Finite Group Databasebeta

← #49170318

Subgroups — #49170318

Order 37504.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3#34036798C211yesmaximal, characteristic, complemented
4#34036776C211yesmaximal, characteristic, complemented
5#34036517C211yesmaximal, characteristic, complemented
6#34036374C211yesmaximal, characteristic, complemented
7#34036363C211yesmaximal, characteristic, complemented
8#34036349C211yesmaximal, characteristic, complemented
9#34036323C211yesmaximal, characteristic, complemented
10#34036294C211yesmaximal, characteristic, complemented
11#34036276C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036232C211yesmaximal, characteristic, complemented
14#34036197C211yesmaximal, characteristic, complemented
15#449×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989802C2211yescharacteristic
18C2×D4×D293C2211yescharacteristic
19#22627651—24no—
20#22627644—24no—
21C2×#17989792C2211yescharacteristic, complemented
22C2×#17989791C2211yescharacteristic, complemented
23C2×#17989790C2211yescharacteristic, complemented
24C2×#17989789C2211yescharacteristic
25C2×D2344C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989785C2211yescharacteristic
29C22×#14556873C2211yescharacteristic, complemented
30#22627620—21no—
31#22627620C2211yescharacteristic, complemented
32#22627611C2211yescharacteristic
33#22627606—21no—
34#22627604C2211yescharacteristic
35#22627598—21no—
36#22627598C2211yescharacteristic, complemented
37#22627595—21no—
38#22627595—21no—
39#22627595C2211yescharacteristic
40#22627592—21no—
41Q8×Dic293C2211yescharacteristic, complemented
42#22627586C2211yescharacteristic
43C4×C22⋊D293—21no—
44C4×C22⋊D293C2211yescharacteristic
45C22⋊C4×D293—21no—
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627546C2211yescharacteristic
50#22627545C2211yescharacteristic
51#22627544C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic
53#22627534C2211yescharacteristic, complemented
54C2×#41×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic
56C2×D8×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic
58#103×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic, complemented
60C22⋊C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989803—22no—
63#17989803—41no—
64#17989802—22no—
65#17989802—41no—
66C2×Dic586C2311yescharacteristic
67C2×D1172—21no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293D411yescharacteristic
77D4×D293—22no—
78D4×D293—41no—
79C2×C4×D293—21no—
80C2×C4×D293—21no—
81C2×C4×D293—21no—
82C2×C4×D293C2311yescharacteristic
83#17989792—22no—
84#17989791—41no—
85#17989790—22no—
86#17989789D414yescomplemented
87D2344—22no—
88#17989787—22no—
89#17989787—22no—
90#17989785—22no—
91C2×#14556873—22no—
92C2×#14556873C2311yescharacteristic
93C2×#14556873C2311yescharacteristic
94C4⋊Dic293—21no—
95C4⋊Dic293—21no—
96C4⋊Dic293C2311yescharacteristic
97#17989781D411yescharacteristic
98C22⋊Dic293—21no—
99C22⋊Dic293—21no—
100C22⋊Dic293—21no—
101C22⋊Dic293D411yescharacteristic
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779D411yescharacteristic
106C4×Dic293—21no—
107C4×Dic293C2311yescharacteristic
108#41×C293—22no—
109#41×C293—41no—
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293—21no—
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293—21no—
116C22×C4×C293C2311yescharacteristic
117SD16×C293—22no—
118D8×C293—41no—
119C2×C8×C293C2311yescharacteristic
120C2×C8×C293C2311yescharacteristic
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586—21no—
131D1172—21no—
132D1172—21no—
133D1172—21no—
134D1172—42no—
135D1172C2×D412yes—
136C22⋊D293—41no—
137C22⋊D293—42no—
138C22⋊D293—42no—
139C22⋊D293—42no—
140C22⋊D293—81no—
141C22⋊D293—81no—
142C22⋊D293—81no—
143C2×Dic293—21no—
144C2×Dic293—21no—
145C2×Dic293C2×D411yescharacteristic
146C2×Dic293C2×D411yescharacteristic
147C2×Dic293C2×D411yescharacteristic
148C4×D293—21no—
149C4×D293—21no—
150C4×D293—42no—
151C4×D293—42no—
152C4×D293—81no—
153#14556873—21no—
154#14556873C2×D412yes—
155C23×C293—21no—
156C23×C293—21no—
157C23×C293C2×D411yescharacteristic
158Q8×C293—21no—
159Q8×C293C2×D412yes—
160D4×C293—21no—
161D4×C293—21no—
162D4×C293—21no—
163D4×C293—21no—
164D4×C293—42no—
165D4×C293—81no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—42no—
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293—21no—
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—21no—
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293C22×D411yescharacteristic
185C2×D293—21no—
186C2×D293—22no—
187C2×D293—21no—
188C2×D293—21no—
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—84no—
195C4×C293—21no—
196C4×C293—22no—
197C4×C293—41no—
198C4×C293C22×D411yescharacteristic
199C4×C293H(5,𝔽2)11yescharacteristic
200Dic293—21no—
201Dic293—21no—
202Dic293—21no—
203Dic293—41no—
204Dic293—41no—
205C2×C293—41no—
206C2×C293—81no—
207C2×C293—81no—
208C2×C293D4211yescharacteristic
209C2×C293#57511yescharacteristic
210C2×C293#57611yescharacteristic
211D293—41no—
212D293—42no—
213D293—81no—
214C293#326811yescharacteristic, complemented, Sylow
215#3268—2931nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2931no—
217C2×#135—2931no—
218#550—2931no—
219#546—2931no—
220#496—2931no—
221#468—2931no—
222#466—2931no—
223#464—2931no—
224#459—2931no—
225#452—2931no—
226#449—2931no—
227#449D29311yescharacteristic, complemented
228#447—2931no—
229#440—2931no—
230#435—2931no—
231C2×#41—2931no—
232C2×#41—2931no—
233C2×#41C2×D29311yescharacteristic, complemented
234C22×D4—2931no—
235AGL(1,ℤ/8)—5864no—
236#135—5864no—
237C2×Q16—2931no—
238C2×SD16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16C2×D29311yescharacteristic
242C2×D8—2931no—
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8C2×D29311yescharacteristic
246C2×M16—2931no—
247C22×C8—2931no—
248#127—2931no—
249#127—5861no—
250#124—2931no—
251#123—5861no—
252C22⋊Q8—2931no—
253#121—2931no—
254#121—2931no—
255#121—5861no—
256#121—5861no—
257#121—5861no—
258#121C2×D29311yescharacteristic
259#120—5861no—
260C4×Q8—2931no—
261C4×D4—2931no—
262C4×D4—2931no—
263C4×D4—5861no—
264C2×C22⋊C4—5861no—
265#107—2931no—
266C4⋊C8—2931no—
267#103—2931no—
268#103C2×D29311yescharacteristic, complemented
269#102—2931no—
270#102—2931no—
271#102—2931no—
272#102C2×D29311yescharacteristic, complemented
273C22⋊C8—2931no—
274C22⋊C8C2×D29311yescharacteristic, complemented
275#97—2931no—
276C24—5861no—
277#41—22no—
278#41—41no—
279#41—5862no—
280#41—5862no—
281#41—11721no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2931no—
288C2×D4—2931no—
289C2×D4—2931no—
290C2×D4—5861no—
291C2×D4—5861no—
292C2×D4—5861no—
293C2×D4—5861no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—11721no—
299C2×D4C22×D29311yescharacteristic
300C2×D4C22×D29311yescharacteristic
301C22×C4—21no—
302C22×C4—2931no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4—5861no—
306C22×C4C22×D29311yescharacteristic
307Q16—5862no—
308SD16—22no—
309SD16—2934no—
310SD16—5862no—
311SD16—11721no—
312D8—41no—
313D8—5862no—
314D8—5862no—
315D8—5862no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8—5862no—
320C2×C8C22×D29311yescharacteristic
321C2×C8C22×D29311yescharacteristic
322C4⋊C4—2931no—
323C4⋊C4—2931no—
324C4⋊C4—5861no—
325C4⋊C4—5861no—
326C4⋊C4C22×D29311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2931no—
329C22⋊C4—2931no—
330C22⋊C4—5861no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C42—2931no—
337C42—5861no—
338C23—21no—
339C23—21no—
340C23—2931no—
341C23—23441no—
342C23—5861no—
343C23—5861no—
344C23—11722no—
345C23—11722no—
346C23D4×D29311yescharacteristic
347Q8—21no—
348Q8—5861no—
349Q8—5861no—
350Q8D4×D29312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—2932no—
358D4—23441no—
359D4—23441no—
360D4—23441no—
361D4—5861no—
362D4—5861no—
363D4—5861no—
364D4—11722no—
365D4—11721no—
366D4—11722no—
367D4—11722no—
368D4—11722no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2931no—
375C2×C4—2931no—
376C2×C4—2931no—
377C2×C4—23441no—
378C2×C4—5861no—
379C2×C4—5861no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—11722no—
383C2×C4—11722no—
384C2×C4D4×D29311yescharacteristic
385C2×C4C23×D29311yescharacteristic
386C8—21no—
387C8—21no—
388C8—2932no—
389C8—5861no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—23444no—
400C22—5861no—
401C22—5862no—
402C22—5861no—
403C22—5861no—
404C22—11721no—
405C22—11721no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22C2×D4×D29311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—5861no—
414C4—5861no—
415C4—5861no—
416C4—11721no—
417C4—11721no—
418C4C2×D4×D29311yescharacteristic
419C4#2262766911yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—23441no—
424C2—11721no—
425C2—11722no—
426C2#3403676111yescharacteristic
427C2#3403692611yescharacteristic
428C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#449×C293C2
socleC22×C293C22×D4
hypercenter#449D293
hypocenterC293#3268
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627644C2semidirect
3. #34036798C2semidirect
4. #34036776C2semidirect
5. #34036517C2semidirect
6. #34036374C2semidirect
7. #34036363C2semidirect
8. #34036349C2semidirect
9. #34036323C2semidirect
10. #34036294C2semidirect
11. #34036276C2semidirect
12. #34036261C2semidirect
13. #34036232C2semidirect
14. #34036197C2semidirect
15. #449×C293C2semidirect
21. C2×#17989792C22semidirect
22. C2×#17989791C22semidirect
23. C2×#17989790C22semidirect
29. C22×#14556873C22semidirect
31. #22627620C22semidirect
36. #22627598C22semidirect
41. Q8×Dic293C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
48. #22627549C22semidirect
51. #22627544C22semidirect
53. #22627534C22semidirect
54. C2×#41×C293C22semidirect
58. #103×C293C22semidirect
59. #102×C293C22semidirect
60. C22⋊C8×C293C22semidirect
86. #17989789D4semidirect
214. C293#3268semidirect
227. #449D293semidirect
233. C2×#41C2×D293semidirect
268. #103C2×D293semidirect
272. #102C2×D293semidirect
274. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276442
#340367982
#340367762
#340365172
#340363742
#340363632
#340363492
#340363232
#340362942
#340362762
#340362612
#340362322
#340361972
#449×C2932
#3268293