Finite Group Databasebeta

← #49169269

Subgroups — #49169269

Order 37504.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627648C211yesmaximal, characteristic, complemented
3C2×#22627645C211yesmaximal, characteristic, complemented
4C2×D8×D293C211yesmaximal, characteristic, complemented
5#34036697C211yesmaximal, characteristic, complemented
6#34036692C211yesmaximal, characteristic, complemented
7#34036518C211yesmaximal, characteristic, complemented
8#34036517C211yesmaximal, characteristic, complemented
9#34036515C211yesmaximal, characteristic, complemented
10#34036511C211yesmaximal, characteristic, complemented
11#34036510C211yesmaximal, characteristic, complemented
12#34036505C211yesmaximal, characteristic, complemented
13#34036504C211yesmaximal, characteristic, complemented
14#34036086C211yesmaximal, characteristic, complemented
15#496×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×D4×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627651—24no—
21#22627648—24no—
22#22627645—24no—
23C2×#17989792C2211yescharacteristic, complemented
24C2×#17989791C2211yescharacteristic, complemented
25C2×#17989791C2211yescharacteristic, complemented
26C2×#17989790C2211yescharacteristic, complemented
27C2×#17989789C2211yescharacteristic, complemented
28C2×D2344C2211yescharacteristic
29C2×D2344C2211yescharacteristic, complemented
30C2×#17989787C2211yescharacteristic
31C2×#17989787C2211yescharacteristic
32C2×#17989787C2211yescharacteristic, complemented
33C2×#17989787C2211yescharacteristic, complemented
34C2×#17989787C2211yescharacteristic, complemented
35D8×D293—24no—
36C2×#17989785C2211yescharacteristic
37C2×C8×D293C2211yescharacteristic
38#22627621C2211yescharacteristic, complemented
39#22627620C2211yescharacteristic, complemented
40#22627620C2211yescharacteristic, complemented
41#22627613C2211yescharacteristic, complemented
42#22627612C2211yescharacteristic, complemented
43#22627611C2211yescharacteristic, complemented
44#22627597—22no—
45#22627596—22no—
46#22627594—22no—
47#22627592—22no—
48#22627581C2211yescharacteristic
49#22627534C2211yescharacteristic, complemented
50C4×#14556873C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic, complemented
53C2×SD16×C293C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic
55C2×D8×C293C2211yescharacteristic, complemented
56#127×C293C2211yescharacteristic, complemented
57#124×C293C2211yescharacteristic, complemented
58#97×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60C23×D293—21no—
61#17989803—22no—
62#17989803—22no—
63#17989801—22no—
64#17989801—22no—
65C2×Dic586C2311yescharacteristic
66C2×D1172—21no—
67C2×D1172—22no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—22no—
75C2×C22⋊D293—22no—
76D4×D293—22no—
77D4×D293—22no—
78D4×D293—22no—
79D4×D293—22no—
80C22×Dic293—21no—
81C2×C4×D293—21no—
82C2×C4×D293C2311yescharacteristic
83#17989792—22no—
84#17989791—22no—
85#17989791—22no—
86#17989790—22no—
87#17989789—22no—
88D2344—22no—
89D2344—22no—
90#17989787—22no—
91#17989787—22no—
92#17989787—22no—
93#17989787—22no—
94#17989787—22no—
95#17989785D414yescomplemented
96C2×#14556873C2311yescharacteristic
97C2×#14556873C2311yescharacteristic
98C8×D293D414yescomplemented
99#17989781D412yes—
100C22⋊Dic293—22no—
101#17989779—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—21no—
120C22×D293—21no—
121C22×D293—42no—
122C22×D293—42no—
123C22×D293—82no—
124C22×D293—82no—
125C22×D293C2×D411yescharacteristic
126Dic586—21no—
127Dic586—21no—
128D1172—21no—
129D1172—21no—
130D1172—21no—
131D1172—21no—
132D1172—21no—
133D1172—21no—
134D1172—42no—
135D1172—82no—
136C22⋊D293—42no—
137C22⋊D293—42no—
138C22⋊D293—42no—
139C22⋊D293—82no—
140C22⋊D293—82no—
141C2×Dic293—21no—
142C2×Dic293—42no—
143C2×Dic293C2×D412yes—
144C2×Dic293C2×D411yescharacteristic
145C4×D293—42no—
146C4×D293C2×D412yes—
147C4×D293C2×D412yes—
148#14556873C2×D412yes—
149#14556873C2×D412yes—
150C23×C293—21no—
151C23×C293—21no—
152C23×C293—21no—
153Q8×C293—21no—
154Q8×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—21no—
158D4×C293—21no—
159D4×C293—21no—
160D4×C293—21no—
161D4×C293—82no—
162C2×C4×C293—21no—
163C2×C4×C293C2×D412yes—
164C2×C4×C293C2411yescharacteristic
165C8×C293C2×D412yes—
166C8×C293C2×D412yes—
167C22×C293—41no—
168C22×C293—41no—
169C22×C293—41no—
170C22×C293—42no—
171C22×C293—42no—
172C22×C293—42no—
173C22×C293C22×D411yescharacteristic
174C2×D293—22no—
175C2×D293—41no—
176C2×D293—41no—
177C2×D293—41no—
178C2×D293—42no—
179C2×D293—42no—
180C2×D293—42no—
181C2×D293—84no—
182C2×D293—84no—
183C2×D293#12712yes—
184C4×C293—22no—
185C4×C293—41no—
186C4×C293C22×D411yescharacteristic
187C4×C293C22×D411yescharacteristic
188Dic293—22no—
189Dic293—41no—
190Dic293#12712yes—
191C2×C293—81no—
192C2×C293—81no—
193C2×C293—81no—
194C2×C293C2×#12711yescharacteristic
195C2×C293#57612yes—
196D293—22no—
197D293—81no—
198D293—81no—
199D293—81no—
200C293#315511yescharacteristic, complemented, Sylow
201#3155—2931nomaximal, Sylow
202C2×AGL(1,ℤ/8)—2931no—
203C2×AGL(1,ℤ/8)—2931no—
204C2×#135—2931no—
205C22×D8—2931no—
206#535—2931no—
207#535—2931no—
208#496—2931no—
209#496—2931no—
210#496—2931no—
211#496D29311yescharacteristic, complemented
212#495—2931no—
213#495—2931no—
214#493—2931no—
215#493—2931no—
216#405—2931no—
217C2×#41—2931no—
218C2×#41—2931no—
219C22×D4—2931no—
220C22×D4—2931no—
221AGL(1,ℤ/8)—5864no—
222AGL(1,ℤ/8)—5864no—
223#135—5864no—
224C2×Q16—2931no—
225C2×SD16—2931no—
226C2×SD16—2931no—
227C2×SD16—2931no—
228C2×SD16—2931no—
229C2×SD16C2×D29311yescharacteristic, complemented
230C2×SD16C2×D29311yescharacteristic, complemented
231C2×D8—2931no—
232C2×D8—2931no—
233C2×D8—2931no—
234C2×D8—2931no—
235C2×D8—2931no—
236C2×D8—2931no—
237C2×D8—2931no—
238C2×D8—5864no—
239C2×D8C2×D29311yescharacteristic
240C2×D8C2×D29311yescharacteristic, complemented
241C2×M16—2931no—
242C22×C8—2931no—
243#127—2931no—
244#127—2931no—
245#127—2931no—
246#127C2×D29311yescharacteristic, complemented
247#124—2931no—
248#124—2931no—
249#124—2931no—
250#124C2×D29311yescharacteristic, complemented
251#121—5862no—
252#121—5862no—
253#120—5862no—
254#120—5862no—
255#117—2931no—
256#97—2931no—
257#97C2×D29311yescharacteristic, complemented
258C4×C8—2931no—
259C4×C8—2931no—
260C24—5861no—
261C24—5861no—
262#41—5862no—
263#41—5862no—
264#41—5862no—
265#41—5862no—
266C2×Q8—2931no—
267C2×Q8C22×D29311yescharacteristic
268C2×D4—22no—
269C2×D4—2931no—
270C2×D4—2931no—
271C2×D4—2931no—
272C2×D4—5861no—
273C2×D4—5862no—
274C2×D4—5862no—
275C2×D4—5862no—
276C2×D4—5862no—
277C2×D4—5862no—
278C2×D4—5862no—
279C2×D4—5861no—
280C2×D4—5861no—
281C2×D4—5861no—
282C2×D4—5862no—
283C2×D4C22×D29311yescharacteristic
284C2×D4C22×D29311yescharacteristic
285C2×D4C22×D29311yescharacteristic
286C22×C4—2931no—
287C22×C4—5861no—
288C22×C4—5861no—
289Q16—5862no—
290SD16—22no—
291SD16—22no—
292SD16—5862no—
293SD16—5862no—
294SD16—5862no—
295SD16—5862no—
296D8—22no—
297D8—22no—
298D8—5862no—
299D8—5862no—
300D8—5862no—
301D8—5862no—
302D8—5862no—
303D8—5862no—
304D8—5862no—
305M16—2934no—
306C2×C8—2931no—
307C2×C8—2931no—
308C2×C8—2934no—
309C2×C8C22×D29311yescharacteristic
310C2×C8C22×D29311yescharacteristic
311C4⋊C4—2932no—
312C22⋊C4—22no—
313C22⋊C4—2932no—
314C22⋊C4—5862no—
315C22⋊C4—5862no—
316C22⋊C4—5862no—
317C42—2931no—
318C42C22×D29311yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—2931no—
323C23—23442no—
324C23—23442no—
325C23—5861no—
326C23—5861no—
327C23—5861no—
328C23—11722no—
329C23—11722no—
330Q8—21no—
331Q8—21no—
332Q8—5861no—
333Q8—5861no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—23442no—
342D4—23442no—
343D4—23442no—
344D4—5861no—
345D4—5861no—
346D4—5861no—
347D4—5861no—
348D4—5861no—
349D4—5861no—
350D4—11722no—
351D4—11722no—
352D4—11722no—
353D4—11722no—
354C2×C4—21no—
355C2×C4—2931no—
356C2×C4—2932no—
357C2×C4—2932no—
358C2×C4—2932no—
359C2×C4—5861no—
360C2×C4—11722no—
361C2×C4—11722no—
362C2×C4D4×D29312yes—
363C2×C4C23×D29311yescharacteristic
364C8—2932no—
365C8—2932no—
366C8D4×D29312yes—
367C8D4×D29312yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—2932no—
375C22—23444no—
376C22—23444no—
377C22—5862no—
378C22—11721no—
379C22—11721no—
380C22—11721no—
381C22—11722no—
382C22—11722no—
383C22—11722no—
384C22C2×D4×D29311yescharacteristic
385C4—22no—
386C4—41no—
387C4—2932no—
388C4—5862no—
389C4—11721no—
390C4C2×D4×D29311yescharacteristic
391C4C2×D4×D29311yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—23441no—
396C2—23441no—
397C2—23441no—
398C2—5862no—
399C2#127×D29311yescharacteristic
400C2#3403693312yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3155
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×#22627651C2semidirect
2. C2×#22627648C2semidirect
3. C2×#22627645C2semidirect
4. C2×D8×D293C2semidirect
5. #34036697C2semidirect
6. #34036692C2semidirect
7. #34036518C2semidirect
8. #34036517C2semidirect
9. #34036515C2semidirect
10. #34036511C2semidirect
11. #34036510C2semidirect
12. #34036505C2semidirect
13. #34036504C2semidirect
14. #34036086C2semidirect
15. #496×C293C2semidirect
23. C2×#17989792C22semidirect
24. C2×#17989791C22semidirect
25. C2×#17989791C22semidirect
26. C2×#17989790C22semidirect
27. C2×#17989789C22semidirect
29. C2×D2344C22semidirect
32. C2×#17989787C22semidirect
33. C2×#17989787C22semidirect
34. C2×#17989787C22semidirect
38. #22627621C22semidirect
39. #22627620C22semidirect
40. #22627620C22semidirect
41. #22627613C22semidirect
42. #22627612C22semidirect
43. #22627611C22semidirect
49. #22627534C22semidirect
50. C4×#14556873C22semidirect
51. C8×Dic293C22semidirect
52. C2×SD16×C293C22semidirect
53. C2×SD16×C293C22semidirect
55. C2×D8×C293C22semidirect
56. #127×C293C22semidirect
57. #124×C293C22semidirect
58. #97×C293C22semidirect
95. #17989785D4semidirect
98. C8×D293D4semidirect
200. C293#3155semidirect
211. #496D293semidirect
229. C2×SD16C2×D293semidirect
230. C2×SD16C2×D293semidirect
240. C2×D8C2×D293semidirect
246. #127C2×D293semidirect
250. #124C2×D293semidirect
257. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276482
C2×#226276452
C2×D8×D2932
#340366972
#340366922
#340365182
#340365172
#340365152
#340365112
#340365102
#340365052
#340365042
#340360862
#496×C2932
#3155293