Finite Group Databasebeta

← #49170316

Subgroups — #49170316

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×#22627645C211yesmaximal, characteristic, complemented
3#34036782C211yesmaximal, characteristic, complemented
4#34036774C211yesmaximal, characteristic, complemented
5#34036521C211yesmaximal, characteristic, complemented
6#34036386C211yesmaximal, characteristic, complemented
7#34036361C211yesmaximal, characteristic, complemented
8#34036345C211yesmaximal, characteristic, complemented
9#34036294C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11#34036278C211yesmaximal, characteristic, complemented
12#34036277C211yesmaximal, characteristic, complemented
13#34036239C211yesmaximal, characteristic, complemented
14#34036199C211yesmaximal, characteristic, complemented
15#461×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989803C2211yescharacteristic, complemented
18C2×#17989801C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic
20#22627651—24no—
21#22627645—24no—
22C2×#17989792C2211yescharacteristic, complemented
23C2×#17989792C2211yescharacteristic, complemented
24C2×#17989791C2211yescharacteristic, complemented
25C2×#17989789C2211yescharacteristic
26C2×#17989789C2211yescharacteristic
27C2×D2344C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989785C2211yescharacteristic
30C2×C8×D293C2211yescharacteristic, complemented
31#22627612C2211yescharacteristic
32#22627609—21no—
33#22627609—21no—
34#22627608—21no—
35#22627604C2211yescharacteristic
36#22627600C2211yescharacteristic
37#22627600C2211yescharacteristic, complemented
38#22627599—21no—
39#22627599C2211yescharacteristic
40#22627598—21no—
41#22627594—21no—
42C4×D1172C2211yescharacteristic
43C4×D1172C2211yescharacteristic, complemented
44C4⋊C4×D293—21no—
45C2×#17989779—21no—
46#22627555C2211yescharacteristic, complemented
47#22627549C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic
49#22627545C2211yescharacteristic
50#22627544C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic
52#22627536C2211yescharacteristic, complemented
53#22627535C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56C4⋊Q8×C293C2211yescharacteristic, complemented
57C4×D4×C293C2211yescharacteristic
58C4⋊C8×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60#103×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989803—22no—
63#17989803—22no—
64#17989803—41no—
65#17989803—41no—
66#17989801—22no—
67#17989801—41no—
68C2×Dic586C2311yescharacteristic
69C2×D1172—21no—
70C2×D1172—21no—
71C2×D1172—21no—
72C2×D1172D411yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×D1172C2311yescharacteristic
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77D4×D293—22no—
78D4×D293—41no—
79C22×Dic293—21no—
80C2×C4×D293—21no—
81C2×C4×D293—21no—
82C2×C4×D293—21no—
83C2×C4×D293C2311yescharacteristic
84C2×C4×D293C2311yescharacteristic
85#17989792—22no—
86#17989792—41no—
87#17989791—22no—
88#17989789—22no—
89#17989789—41no—
90D2344—22no—
91#17989787D414yescomplemented
92#17989785—22no—
93C2×#14556873C2311yescharacteristic
94C2×#14556873C2311yescharacteristic
95C8×D293—22no—
96C4⋊Dic293—21no—
97C4⋊Dic293D411yescharacteristic
98C4⋊Dic293C2311yescharacteristic
99#17989781—21no—
100#17989781—21no—
101#17989779—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779D411yescharacteristic
109#17989779D411yescharacteristic
110C2×Q8×C293C2311yescharacteristic
111C2×Q8×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114SD16×C293—22no—
115SD16×C293—22no—
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C4⋊C4×C293—21no—
119C4⋊C4×C293—21no—
120C4⋊C4×C293C2311yescharacteristic
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
130Dic586—21no—
131Dic586—21no—
132D1172—21no—
133D1172—21no—
134D1172—21no—
135D1172—41no—
136D1172—42no—
137D1172—42no—
138D1172—81no—
139D1172C2×D412yes—
140C22⋊D293—42no—
141C22⋊D293—42no—
142C2×Dic293—21no—
143C2×Dic293—21no—
144C2×Dic293—42no—
145C2×Dic293C2×D411yescharacteristic
146C2×Dic293C2×D411yescharacteristic
147C4×D293—21no—
148C4×D293—21no—
149C4×D293—21no—
150C4×D293—21no—
151C4×D293—42no—
152C4×D293—42no—
153C4×D293—42no—
154#14556873—21no—
155#14556873C2×D412yes—
156C23×C293—21no—
157Q8×C293—21no—
158Q8×C293—21no—
159Q8×C293—21no—
160Q8×C293—21no—
161D4×C293—21no—
162D4×C293C2×D412yes—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—81no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2411yescharacteristic
170C8×C293—21no—
171C8×C293—21no—
172C22×C293—22no—
173C22×C293—41no—
174C22×C293—41no—
175C22×C293C22×D411yescharacteristic
176C2×D293—21no—
177C2×D293—21no—
178C2×D293—22no—
179C2×D293—21no—
180C2×D293—21no—
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—41no—
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—42no—
201C2×C293D4211yescharacteristic
202C2×C293#57511yescharacteristic
203C2×C293#57711yescharacteristic
204D293—41no—
205D293—41no—
206D293—42no—
207D293—81no—
208C293#326711yescharacteristic, complemented, Sylow
209#3267—2931nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2931no—
211C2×#135—2931no—
212#547—2931no—
213#546—2931no—
214#497—2931no—
215#470—2931no—
216#465—2931no—
217#463—2931no—
218#461D29311yescharacteristic, complemented
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×Q16—2931no—
232C2×Q16—2931no—
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#124—2931no—
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×D4C2×D29311yescharacteristic
257C2×C4⋊C4—5861no—
258C2×C22⋊C4—5861no—
259#106—2931no—
260C4⋊C8C2×D29311yescharacteristic, complemented
261#103—2931no—
262#103C2×D29311yescharacteristic, complemented
263#103C2×D29311yescharacteristic, complemented
264#102—2931no—
265#102—2931no—
266#102—2931no—
267C22⋊C8—2931no—
268C22⋊C8—2931no—
269#97—2931no—
270C24—5861no—
271#41—5862no—
272#41—5862no—
273#41—5862no—
274#41—11721no—
275#41—11721no—
276#41—11721no—
277C2×Q8—2931no—
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—21no—
292C22×C4—2931no—
293C22×C4—2931no—
294C22×C4—5861no—
295C22×C4—5861no—
296C22×C4—5861no—
297C22×C4—5861no—
298Q16—5862no—
299Q16—11721no—
300SD16—22no—
301SD16—22no—
302SD16—5862no—
303SD16—5862no—
304SD16—11721no—
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—5861no—
318C4⋊C4—5861no—
319C4⋊C4—5861no—
320C4⋊C4C22×D29311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2931no—
323C22⋊C4—2931no—
324C22⋊C4—5861no—
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—5861no—
345Q8—5861no—
346D4—21no—
347D4—2932no—
348D4—23441no—
349D4—5861no—
350D4—5861no—
351D4—5861no—
352D4—11721no—
353D4—11722no—
354D4—11722no—
355D4—11722no—
356D4—11722no—
357D4D4×D29312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
362C2×C4—2931no—
363C2×C4—2931no—
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—21no—
379C8—2932no—
380C8—5861no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—23444no—
385C22—5861no—
386C22—5861no—
387C22—5862no—
388C22—5861no—
389C22—5861no—
390C22—5861no—
391C22—5861no—
392C22—11721no—
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#2262767011yescharacteristic
409C2—42no—
410C2—23441no—
411C2—11721no—
412C2—11721no—
413C2—11722no—
414C2#3403676311yescharacteristic
415C2#3403692811yescharacteristic
416C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#461×C293C2
socleC22×C293C22×D4
hypercenter#461D293
hypocenterC293#3267
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627645C2semidirect
3. #34036782C2semidirect
4. #34036774C2semidirect
5. #34036521C2semidirect
6. #34036386C2semidirect
7. #34036361C2semidirect
8. #34036345C2semidirect
9. #34036294C2semidirect
10. #34036288C2semidirect
11. #34036278C2semidirect
12. #34036277C2semidirect
13. #34036239C2semidirect
14. #34036199C2semidirect
15. #461×C293C2semidirect
17. C2×#17989803C22semidirect
22. C2×#17989792C22semidirect
23. C2×#17989792C22semidirect
24. C2×#17989791C22semidirect
30. C2×C8×D293C22semidirect
37. #22627600C22semidirect
43. C4×D1172C22semidirect
46. #22627555C22semidirect
47. #22627549C22semidirect
50. #22627544C22semidirect
52. #22627536C22semidirect
53. #22627535C22semidirect
56. C4⋊Q8×C293C22semidirect
58. C4⋊C8×C293C22semidirect
59. #103×C293C22semidirect
60. #103×C293C22semidirect
91. #17989787D4semidirect
208. C293#3267semidirect
218. #461D293semidirect
242. C4⋊Q8C2×D293semidirect
260. C4⋊C8C2×D293semidirect
262. #103C2×D293semidirect
263. #103C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276452
#340367822
#340367742
#340365212
#340363862
#340363612
#340363452
#340362942
#340362882
#340362782
#340362772
#340362392
#340361992
#461×C2932
#3267293