Finite Group Databasebeta

← #49169260

Subgroups — #49169260

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×#22627656C211yesmaximal, characteristic, complemented
2C2×#22627651C211yesmaximal, characteristic, complemented
3C2×#22627647C211yesmaximal, characteristic, complemented
4C2×SD16×D293C211yesmaximal, characteristic, complemented
5#34036704C211yesmaximal, characteristic, complemented
6#34036697C211yesmaximal, characteristic, complemented
7#34036522C211yesmaximal, characteristic, complemented
8#34036518C211yesmaximal, characteristic, complemented
9#34036515C211yesmaximal, characteristic, complemented
10#34036514C211yesmaximal, characteristic, complemented
11#34036511C211yesmaximal, characteristic, complemented
12#34036503C211yesmaximal, characteristic, complemented
13#34036500C211yesmaximal, characteristic, complemented
14#34036086C211yesmaximal, characteristic, complemented
15#497×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989803C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627656—24no—
21#22627651—24no—
22#22627647—24no—
23C2×#17989792C2211yescharacteristic, complemented
24C2×#17989791C2211yescharacteristic
25C2×#17989791C2211yescharacteristic, complemented
26C2×#17989790C2211yescharacteristic
27C2×#17989789C2211yescharacteristic
28C2×#17989789C2211yescharacteristic
29C2×#17989789C2211yescharacteristic, complemented
30C2×#17989789C2211yescharacteristic, complemented
31C2×#17989789C2211yescharacteristic, complemented
32SD16×D293—24no—
33C2×D2344C2211yescharacteristic
34C2×D2344C2211yescharacteristic, complemented
35C2×#17989787C2211yescharacteristic
36C2×#17989785C2211yescharacteristic
37C2×C8×D293C2211yescharacteristic
38#22627622C2211yescharacteristic, complemented
39#22627621C2211yescharacteristic
40#22627620C2211yescharacteristic
41#22627613C2211yescharacteristic
42#22627613C2211yescharacteristic, complemented
43#22627612C2211yescharacteristic, complemented
44#22627609—22no—
45#22627602—22no—
46#22627597—22no—
47#22627594—22no—
48#22627581C2211yescharacteristic
49#22627534C2211yescharacteristic, complemented
50C4×#14556873C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C2×Q16×C293C2211yescharacteristic, complemented
53C2×Q16×C293C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56C4⋊Q8×C293C2211yescharacteristic, complemented
57#124×C293C2211yescharacteristic
58#97×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989803—22no—
61#17989803—22no—
62#17989803—22no—
63#17989803—22no—
64C2×Dic586—21no—
65C2×Dic586C2311yescharacteristic
66Q8×D293—22no—
67Q8×D293—22no—
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172—22no—
71C2×D1172C2311yescharacteristic
72C2×D1172C2311yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×C22⋊D293—21no—
75C2×C22⋊D293—22no—
76D4×D293—22no—
77D4×D293—22no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293—21no—
81C2×C4×D293C2311yescharacteristic
82#17989792—22no—
83#17989791—22no—
84#17989791—22no—
85#17989790—22no—
86#17989789—22no—
87#17989789—22no—
88#17989789—22no—
89#17989789—22no—
90#17989789—22no—
91D2344—22no—
92D2344—22no—
93#17989787—22no—
94#17989785D414yescomplemented
95C2×#14556873C2311yescharacteristic
96C2×#14556873C2311yescharacteristic
97C8×D293D414yescomplemented
98#17989781—22no—
99#17989781D412yes—
100#17989779—22no—
101#17989779—22no—
102#17989779—22no—
103#17989779D412yes—
104C4×Dic293C2311yescharacteristic
105C2×Q8×C293C2311yescharacteristic
106C2×Q8×C293C2311yescharacteristic
107C2×Q8×C293C2311yescharacteristic
108C2×D4×C293C2311yescharacteristic
109Q16×C293—22no—
110Q16×C293—22no—
111SD16×C293—22no—
112SD16×C293—22no—
113C2×C8×C293C2311yescharacteristic
114C2×C8×C293C2311yescharacteristic
115C4⋊C4×C293—22no—
116C22⋊C4×C293—22no—
117C42×C293C2311yescharacteristic
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—21no—
121C22×D293—42no—
122C22×D293—82no—
123C22×D293C2×D411yescharacteristic
124Dic586—21no—
125Dic586—21no—
126Dic586—42no—
127D1172—21no—
128D1172—21no—
129D1172—21no—
130D1172—21no—
131D1172—21no—
132D1172—21no—
133D1172—42no—
134D1172—42no—
135D1172—82no—
136C22⋊D293—42no—
137C22⋊D293—82no—
138C2×Dic293—21no—
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—
149Q8×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152Q8×C293—21no—
153Q8×C293—21no—
154Q8×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157C2×C4×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—42no—
166C22×C293C22×D411yescharacteristic
167C2×D293—22no—
168C2×D293—41no—
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—42no—
172C2×D293—42no—
173C2×D293—42no—
174C2×D293—84no—
175C2×D293#12712yes—
176C4×C293—22no—
177C4×C293—41no—
178C4×C293—41no—
179C4×C293—41no—
180C4×C293C22×D411yescharacteristic
181C4×C293C22×D411yescharacteristic
182Dic293—22no—
183Dic293—41no—
184Dic293#12712yes—
185C2×C293—81no—
186C2×C293C2×#12711yescharacteristic
187C2×C293#57712yes—
188D293—22no—
189D293—81no—
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#497D29311yescharacteristic, complemented
202#496—2931no—
203#496—2931no—
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×Q16C2×D29311yescharacteristic, complemented
218C2×Q16C2×D29311yescharacteristic, complemented
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
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#127—2931no—
239#124—2931no—
240#124—2931no—
241#124—2931no—
242#124C2×D29311yescharacteristic
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—5861no—
259C2×Q8—5862no—
260C2×Q8—5862no—
261C2×Q8C22×D29311yescharacteristic
262C2×Q8C22×D29311yescharacteristic
263C2×Q8C22×D29311yescharacteristic
264C2×D4—2931no—
265C2×D4—2931no—
266C2×D4—2931no—
267C2×D4—5861no—
268C2×D4—5861no—
269C2×D4—5862no—
270C2×D4—5862no—
271C2×D4—5862no—
272C2×D4—5861no—
273C2×D4—5862no—
274C2×D4C22×D29311yescharacteristic
275C22×C4—2931no—
276C22×C4—5861no—
277C22×C4—5861no—
278C22×C4—5861no—
279Q16—22no—
280Q16—22no—
281Q16—5862no—
282SD16—22no—
283SD16—22no—
284SD16—5862no—
285SD16—5862no—
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—22no—
305C22⋊C4—2932no—
306C22⋊C4—5862no—
307C22⋊C4—5862no—
308C22⋊C4—5862no—
309C42—2931no—
310C42C22×D29311yescharacteristic
311C23—21no—
312C23—2931no—
313C23—23442no—
314C23—5861no—
315C23—5861no—
316C23—5861no—
317C23—11722no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—5861no—
325Q8—5861no—
326Q8—11722no—
327D4—21no—
328D4—21no—
329D4—23442no—
330D4—23442no—
331D4—5861no—
332D4—5861no—
333D4—5861no—
334D4—5861no—
335D4—5861no—
336D4—5861no—
337D4—11722no—
338D4—11722no—
339D4—11722no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—2931no—
344C2×C4—2932no—
345C2×C4—2932no—
346C2×C4—2932no—
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—42no—
359C22—2932no—
360C22—23444no—
361C22—5862no—
362C22—11721no—
363C22—11721no—
364C22—11721no—
365C22—11722no—
366C22—11722no—
367C22—11722no—
368C22C2×D4×D29311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2932no—
374C4—5862no—
375C4—11721no—
376C4C2×D4×D29311yescharacteristic
377C4C2×D4×D29311yescharacteristic
378C2—81no—
379C2—23441no—
380C2—23441no—
381C2—23441no—
382C2—5862no—
383C2#127×D29311yescharacteristic
384C2#3403693812yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#497×C293C2
socleC22×C293C22×D4
hypercenter#497D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22627656C2semidirect
2. C2×#22627651C2semidirect
3. C2×#22627647C2semidirect
4. C2×SD16×D293C2semidirect
5. #34036704C2semidirect
6. #34036697C2semidirect
7. #34036522C2semidirect
8. #34036518C2semidirect
9. #34036515C2semidirect
10. #34036514C2semidirect
11. #34036511C2semidirect
12. #34036503C2semidirect
13. #34036500C2semidirect
14. #34036086C2semidirect
15. #497×C293C2semidirect
23. C2×#17989792C22semidirect
25. C2×#17989791C22semidirect
29. C2×#17989789C22semidirect
30. C2×#17989789C22semidirect
31. C2×#17989789C22semidirect
34. C2×D2344C22semidirect
38. #22627622C22semidirect
42. #22627613C22semidirect
43. #22627612C22semidirect
49. #22627534C22semidirect
50. C4×#14556873C22semidirect
51. C8×Dic293C22semidirect
52. C2×Q16×C293C22semidirect
53. C2×Q16×C293C22semidirect
56. C4⋊Q8×C293C22semidirect
58. #97×C293C22semidirect
94. #17989785D4semidirect
97. C8×D293D4semidirect
192. C293#3154semidirect
201. #497D293semidirect
217. C2×Q16C2×D293semidirect
218. C2×Q16C2×D293semidirect
236. C4⋊Q8C2×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×#226276562
C2×#226276512
C2×#226276472
C2×SD16×D2932
#340367042
#340366972
#340365222
#340365182
#340365152
#340365142
#340365112
#340365032
#340365002
#340360862
#497×C2932
#3154293