Finite Group Databasebeta

← #49169258

Subgroups — #49169258

Order 37504.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627653C211yesmaximal, characteristic, complemented
2C2×#22627648C211yesmaximal, characteristic, complemented
3C2×#22627643C211yesmaximal, characteristic, complemented
4C2×SD16×D293C211yesmaximal, characteristic, complemented
5#34036699C211yesmaximal, characteristic, complemented
6#34036692C211yesmaximal, characteristic, complemented
7#34036523C211yesmaximal, characteristic, complemented
8#34036520C211yesmaximal, characteristic, complemented
9#34036517C211yesmaximal, characteristic, complemented
10#34036512C211yesmaximal, characteristic, complemented
11#34036511C211yesmaximal, characteristic, complemented
12#34036501C211yesmaximal, characteristic, complemented
13#34036500C211yesmaximal, characteristic, complemented
14#34036086C211yesmaximal, characteristic, complemented
15#496×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627653—24no—
21#22627648—24no—
22#22627643—24no—
23C2×Dic1172C2211yescharacteristic, complemented
24C2×Dic1172C2211yescharacteristic, complemented
25C2×#17989792C2211yescharacteristic, complemented
26C2×#17989791C2211yescharacteristic
27C2×#17989791C2211yescharacteristic
28C2×#17989790C2211yescharacteristic
29C2×#17989790C2211yescharacteristic
30C2×#17989790C2211yescharacteristic, complemented
31C2×#17989790C2211yescharacteristic, complemented
32C2×#17989790C2211yescharacteristic, complemented
33C2×#17989789C2211yescharacteristic
34SD16×D293—24no—
35C2×#17989787C2211yescharacteristic
36C2×#17989785C2211yescharacteristic
37C2×C8×D293C2211yescharacteristic
38#22627624C2211yescharacteristic, complemented
39#22627622C2211yescharacteristic, complemented
40#22627620C2211yescharacteristic
41#22627612C2211yescharacteristic
42#22627611C2211yescharacteristic
43#22627611C2211yescharacteristic, complemented
44#22627606—22no—
45#22627601—22no—
46#22627596—22no—
47#22627592—22no—
48#22627581C2211yescharacteristic
49#22627534C2211yescharacteristic, complemented
50C4×#14556873C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic
53C2×SD16×C293C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic
55C2×D8×C293C2211yescharacteristic, complemented
56#127×C293C2211yescharacteristic
57#124×C293C2211yescharacteristic, complemented
58#97×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989801—22no—
61#17989801—22no—
62#17989801—22no—
63#17989801—22no—
64C2×Dic586—21no—
65C2×Dic586C2311yescharacteristic
66C2×Dic586C2311yescharacteristic
67C2×Dic586C2311yescharacteristic
68Q8×D293—22no—
69Q8×D293—22no—
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—22no—
75D4×D293—22no—
76D4×D293—22no—
77C22×Dic293—21no—
78C22×Dic293—21no—
79C2×C4×D293—21no—
80C2×C4×D293C2311yescharacteristic
81Dic1172—22no—
82Dic1172—22no—
83#17989792—22no—
84#17989791—22no—
85#17989791—22no—
86#17989790—22no—
87#17989790—22no—
88#17989790—22no—
89#17989790—22no—
90#17989790—22no—
91#17989789—22no—
92#17989787—22no—
93#17989785D414yescomplemented
94C2×#14556873C2311yescharacteristic
95C2×#14556873C2311yescharacteristic
96C8×D293D414yescomplemented
97C4⋊Dic293—22no—
98#17989781—22no—
99#17989781D412yes—
100C22⋊Dic293—22no—
101C22⋊Dic293—22no—
102#17989779—22no—
103#17989779D412yes—
104C4×Dic293C2311yescharacteristic
105C2×Q8×C293C2311yescharacteristic
106C2×D4×C293—22no—
107C2×D4×C293C2311yescharacteristic
108C2×D4×C293C2311yescharacteristic
109C2×D4×C293C2311yescharacteristic
110SD16×C293—22no—
111SD16×C293—22no—
112D8×C293—22no—
113D8×C293—22no—
114C2×C8×C293C2311yescharacteristic
115C2×C8×C293C2311yescharacteristic
116C22⋊C4×C293—22no—
117C42×C293C2311yescharacteristic
118C22×D293—21no—
119C22×D293—42no—
120C22×D293—82no—
121C22×D293C2×D411yescharacteristic
122Dic586—21no—
123Dic586—21no—
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—21no—
128Dic586—42no—
129D1172—21no—
130D1172—21no—
131C22⋊D293—42no—
132C22⋊D293—42no—
133C22⋊D293—42no—
134C22⋊D293—82no—
135C2×Dic293—21no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—42no—
139C2×Dic293—42no—
140C2×Dic293C2×D412yes—
141C2×Dic293C2×D411yescharacteristic
142C4×D293—42no—
143C4×D293C2×D412yes—
144C4×D293C2×D412yes—
145#14556873C2×D412yes—
146#14556873C2×D412yes—
147C23×C293—21no—
148C23×C293—21no—
149C23×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152D4×C293—21no—
153D4×C293—21no—
154D4×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—21no—
158D4×C293—82no—
159C2×C4×C293—21no—
160C2×C4×C293C2×D412yes—
161C2×C4×C293C2411yescharacteristic
162C8×C293C2×D412yes—
163C8×C293C2×D412yes—
164C22×C293—41no—
165C22×C293—41no—
166C22×C293—41no—
167C22×C293—42no—
168C22×C293—42no—
169C22×C293—42no—
170C22×C293C22×D411yescharacteristic
171C2×D293—22no—
172C2×D293—41no—
173C2×D293—42no—
174C2×D293—84no—
175C2×D293#12712yes—
176C4×C293—22no—
177C4×C293—41no—
178C4×C293C22×D411yescharacteristic
179C4×C293C22×D411yescharacteristic
180Dic293—22no—
181Dic293—41no—
182Dic293—41no—
183Dic293—41no—
184Dic293#12712yes—
185C2×C293—81no—
186C2×C293—81no—
187C2×C293—81no—
188C2×C293C2×#12711yescharacteristic
189C2×C293#57712yes—
190D293—22no—
191D293—81no—
192C293#315411yescharacteristic, complemented, Sylow
193#3154—2931nomaximal, Sylow
194C2×#137—2931no—
195C2×AGL(1,ℤ/8)—2931no—
196C2×#135—2931no—
197C22×SD16—2931no—
198#536—2931no—
199#535—2931no—
200#497—2931no—
201#497—2931no—
202#496—2931no—
203#496D29311yescharacteristic, complemented
204#495—2931no—
205#495—2931no—
206#492—2931no—
207#492—2931no—
208#405—2931no—
209C2×#41—2931no—
210C2×#41—2931no—
211C22×Q8—2931no—
212C22×D4—2931no—
213#137—5864no—
214AGL(1,ℤ/8)—5864no—
215#135—5864no—
216C2×Q16—2931no—
217C2×Q16—2931no—
218C2×Q16—2931no—
219C2×SD16—2931no—
220C2×SD16—2931no—
221C2×SD16—2931no—
222C2×SD16—2931no—
223C2×SD16—2931no—
224C2×SD16—2931no—
225C2×SD16—2931no—
226C2×SD16—2931no—
227C2×SD16—5864no—
228C2×SD16C2×D29311yescharacteristic
229C2×SD16C2×D29311yescharacteristic, complemented
230C2×D8—2931no—
231C2×D8C2×D29311yescharacteristic
232C2×D8C2×D29311yescharacteristic, complemented
233C2×M16—2931no—
234C22×C8—2931no—
235C4⋊Q8—2931no—
236C4⋊Q8—2931no—
237#127—2931no—
238#127C2×D29311yescharacteristic
239#124—2931no—
240#124—2931no—
241#124—2931no—
242#124C2×D29311yescharacteristic, complemented
243#123—5862no—
244C22⋊Q8—5862no—
245#121—5862no—
246#120—5862no—
247#117—2931no—
248#97—2931no—
249#97C2×D29311yescharacteristic, complemented
250C4×C8—2931no—
251C4×C8—2931no—
252C24—5861no—
253#41—5862no—
254#41—5862no—
255#41—5862no—
256#41—5862no—
257C2×Q8—2931no—
258C2×Q8—2931no—
259C2×Q8—2931no—
260C2×Q8—5861no—
261C2×Q8—5862no—
262C2×Q8—5862no—
263C2×Q8C22×D29311yescharacteristic
264C2×D4—22no—
265C2×D4—2931no—
266C2×D4—5862no—
267C2×D4—5862no—
268C2×D4—5862no—
269C2×D4—5861no—
270C2×D4—5861no—
271C2×D4—5861no—
272C2×D4C22×D29311yescharacteristic
273C2×D4C22×D29311yescharacteristic
274C2×D4C22×D29311yescharacteristic
275C22×C4—2931no—
276C22×C4—5861no—
277C22×C4—5861no—
278C22×C4—5861no—
279Q16—5862no—
280Q16—5862no—
281Q16—5862no—
282SD16—22no—
283SD16—22no—
284SD16—5862no—
285SD16—5862no—
286SD16—5862no—
287SD16—5862no—
288SD16—5862no—
289SD16—5862no—
290SD16—5862no—
291SD16—5862no—
292D8—22no—
293D8—22no—
294D8—5862no—
295M16—2934no—
296C2×C8—2931no—
297C2×C8—2931no—
298C2×C8—2934no—
299C2×C8C22×D29311yescharacteristic
300C2×C8C22×D29311yescharacteristic
301C4⋊C4—2932no—
302C4⋊C4—5862no—
303C4⋊C4—5862no—
304C22⋊C4—22no—
305C22⋊C4—2932no—
306C22⋊C4—5862no—
307C22⋊C4—5862no—
308C22⋊C4—5862no—
309C42—2931no—
310C42C22×D29311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—2931no—
315C23—23442no—
316C23—5861no—
317C23—11722no—
318Q8—21no—
319Q8—21no—
320Q8—5861no—
321Q8—5861no—
322Q8—5861no—
323Q8—5861no—
324Q8—5861no—
325Q8—5861no—
326Q8—11722no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—82no—
334D4—23442no—
335D4—5861no—
336D4—5861no—
337D4—11722no—
338D4—11722no—
339D4—11722no—
340C2×C4—21no—
341C2×C4—2931no—
342C2×C4—2932no—
343C2×C4—2932no—
344C2×C4—2932no—
345C2×C4—5861no—
346C2×C4—5861no—
347C2×C4—5861no—
348C2×C4—11722no—
349C2×C4—11722no—
350C2×C4—11722no—
351C2×C4D4×D29312yes—
352C2×C4C23×D29311yescharacteristic
353C8—2932no—
354C8—2932no—
355C8D4×D29312yes—
356C8D4×D29312yes—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—2932no—
364C22—23444no—
365C22—5862no—
366C22—11721no—
367C22—11722no—
368C22C2×D4×D29311yescharacteristic
369C4—22no—
370C4—41no—
371C4—2932no—
372C4—5862no—
373C4—11721no—
374C4—11721no—
375C4—11721no—
376C4C2×D4×D29311yescharacteristic
377C4C2×D4×D29311yescharacteristic
378C2—81no—
379C2—81no—
380C2—81no—
381C2—23441no—
382C2—5862no—
383C2#127×D29311yescharacteristic
384C2#3403693912yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22627653C2semidirect
2. C2×#22627648C2semidirect
3. C2×#22627643C2semidirect
4. C2×SD16×D293C2semidirect
5. #34036699C2semidirect
6. #34036692C2semidirect
7. #34036523C2semidirect
8. #34036520C2semidirect
9. #34036517C2semidirect
10. #34036512C2semidirect
11. #34036511C2semidirect
12. #34036501C2semidirect
13. #34036500C2semidirect
14. #34036086C2semidirect
15. #496×C293C2semidirect
23. C2×Dic1172C22semidirect
24. C2×Dic1172C22semidirect
25. C2×#17989792C22semidirect
30. C2×#17989790C22semidirect
31. C2×#17989790C22semidirect
32. C2×#17989790C22semidirect
38. #22627624C22semidirect
39. #22627622C22semidirect
43. #22627611C22semidirect
49. #22627534C22semidirect
50. C4×#14556873C22semidirect
51. C8×Dic293C22semidirect
53. C2×SD16×C293C22semidirect
55. C2×D8×C293C22semidirect
57. #124×C293C22semidirect
58. #97×C293C22semidirect
93. #17989785D4semidirect
96. C8×D293D4semidirect
192. C293#3154semidirect
203. #496D293semidirect
229. C2×SD16C2×D293semidirect
232. C2×D8C2×D293semidirect
242. #124C2×D293semidirect
249. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276532
C2×#226276482
C2×#226276432
C2×SD16×D2932
#340366992
#340366922
#340365232
#340365202
#340365172
#340365122
#340365112
#340365012
#340365002
#340360862
#496×C2932
#3154293