Finite Group Databasebeta

← #49170309

Subgroups — #49170309

Order 37504.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627647C211yesmaximal, characteristic, complemented
3#34036782C211yesmaximal, characteristic, complemented
4#34036772C211yesmaximal, characteristic, complemented
5#34036384C211yesmaximal, characteristic, complemented
6#34036355C211yesmaximal, characteristic, complemented
7#34036341C211yesmaximal, characteristic, complemented
8#34036331C211yesmaximal, characteristic, complemented
9#34036290C211yesmaximal, characteristic, complemented
10#34036290C211yesmaximal, characteristic, complemented
11#34036278C211yesmaximal, characteristic, complemented
12#34036273C211yesmaximal, characteristic, complemented
13#34036242C211yesmaximal, characteristic, complemented
14#34036202C211yesmaximal, characteristic, complemented
15#497×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989803C2211yescharacteristic
18C2×#17989803C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic
20#22627651—24no—
21#22627647—24no—
22C2×#17989789C2211yescharacteristic
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic, complemented
25C2×D2344C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989785C2211yescharacteristic
28C2×C8×D293C2211yescharacteristic, complemented
29#22627609—21no—
30#22627609—21no—
31#22627606—21no—
32#22627600C2211yescharacteristic
33#22627600C2211yescharacteristic
34#22627600C2211yescharacteristic, complemented
35#22627599—21no—
36#22627597—21no—
37#22627595C2211yescharacteristic
38#22627594—21no—
39C4×D1172C2211yescharacteristic
40C4×D1172C2211yescharacteristic
41C4×D1172C2211yescharacteristic, complemented
42C4⋊C4×D293—21no—
43C22⋊C4×D293—21no—
44#22627556C2211yescharacteristic, complemented
45C4⋊#14556873C2211yescharacteristic, complemented
46#22627548C2211yescharacteristic, complemented
47#22627548C2211yescharacteristic, complemented
48#22627548C2211yescharacteristic, complemented
49#22627546C2211yescharacteristic
50#22627546C2211yescharacteristic, complemented
51#22627545C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627536C2211yescharacteristic, complemented
54C2×Q16×C293C2211yescharacteristic, complemented
55C2×Q16×C293C2211yescharacteristic, complemented
56C2×SD16×C293C2211yescharacteristic
57C2×SD16×C293C2211yescharacteristic
58C4⋊Q8×C293C2211yescharacteristic, complemented
59#124×C293C2211yescharacteristic
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989803—22no—
63#17989803—22no—
64#17989803—22no—
65#17989803—41no—
66#17989803—41no—
67#17989803—41no—
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172—21no—
71C2×D1172D411yescharacteristic
72C2×D1172C2311yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76D4×D293—22no—
77D4×D293—41no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293—21no—
81C2×C4×D293—21no—
82C2×C4×D293—21no—
83C2×C4×D293C2311yescharacteristic
84C2×C4×D293C2311yescharacteristic
85#17989789—22no—
86#17989789—22no—
87#17989789—22no—
88D2344D414yescomplemented
89#17989787—22no—
90#17989785—22no—
91C2×#14556873C2311yescharacteristic
92C2×#14556873C2311yescharacteristic
93C8×D293—22no—
94C4⋊Dic293D411yescharacteristic
95C4⋊Dic293C2311yescharacteristic
96C4⋊Dic293C2311yescharacteristic
97#17989781—21no—
98#17989781—21no—
99#17989781—21no—
100C22⋊Dic293—21no—
101#17989779—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107#17989779D411yescharacteristic
108C2×Q8×C293C2311yescharacteristic
109C2×Q8×C293C2311yescharacteristic
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293C2311yescharacteristic
112Q16×C293—22no—
113Q16×C293—41no—
114SD16×C293—22no—
115SD16×C293—41no—
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C4⋊C4×C293—21no—
119C4⋊C4×C293—21no—
120C22⋊C4×C293—21no—
121C22⋊C4×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128C22×D293C2×D411yescharacteristic
129C22×D293C2×D411yescharacteristic
130D1172—21no—
131D1172—21no—
132D1172—41no—
133D1172—42no—
134D1172—42no—
135D1172—42no—
136D1172C2×D412yes—
137D1172C2×D412yes—
138C22⋊D293—42no—
139C22⋊D293—81no—
140C2×Dic293—21no—
141C2×Dic293—21no—
142C2×Dic293C2×D411yescharacteristic
143C2×Dic293C2×D411yescharacteristic
144C4×D293—21no—
145C4×D293—21no—
146C4×D293—21no—
147C4×D293—21no—
148C4×D293—42no—
149C4×D293—42no—
150C4×D293—42no—
151C4×D293—42no—
152C4×D293—81no—
153#14556873—21no—
154#14556873—21no—
155C23×C293—21no—
156Q8×C293—21no—
157Q8×C293—21no—
158Q8×C293—21no—
159Q8×C293—21no—
160Q8×C293—21no—
161Q8×C293—21no—
162D4×C293—21no—
163D4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2411yescharacteristic
170C8×C293—21no—
171C8×C293C2×D412yes—
172C22×C293—41no—
173C22×C293—41no—
174C22×C293—41no—
175C22×C293C22×D411yescharacteristic
176C2×D293—21no—
177C2×D293—21no—
178C2×D293—22no—
179C2×D293—22no—
180C2×D293—21no—
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—41no—
185C2×D293—41no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—84no—
189C4×C293—21no—
190C4×C293—21no—
191C4×C293—41no—
192C4×C293—41no—
193C4×C293—41no—
194C4×C293C22×D411yescharacteristic
195C4×C293H(5,𝔽2)11yescharacteristic
196Dic293—21no—
197Dic293—21no—
198Dic293—41no—
199Dic293—41no—
200C2×C293—81no—
201C2×C293D4211yescharacteristic
202C2×C293#57511yescharacteristic
203C2×C293#57711yescharacteristic
204D293—41no—
205D293—41no—
206D293—42no—
207D293—42no—
208C293#326711yescharacteristic, complemented, Sylow
209#3267—2931nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2931no—
211C2×#135—2931no—
212#547—2931no—
213#546—2931no—
214#497D29311yescharacteristic, complemented
215#470—2931no—
216#465—2931no—
217#463—2931no—
218#461—2931no—
219#452—2931no—
220#452—2931no—
221#450—2931no—
222#449—2931no—
223#442—2931no—
224#435—2931no—
225C2×#41—2931no—
226C2×#41—2931no—
227C2×#41—2931no—
228C22×D4—2931no—
229AGL(1,ℤ/8)—5864no—
230#135—5864no—
231C2×Q16C2×D29311yescharacteristic, complemented
232C2×Q16C2×D29311yescharacteristic, complemented
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—2931no—
236C2×SD16C2×D29311yescharacteristic
237C2×SD16C2×D29311yescharacteristic
238C2×D8—2931no—
239C2×D8—2931no—
240C2×M16—2931no—
241C22×C8—2931no—
242C4⋊Q8C2×D29311yescharacteristic, complemented
243#124C2×D29311yescharacteristic
244#123—5861no—
245#123—5861no—
246#123—5861no—
247C22⋊Q8—2931no—
248C22⋊Q8—2931no—
249C22⋊Q8—2931no—
250#121—2931no—
251#121—5861no—
252#121—5861no—
253#120—5861no—
254C4×D4—2931no—
255C4×D4—2931no—
256C4×D4—2931no—
257C2×C4⋊C4—5861no—
258C2×C22⋊C4—5861no—
259#106—2931no—
260C4⋊C8—2931no—
261#103—2931no—
262#103—2931no—
263#103—2931no—
264#102—2931no—
265#102—2931no—
266#102—2931no—
267C22⋊C8—2931no—
268C22⋊C8—2931no—
269#97C2×D29311yescharacteristic, complemented
270C24—5861no—
271#41—5862no—
272#41—5862no—
273#41—5862no—
274#41—11721no—
275#41—11721no—
276#41—11721no—
277C2×Q8C22×D29311yescharacteristic
278C2×Q8C22×D29311yescharacteristic
279C2×Q8C22×D29311yescharacteristic
280C2×D4—2931no—
281C2×D4—2931no—
282C2×D4—2931no—
283C2×D4—5861no—
284C2×D4—5861no—
285C2×D4—5861no—
286C2×D4—5861no—
287C2×D4—5862no—
288C2×D4—5861no—
289C2×D4—11721no—
290C2×D4C22×D29311yescharacteristic
291C22×C4—2931no—
292C22×C4—2931no—
293C22×C4—5861no—
294C22×C4—5861no—
295C22×C4—5861no—
296C22×C4—5861no—
297C22×C4—5861no—
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—41no—
302SD16—5862no—
303SD16—5862no—
304SD16—5862no—
305D8—2934no—
306D8—5862no—
307M16—5862no—
308C2×C8—2931no—
309C2×C8—2931no—
310C2×C8—5862no—
311C2×C8C22×D29311yescharacteristic
312C2×C8C22×D29311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2931no—
316C4⋊C4—2931no—
317C4⋊C4—2931no—
318C4⋊C4—5861no—
319C4⋊C4—5861no—
320C4⋊C4—5861no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2931no—
324C22⋊C4—2931no—
325C22⋊C4—5861no—
326C22⋊C4—5861no—
327C22⋊C4—5861no—
328C22⋊C4—5861no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C42C22×D29311yescharacteristic
332C23—21no—
333C23—2931no—
334C23—2931no—
335C23—23441no—
336C23—5861no—
337C23—5861no—
338C23—11722no—
339C23—11722no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—2932no—
349D4—2932no—
350D4—23441no—
351D4—5861no—
352D4—5861no—
353D4—11721no—
354D4—11722no—
355D4—11722no—
356D4—11722no—
357D4—11722no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—2931no—
362C2×C4—2931no—
363C2×C4—23441no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—5861no—
367C2×C4—5861no—
368C2×C4—5861no—
369C2×C4—5861no—
370C2×C4—11722no—
371C2×C4—11722no—
372C2×C4—11722no—
373C2×C4—11722no—
374C2×C4D4×D29311yescharacteristic
375C2×C4D4×D29311yescharacteristic
376C2×C4C23×D29311yescharacteristic
377C8—21no—
378C8—5861no—
379C8—5861no—
380C8D4×D29312yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—23444no—
385C22—5861no—
386C22—5861no—
387C22—5862no—
388C22—5862no—
389C22—5861no—
390C22—5861no—
391C22—5861no—
392C22—5861no—
393C22—11721no—
394C22—11721no—
395C22—11721no—
396C22—11721no—
397C22C2×D4×D29311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5861no—
404C4—5861no—
405C4—11721no—
406C4—11721no—
407C4C2×D4×D29311yescharacteristic
408C4#2262766911yescharacteristic
409C2—81no—
410C2—11721no—
411C2—11722no—
412C2—11721no—
413C2—11722no—
414C2#3403676411yescharacteristic
415C2#3403692811yescharacteristic
416C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#497×C293C2
socleC22×C293C22×D4
hypercenter#497D293
hypocenterC293#3267
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×#22627651C2semidirect
2. C2×#22627647C2semidirect
3. #34036782C2semidirect
4. #34036772C2semidirect
5. #34036384C2semidirect
6. #34036355C2semidirect
7. #34036341C2semidirect
8. #34036331C2semidirect
9. #34036290C2semidirect
10. #34036290C2semidirect
11. #34036278C2semidirect
12. #34036273C2semidirect
13. #34036242C2semidirect
14. #34036202C2semidirect
15. #497×C293C2semidirect
18. C2×#17989803C22semidirect
24. C2×#17989789C22semidirect
28. C2×C8×D293C22semidirect
34. #22627600C22semidirect
41. C4×D1172C22semidirect
44. #22627556C22semidirect
45. C4⋊#14556873C22semidirect
46. #22627548C22semidirect
47. #22627548C22semidirect
48. #22627548C22semidirect
50. #22627546C22semidirect
53. #22627536C22semidirect
54. C2×Q16×C293C22semidirect
55. C2×Q16×C293C22semidirect
58. C4⋊Q8×C293C22semidirect
60. #97×C293C22semidirect
88. D2344D4semidirect
208. C293#3267semidirect
214. #497D293semidirect
231. C2×Q16C2×D293semidirect
232. C2×Q16C2×D293semidirect
242. C4⋊Q8C2×D293semidirect
269. #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×#226276472
#340367822
#340367722
#340363842
#340363552
#340363412
#340363312
#340362902
#340362782
#340362732
#340362422
#340362022
#497×C2932
#3267293