Finite Group Databasebeta

← #49169246

Subgroups — #49169246

Order 37504.

572 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×#22627656C211yesmaximal, characteristic, complemented
2C2×#22627653C211yesmaximal, characteristic, complemented
3C2×#22627651C211yesmaximal, characteristic, complemented
4C2×#22627648C211yesmaximal, characteristic, complemented
5#34036678C211yesmaximal, characteristic, complemented
6#124×D293C211yesmaximal, characteristic, complemented
7#34036522C211yesmaximal, characteristic, complemented
8#34036520C211yesmaximal, characteristic, complemented
9#34036517C211yesmaximal, characteristic, complemented
10#34036515C211yesmaximal, characteristic, complemented
11#34036513C211yesmaximal, characteristic, complemented
12#34036511C211yesmaximal, characteristic, complemented
13#34036511C211yesmaximal, characteristic, complemented
14C4×#17989785C211yesmaximal, characteristic, complemented
15#495×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627656—24no—
21#22627653—24no—
22#22627651—24no—
23#22627648—24no—
24C2×Dic1172C2211yescharacteristic, complemented
25C2×#17989792C2211yescharacteristic, complemented
26C2×#17989792C2211yescharacteristic, complemented
27C2×#17989791C2211yescharacteristic
28C2×#17989791C2211yescharacteristic, complemented
29C2×#17989790C2211yescharacteristic
30C2×#17989790C2211yescharacteristic, complemented
31C2×#17989789C2211yescharacteristic
32C2×#17989789C2211yescharacteristic, complemented
33C2×D2344C2211yescharacteristic
34C2×#17989787C2211yescharacteristic
35C2×#17989787C2211yescharacteristic
36C2×#17989785C2211yescharacteristic
37C2×#17989785C2211yescharacteristic
38#22627622C2211yescharacteristic, complemented
39#22627620C2211yescharacteristic, complemented
40#22627613C2211yescharacteristic
41#22627612C2211yescharacteristic
42#22627612C2211yescharacteristic, complemented
43#22627611C2211yescharacteristic
44#22627610—22no—
45#22627597—22no—
46#22627584—22no—
47C22⋊C4×D293—22no—
48C42×D293C2211yescharacteristic
49#22627534C2211yescharacteristic, complemented
50#22627534C2211yescharacteristic, complemented
51C4×#14556873C2211yescharacteristic, complemented
52C2×Q16×C293C2211yescharacteristic, complemented
53C2×SD16×C293C2211yescharacteristic
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×D8×C293C2211yescharacteristic
56#124×C293C2211yescharacteristic
57#124×C293C2211yescharacteristic, complemented
58C4×C8×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989803—22no—
61#17989803—22no—
62#17989801—22no—
63#17989801—22no—
64C2×Dic586—21no—
65C2×Dic586C2311yescharacteristic
66C2×Dic586C2311yescharacteristic
67Q8×D293—22no—
68Q8×D293—22no—
69C2×D1172—21no—
70C2×D1172C2311yescharacteristic
71C2×D1172C2311yescharacteristic
72C2×C22⋊D293—21no—
73C2×C22⋊D293—22no—
74C2×C22⋊D293—21no—
75D4×D293—22no—
76D4×D293—22no—
77C22×Dic293—21no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293D412yes—
81C2×C4×D293C2311yescharacteristic
82Dic1172—22no—
83#17989792—22no—
84#17989792—22no—
85#17989791—22no—
86#17989791—22no—
87#17989790—22no—
88#17989790—22no—
89#17989789—22no—
90#17989789—22no—
91D2344—22no—
92#17989787—22no—
93#17989787—22no—
94#17989785D414yescomplemented
95#17989785D414yescomplemented
96C2×#14556873C2311yescharacteristic
97C2×#14556873C2311yescharacteristic
98#17989781—22no—
99C22⋊Dic293—22no—
100#17989779—22no—
101#17989779—22no—
102#17989779—22no—
103C4×Dic293D412yes—
104C4×Dic293C2311yescharacteristic
105C2×Q8×C293C2311yescharacteristic
106C2×Q8×C293C2311yescharacteristic
107C2×D4×C293C2311yescharacteristic
108C2×D4×C293C2311yescharacteristic
109Q16×C293—22no—
110SD16×C293—22no—
111SD16×C293—22no—
112D8×C293—22no—
113C2×C8×C293C2311yescharacteristic
114C2×C8×C293C2311yescharacteristic
115C22⋊C4×C293—22no—
116C22⋊C4×C293—22no—
117C42×C293C2311yescharacteristic
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—44no—
121C22×D293—42no—
122C22×D293C2×D411yescharacteristic
123Dic586—21no—
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—42no—
128D1172—21no—
129D1172—21no—
130D1172—21no—
131D1172—21no—
132D1172—42no—
133C22⋊D293—44no—
134C22⋊D293—42no—
135C22⋊D293—42no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—42no—
139C2×Dic293C2×D412yes—
140C2×Dic293C2×D411yescharacteristic
141C4×D293—42no—
142C4×D293—42no—
143C4×D293—42no—
144C4×D293C2×D412yes—
145C4×D293C2×D412yes—
146#14556873C2×D412yes—
147#14556873C2×D412yes—
148C23×C293—21no—
149C23×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152Q8×C293—21no—
153Q8×C293—21no—
154D4×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—21no—
158C2×C4×C293—21no—
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—42no—
167C22×C293—42no—
168C22×C293C22×D411yescharacteristic
169C2×D293—22no—
170C2×D293—41no—
171C2×D293—41no—
172C2×D293—42no—
173C2×D293—42no—
174C2×D293—84no—
175C2×D293#12712yes—
176C4×C293—22no—
177C4×C293—41no—
178C4×C293—41no—
179C4×C293C22×D411yescharacteristic
180C4×C293C22×D411yescharacteristic
181Dic293—22no—
182Dic293—41no—
183Dic293—41no—
184Dic293#12712yes—
185C2×C293—81no—
186C2×C293—81no—
187C2×C293C2×#12711yescharacteristic
188C2×C293#57512yes—
189D293—22no—
190D293—81no—
191D293—81no—
192C293#315311yescharacteristic, complemented, Sylow
193#3153—2931nomaximal, Sylow
194C2×#137—2931no—
195C2×#137—2931no—
196C2×AGL(1,ℤ/8)—2931no—
197C2×AGL(1,ℤ/8)—2931no—
198#532—2931no—
199C2×#124—2931no—
200#497—2931no—
201#497—2931no—
202#496—2931no—
203#496—2931no—
204#495—2931no—
205#495—2931no—
206#495—2931no—
207#495D29311yescharacteristic, complemented
208C4×M16—2931no—
209C2×#41—2931no—
210C2×#41—2931no—
211C22×Q8—2931no—
212C22×D4—2931no—
213#137—5864no—
214#137—5864no—
215AGL(1,ℤ/8)—5864no—
216AGL(1,ℤ/8)—5864no—
217C2×Q16—2931no—
218C2×Q16—2931no—
219C2×Q16—2931no—
220C2×Q16C2×D29311yescharacteristic, complemented
221C2×SD16—2931no—
222C2×SD16—2931no—
223C2×SD16—2931no—
224C2×SD16—2931no—
225C2×SD16—2931no—
226C2×SD16—2931no—
227C2×SD16C2×D29311yescharacteristic
228C2×SD16C2×D29311yescharacteristic, complemented
229C2×D8—2931no—
230C2×D8—2931no—
231C2×D8—2931no—
232C2×D8C2×D29311yescharacteristic
233C2×M16—2931no—
234C2×M16—2931no—
235C4⋊Q8—2931no—
236#127—2931no—
237#124—2931no—
238#124—2931no—
239#124—2931no—
240#124—2931no—
241#124—5862no—
242#124C2×D29311yescharacteristic
243#124C2×D29311yescharacteristic, complemented
244#121—5862no—
245C4×D4—5862no—
246C2×C22⋊C4—5862no—
247C2×C42—2931no—
248#97—2931no—
249#97—2931no—
250C4×C8—2931no—
251C4×C8C2×D29311yescharacteristic, complemented
252C24—5861no—
253#41—5862no—
254#41—5862no—
255#41—5862no—
256#41—5862no—
257C2×Q8—2931no—
258C2×Q8—2931no—
259C2×Q8—5861no—
260C2×Q8—5862no—
261C2×Q8—5862no—
262C2×Q8C22×D29311yescharacteristic
263C2×Q8C22×D29311yescharacteristic
264C2×D4—2931no—
265C2×D4—2931no—
266C2×D4—5861no—
267C2×D4—5862no—
268C2×D4—5862no—
269C2×D4—5862no—
270C2×D4—5861no—
271C2×D4—5861no—
272C2×D4C22×D29311yescharacteristic
273C2×D4C22×D29311yescharacteristic
274C22×C4—2931no—
275C22×C4—2932no—
276C22×C4—5861no—
277C22×C4—5861no—
278C22×C4—5861no—
279Q16—22no—
280Q16—5862no—
281Q16—5862no—
282Q16—5862no—
283SD16—22no—
284SD16—22no—
285SD16—5862no—
286SD16—5862no—
287SD16—5862no—
288SD16—5862no—
289SD16—5862no—
290SD16—5862no—
291D8—22no—
292D8—5862no—
293D8—5862no—
294D8—5862no—
295M16—2934no—
296M16—2934no—
297C2×C8—2931no—
298C2×C8—2931no—
299C2×C8C22×D29311yescharacteristic
300C2×C8C22×D29311yescharacteristic
301C4⋊C4—5862no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—5862no—
305C22⋊C4—5862no—
306C22⋊C4—5862no—
307C22⋊C4—5862no—
308C42—2932no—
309C42—2931no—
310C42C22×D29311yescharacteristic
311C23—21no—
312C23—21no—
313C23—2931no—
314C23—5861no—
315C23—5861no—
316C23—11722no—
317C23—11724no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—5861no—
323Q8—5861no—
324Q8—5861no—
325Q8—5861no—
326Q8—11722no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—5861no—
332D4—5861no—
333D4—5861no—
334D4—5861no—
335D4—11722no—
336D4—11722no—
337D4—11724no—
338D4—11722no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—2931no—
342C2×C4—2932no—
343C2×C4—2932no—
344C2×C4—2932no—
345C2×C4—5861no—
346C2×C4—5861no—
347C2×C4—11722no—
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—42no—
360C22—42no—
361C22—2932no—
362C22—23444no—
363C22—5862no—
364C22—11721no—
365C22—11721no—
366C22—11722no—
367C22—11722no—
368C22C2×D4×D29311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—2932no—
373C4—5862no—
374C4—11721no—
375C4—11721no—
376C4C2×D4×D29311yescharacteristic
377C4C2×D4×D29311yescharacteristic
378C2—81no—
379C2—81no—
380C2—23441no—
381C2—23441no—
382C2—5862no—
383C2#127×D29311yescharacteristic
384C2#3403692612yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#495×C293C2
socleC22×C293C22×D4
hypercenter#495D293
hypocenterC293#3153
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22627656C2semidirect
2. C2×#22627653C2semidirect
3. C2×#22627651C2semidirect
4. C2×#22627648C2semidirect
5. #34036678C2semidirect
6. #124×D293C2semidirect
7. #34036522C2semidirect
8. #34036520C2semidirect
9. #34036517C2semidirect
10. #34036515C2semidirect
11. #34036513C2semidirect
12. #34036511C2semidirect
13. #34036511C2semidirect
14. C4×#17989785C2semidirect
15. #495×C293C2semidirect
24. C2×Dic1172C22semidirect
25. C2×#17989792C22semidirect
26. C2×#17989792C22semidirect
28. C2×#17989791C22semidirect
30. C2×#17989790C22semidirect
32. C2×#17989789C22semidirect
38. #22627622C22semidirect
39. #22627620C22semidirect
42. #22627612C22semidirect
49. #22627534C22semidirect
50. #22627534C22semidirect
51. C4×#14556873C22semidirect
52. C2×Q16×C293C22semidirect
54. C2×SD16×C293C22semidirect
57. #124×C293C22semidirect
58. C4×C8×C293C22semidirect
94. #17989785D4semidirect
95. #17989785D4semidirect
192. C293#3153semidirect
207. #495D293semidirect
220. C2×Q16C2×D293semidirect
228. C2×SD16C2×D293semidirect
243. #124C2×D293semidirect
251. C4×C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276562
C2×#226276532
C2×#226276512
C2×#226276482
#340366782
#124×D2932
#340365222
#340365202
#340365172
#340365152
#340365132
#340365112
C4×#179897852
#495×C2932
#3153293