no—
195C2×D293—84no—
196C4×C293—21no—
197C4×C293—41no—
198C4×C293—41no—
199C4×C293—41no—
200C4×C293C22×D411yescharacteristic
201C4×C293H(5,𝔽2)11yescharacteristic
202Dic293—21no—
203Dic293—21no—
204Dic293—21no—
205Dic293—41no—
206C2×C293—41no—
207C2×C293—42no—
208C2×C293D4211yescharacteristic
209C2×C293#57511yescharacteristic
210C2×C293#57611yescharacteristic
211D293—41no—
212D293—42no—
213D293—81no—
214D293—81no—
215C293#327311yescharacteristic, complemented, Sylow
216#3273—2931nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2931no—
218C2×AGL(1,ℤ/8)—2931no—
219#546—2931no—
220#546—2931no—
221#495—2931no—
222#468—2931no—
223#468—2931no—
224#463—2931no—
225#463—2931no—
226#452—2931no—
227#452D29311yescharacteristic, complemented
228#447—2931no—
229#447—2931no—
230C4×D8—2931no—
231#436—2931no—
232C2×#41—2931no—
233C2×#41C2×D29311yescharacteristic
234C22×D4—2931no—
235C22×D4—2931no—
236AGL(1,ℤ/8)—5864no—
237AGL(1,ℤ/8)—5864no—
238C2×Q16C2×D29311yescharacteristic, complemented
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16—2931no—
242C2×SD16C2×D29311yescharacteristic, complemented
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8—2931no—
246C2×D8—2931no—
247C2×M16—2931no—
248C2×M16—2931no—
249#124—2931no—
250#124—2931no—
251#123—5861no—
252#123—5861no—
253C22⋊Q8—2931no—
254C22⋊Q8C2×D29311yescharacteristic, complemented
255#121—2931no—
256#121—2931no—
257#121—5861no—
258#121—5861no—
259#120—5861no—
260#120—5861no—
261C4×D4—2931no—
262C4×D4—2931no—
263C4×D4—2931no—
264C2×C22⋊C4—5861no—
265C2×C22⋊C4—5861no—
266#107—2931no—
267C4⋊C8—2931no—
268#103—2931no—
269#103C2×D29311yescharacteristic, complemented
270#102—2931no—
271#102—2931no—
272#102—2931no—
273#102C2×D29311yescharacteristic, complemented
274C22⋊C8—2931no—
275C22⋊C8C2×D29311yescharacteristic, complemented
276C4×C8—2931no—
277C24—5861no—
278C24—5861no—
279#41—22no—
280#41—41no—
281#41—5862no—
282#41—11721no—
283C2×Q8C22×D29311yescharacteristic
284C2×Q8C22×D29311yescharacteristic
285C2×D4—21no—
286C2×D4—2931no—
287C2×D4—2931no—
288C2×D4—2931no—
289C2×D4—2931no—
290C2×D4—5862no—
291C2×D4—5861no—
292C2×D4—5861no—
293C2×D4—5861no—
294C2×D4—5861no—
295C2×D4—5862no—
296C2×D4—5861no—
297C2×D4—11721no—
298C2×D4—11721no—
299C2×D4C22×D29311yescharacteristic
300C22×C4—21no—
301C22×C4—2931no—
302C22×C4—5861no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4C22×D29311yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—5862no—
309SD16—5862no—
310SD16—5862no—
311D8—2934no—
312D8—5862no—
313D8—5862no—
314D8—11721no—
315M16—5862no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8C22×D29311yescharacteristic
320C2×C8C22×D29311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—2931no—
323C4⋊C4—2931no—
324C4⋊C4—5861no—
325C4⋊C4C22×D29311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—2931no—
328C22⋊C4—2931no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C42—2931no—
339C23—21no—
340C23—2931no—
341C23—23441no—
342C23—23441no—
343C23—5861no—
344C23—5861no—
345C23—5861no—
346C23—11722no—
347C23—11722no—
348C23—11722no—
349C23—11722no—
350C23D4×D29311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—2932no—
358D4—23441no—
359D4—23441no—
360D4—5861no—
361D4—5861no—
362D4—5861no—
363D4—5861no—
364D4—5861no—
365D4—11722no—
366D4—11722no—
367D4—11721no—
368D4—11722no—
369D4D4×D29312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—2931no—
377C2×C4—2931no—
378C2×C4—2931no—
379C2×C4—23441no—
380C2×C4—23441no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—11722no—
385C2×C4D4×D29311yescharacteristic
386C2×C4C23×D29311yescharacteristic
387C8—21no—
388C8—21no—
389C8—2932no—
390C8—5861no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—23444no—
398C22—23444no—
399C22—5861no—
400C22—5862no—
401C22—5861no—
402C22—5861no—
403C22—11721no—
404C22—11721no—
405C22—11721no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22—11721no—
410C22—11721no—
411C22C2×D4×D29311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—5861no—
417C4—5861no—
418C4—5861no—
419C4—11721no—
420C4C2×D4×D29311yescharacteristic
421C4#2262766911yescharacteristic
422C2—41no—
423C2—42no—
424C2—23441no—
425C2—23441no—
426C2—11721no—
427C2—11722no—
428C2#3403676111yescharacteristic
429C2#3403692811yescharacteristic
430C2#3403693111yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#452×C293C2
socleC22×C293C22×D4
hypercenter#452D293
hypocenterC293#3273
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627649C2semidirect
3. #34036770C2semidirect
4. #34036767C2semidirect
5. #34036514C2semidirect
6. #34036378C2semidirect
7. #34036377C2semidirect
8. #34036347C2semidirect
9. #34036346C2semidirect
10. #34036290C2semidirect
11. #34036265C2semidirect
12. #34036264C2semidirect
13. #34036209C2semidirect
14. #34036204C2semidirect
15. #452×C293C2semidirect
17. C22×D1172C22semidirect
18. C2×D4×D293C22semidirect
21. C2×#17989789C22semidirect
22. C2×#17989789C22semidirect
23. C2×#17989789C22semidirect
28. C2×#17989786C22semidirect
29. C2×#17989785C22semidirect
30. #22627613C22semidirect
31. #22627613C22semidirect
34. #22627600C22semidirect
41. D4×Dic293C22semidirect
42. #22627586C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
48. #22627548C22semidirect
49. #22627546C22semidirect
50. #22627546C22semidirect
51. #22627544C22semidirect
52. #22627536C22semidirect
53. C8×Dic293C22semidirect
55. C2×Q16×C293C22semidirect
56. C2×SD16×C293C22semidirect
57. C22⋊Q8×C293C22semidirect
58. #103×C293C22semidirect
59. #102×C293C22semidirect
60. C22⋊C8×C293C22semidirect
88. #17989787D4semidirect
215. C293#3273semidirect
227. #452D293semidirect
238. C2×Q16C2×D293semidirect
242. C2×SD16C2×D293semidirect
254. C22⋊Q8C2×D293semidirect
269. #103C2×D293semidirect
273. #102C2×D293semidirect
275. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276492
#340367702
#340367672
#340365142
#340363782
#340363772
#340363472
#340363462
#340362902
#340362652
#340362642
#340362092
#340362042
#452×C2932
#3273293