Finite Group Databasebeta

← #49169254

Subgroups — #49169254

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×#22627651C211yesmaximal, characteristic, complemented
3C2×#22627645C211yesmaximal, characteristic, complemented
4C2×SD16×D293C211yesmaximal, characteristic, complemented
5#34036704C211yesmaximal, characteristic, complemented
6#34036692C211yesmaximal, characteristic, complemented
7#34036521C211yesmaximal, characteristic, complemented
8#34036520C211yesmaximal, characteristic, complemented
9#34036516C211yesmaximal, characteristic, complemented
10#34036515C211yesmaximal, characteristic, complemented
11#34036513C211yesmaximal, characteristic, complemented
12#34036510C211yesmaximal, characteristic, complemented
13#34036500C211yesmaximal, characteristic, complemented
14#34036083C211yesmaximal, characteristic, complemented
15#492×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627653—24no—
21#22627651—24no—
22#22627645—24no—
23C2×Dic1172C2211yescharacteristic, complemented
24C2×#17989792C2211yescharacteristic, complemented
25C2×#17989792C2211yescharacteristic, complemented
26C2×#17989791C2211yescharacteristic
27C2×#17989791C2211yescharacteristic, complemented
28C2×#17989790C2211yescharacteristic
29C2×#17989790C2211yescharacteristic, complemented
30C2×#17989789C2211yescharacteristic
31C2×#17989789C2211yescharacteristic
32SD16×D293—24no—
33C2×D2344C2211yescharacteristic
34C2×#17989787C2211yescharacteristic
35C2×#17989787C2211yescharacteristic, complemented
36C2×#17989785C2211yescharacteristic
37C2×C8×D293C2211yescharacteristic
38#22627622C2211yescharacteristic, complemented
39#22627620C2211yescharacteristic
40#22627613C2211yescharacteristic, complemented
41#22627612C2211yescharacteristic
42#22627612C2211yescharacteristic, complemented
43#22627611C2211yescharacteristic
44#22627609—22no—
45#22627602—22no—
46#22627596—22no—
47#22627592—22no—
48#22627581C2211yescharacteristic
49#22627535C2211yescharacteristic, complemented
50#22627534C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic
53C2×SD16×C293C2211yescharacteristic
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic, complemented
56C4⋊Q8×C293C2211yescharacteristic, complemented
57#127×C293C2211yescharacteristic
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—21no—
74C2×C22⋊D293—22no—
75D4×D293—22no—
76D4×D293—22no—
77C22×Dic293—21no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293C2311yescharacteristic
81Dic1172—22no—
82#17989792—22no—
83#17989792—22no—
84#17989791—22no—
85#17989791—22no—
86#17989790—22no—
87#17989790—22no—
88#17989789—22no—
89#17989789—22no—
90D2344—22no—
91#17989787—22no—
92#17989787—22no—
93#17989785D414yescomplemented
94C2×#14556873C2311yescharacteristic
95C2×#14556873C2311yescharacteristic
96C8×D293D414yescomplemented
97#17989781—22no—
98#17989781D412yes—
99C22⋊Dic293—22no—
100#17989779—22no—
101#17989779—22no—
102#17989779—22no—
103#17989779D412yes—
104C4×Dic293C2311yescharacteristic
105C2×Q8×C293C2311yescharacteristic
106C2×Q8×C293C2311yescharacteristic
107C2×D4×C293—22no—
108C2×D4×C293C2311yescharacteristic
109C2×D4×C293C2311yescharacteristic
110SD16×C293—22no—
111SD16×C293—22no—
112SD16×C293—22no—
113SD16×C293—22no—
114C2×C8×C293C2311yescharacteristic
115C2×C8×C293C2311yescharacteristic
116C4⋊C4×C293—22no—
117C42×C293C2311yescharacteristic
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—42no—
121C22×D293—82no—
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—42no—
134C22⋊D293—42no—
135C22⋊D293—82no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—42no—
139C2×Dic293C2×D412yes—
140C2×Dic293C2×D411yescharacteristic
141C4×D293—42no—
142C4×D293—42no—
143C4×D293C2×D412yes—
144C4×D293C2×D412yes—
145#14556873C2×D412yes—
146#14556873C2×D412yes—
147C23×C293—21no—
148C23×C293—21no—
149Q8×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152Q8×C293—21no—
153D4×C293—21no—
154D4×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—82no—
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#57712yes—
189D293—22no—
190D293—81no—
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#496—2931no—
204#495—2931no—
205#495—2931no—
206#492—2931no—
207#492D29311yescharacteristic, complemented
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—5864no—
226C2×SD16C2×D29311yescharacteristic
227C2×SD16C2×D29311yescharacteristic
228C2×SD16C2×D29311yescharacteristic, complemented
229C2×SD16C2×D29311yescharacteristic, complemented
230C2×D8—2931no—
231C2×D8—2931no—
232C2×D8—2931no—
233C2×M16—2931no—
234C22×C8—2931no—
235C4⋊Q8—2931no—
236C4⋊Q8C2×D29311yescharacteristic, complemented
237#127—2931no—
238#127C2×D29311yescharacteristic
239#124—2931no—
240#124—2931no—
241#124—2931no—
242#124—2931no—
243#123—5862no—
244C22⋊Q8—5862no—
245#121—5862no—
246#120—5862no—
247#117—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—22no—
265C2×D4—2931no—
266C2×D4—2931no—
267C2×D4—5861no—
268C2×D4—5862no—
269C2×D4—5862no—
270C2×D4—5862no—
271C2×D4—5861no—
272C2×D4—5861no—
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—22no—
285SD16—22no—
286SD16—5862no—
287SD16—5862no—
288SD16—5862no—
289SD16—5862no—
290SD16—5862no—
291SD16—5862no—
292D8—5862no—
293D8—5862no—
294D8—5862no—
295M16—2934no—
296C2×C8—2931no—
297C2×C8—2931no—
298C2×C8—2934no—
299C2×C8C22×D29311yescharacteristic
300C2×C8C22×D29311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—2932no—
303C4⋊C4—5862no—
304C22⋊C4—2932no—
305C22⋊C4—5862no—
306C22⋊C4—5862no—
307C22⋊C4—5862no—
308C22⋊C4—5862no—
309C42—2931no—
310C42C22×D29311yescharacteristic
311C23—21no—
312C23—21no—
313C23—2931no—
314C23—23442no—
315C23—5861no—
316C23—5861no—
317C23—11722no—
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—82no—
332D4—23442no—
333D4—5861no—
334D4—5861no—
335D4—5861no—
336D4—5861no—
337D4—11722no—
338D4—11722no—
339D4—11722no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2931no—
343C2×C4—2932no—
344C2×C4—2932no—
345C2×C4—2932no—
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—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#3403693512yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#492×C293C2
socleC22×C293C22×D4
hypercenter#492D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22627653C2semidirect
2. C2×#22627651C2semidirect
3. C2×#22627645C2semidirect
4. C2×SD16×D293C2semidirect
5. #34036704C2semidirect
6. #34036692C2semidirect
7. #34036521C2semidirect
8. #34036520C2semidirect
9. #34036516C2semidirect
10. #34036515C2semidirect
11. #34036513C2semidirect
12. #34036510C2semidirect
13. #34036500C2semidirect
14. #34036083C2semidirect
15. #492×C293C2semidirect
23. C2×Dic1172C22semidirect
24. C2×#17989792C22semidirect
25. C2×#17989792C22semidirect
27. C2×#17989791C22semidirect
29. C2×#17989790C22semidirect
35. C2×#17989787C22semidirect
38. #22627622C22semidirect
40. #22627613C22semidirect
42. #22627612C22semidirect
49. #22627535C22semidirect
50. #22627534C22semidirect
51. C8×Dic293C22semidirect
54. C2×SD16×C293C22semidirect
55. C2×SD16×C293C22semidirect
56. C4⋊Q8×C293C22semidirect
58. C4×C8×C293C22semidirect
93. #17989785D4semidirect
96. C8×D293D4semidirect
192. C293#3154semidirect
207. #492D293semidirect
228. C2×SD16C2×D293semidirect
229. C2×SD16C2×D293semidirect
236. C4⋊Q8C2×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×#226276532
C2×#226276512
C2×#226276452
C2×SD16×D2932
#340367042
#340366922
#340365212
#340365202
#340365162
#340365152
#340365132
#340365102
#340365002
#340360832
#492×C2932
#3154293