Finite Group Databasebeta

← #49170331

Subgroups — #49170331

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×#22627648C211yesmaximal, characteristic, complemented
2C2×#22627643C211yesmaximal, characteristic, complemented
3#34036799C211yesmaximal, characteristic, complemented
4#34036774C211yesmaximal, characteristic, complemented
5#34036516C211yesmaximal, characteristic, complemented
6#34036374C211yesmaximal, characteristic, complemented
7#34036364C211yesmaximal, characteristic, complemented
8#34036352C211yesmaximal, characteristic, complemented
9#34036292C211yesmaximal, characteristic, complemented
10#34036275C211yesmaximal, characteristic, complemented
11#34036271C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036229C211yesmaximal, characteristic, complemented
14#34036199C211yesmaximal, characteristic, complemented
15#459×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×#17989801C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic
20#22627648—24no—
21#22627643—24no—
22C2×Dic1172C2211yescharacteristic, complemented
23C2×#17989791C2211yescharacteristic
24C2×#17989790C2211yescharacteristic
25C2×#17989790C2211yescharacteristic, complemented
26C2×#17989790C2211yescharacteristic, complemented
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989785C2211yescharacteristic
30C2×C8×D293C2211yescharacteristic, complemented
31#22627620—21no—
32#22627612C2211yescharacteristic
33#22627608—21no—
34#22627604C2211yescharacteristic
35#22627599C2211yescharacteristic
36#22627598—21no—
37#22627596—21no—
38#22627596—21no—
39#22627595C2211yescharacteristic
40#22627595C2211yescharacteristic, complemented
41#22627594—21no—
42C4×Dic586C2211yescharacteristic, complemented
43C4×D1172C2211yescharacteristic
44D4×Dic293—21no—
45C2×#17989779—21no—
46#22627557C2211yescharacteristic, complemented
47#22627549C2211yescharacteristic, complemented
48#22627547C2211yescharacteristic, complemented
49#22627546C2211yescharacteristic
50#22627545C2211yescharacteristic
51#22627536C2211yescharacteristic
52#22627536C2211yescharacteristic, complemented
53#22627535C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic
55C2×D8×C293C2211yescharacteristic
56#127×C293C2211yescharacteristic, complemented
57C4×D4×C293C2211yescharacteristic
58C4⋊C8×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989801—22no—
63#17989801—22no—
64#17989801—22no—
65#17989801—41no—
66#17989801—41no—
67#17989801—41no—
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70C2×D1172—21no—
71C2×D1172D411yescharacteristic
72C2×D1172C2311yescharacteristic
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78C2×C22⋊D293—21no—
79D4×D293—22no—
80D4×D293—41no—
81C22×Dic293—21no—
82C22×Dic293—21no—
83C22×Dic293—21no—
84C2×C4×D293C2311yescharacteristic
85C2×C4×D293C2311yescharacteristic
86Dic1172—22no—
87#17989791—22no—
88#17989790—22no—
89#17989790—41no—
90#17989790D414yescomplemented
91#17989787—22no—
92#17989787—41no—
93#17989785—22no—
94C2×#14556873C2311yescharacteristic
95C2×#14556873C2311yescharacteristic
96C8×D293—22no—
97C4⋊Dic293—21no—
98C4⋊Dic293D411yescharacteristic
99C4⋊Dic293C2311yescharacteristic
100#17989781—21no—
101C22⋊Dic293—21no—
102C22⋊Dic293—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779D411yescharacteristic
108#17989779D411yescharacteristic
109C4×Dic293—21no—
110C2×D4×C293—21no—
111C2×D4×C293—21no—
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293—21no—
116D8×C293—22no—
117D8×C293—22no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293C2311yescharacteristic
121C22⋊C4×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586—21no—
131Dic586—21no—
132Dic586C2×D412yes—
133D1172—21no—
134D1172—21no—
135D1172—41no—
136D1172—81no—
137C22⋊D293—42no—
138C22⋊D293—42no—
139C22⋊D293—42no—
140C22⋊D293—42no—
141C22⋊D293—81no—
142C22⋊D293—81no—
143C2×Dic293—21no—
144C2×Dic293—21no—
145C2×Dic293—21no—
146C2×Dic293—42no—
147C2×Dic293—42no—
148C2×Dic293—42no—
149C2×Dic293C2×D411yescharacteristic
150C2×Dic293C2×D411yescharacteristic
151C4×D293—21no—
152C4×D293—21no—
153C4×D293—21no—
154C4×D293—21no—
155#14556873—21no—
156#14556873C2×D412yes—
157C23×C293—21no—
158C23×C293—21no—
159C23×C293—21no—
160D4×C293—21no—
161D4×C293—21no—
162D4×C293—21no—
163D4×C293—21no—
164D4×C293—21no—
165D4×C293—42no—
166D4×C293—81no—
167D4×C293C2×D412yes—
168C2×C4×C293—21no—
169C2×C4×C293—81no—
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293—21no—
175C22×C293—22no—
176C22×C293—41no—
177C22×C293—41no—
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—21no—
187C2×D293—21no—
188C2×D293—21no—
189C2×D293—21no—
190C2×D293—21no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—84no—
195C4×C293—21no—
196C4×C293—21no—
197C4×C293—41no—
198C4×C293C22×D411yescharacteristic
199C4×C293H(5,𝔽2)11yescharacteristic
200Dic293—21no—
201Dic293—21no—
202Dic293—22no—
203Dic293—41no—
204Dic293—41no—
205C2×C293—42no—
206C2×C293—81no—
207C2×C293—81no—
208C2×C293D4211yescharacteristic
209C2×C293#57511yescharacteristic
210C2×C293#57611yescharacteristic
211D293—41no—
212D293—41no—
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#459D29311yescharacteristic, complemented
225#452—2931no—
226#449—2931no—
227#449—2931no—
228#447—2931no—
229#440—2931no—
230#435—2931no—
231C2×#41—2931no—
232C2×#41—2931no—
233C2×#41—2931no—
234C22×D4—2931no—
235AGL(1,ℤ/8)—5864no—
236#135—5864no—
237C2×Q16—2931no—
238C2×SD16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16—2931no—
242C2×D8—2931no—
243C2×D8—2931no—
244C2×D8C2×D29311yescharacteristic
245C2×D8C2×D29311yescharacteristic
246C2×M16—2931no—
247C22×C8—2931no—
248#127—5861no—
249#127C2×D29311yescharacteristic, complemented
250#124—2931no—
251#123—5861no—
252C22⋊Q8—2931no—
253#121—2931no—
254#121—2931no—
255#121—2931no—
256#121—5861no—
257#121—5861no—
258#121—5861no—
259#120—5861no—
260C4×Q8—2931no—
261C4×D4—2931no—
262C4×D4—5861no—
263C4×D4C2×D29311yescharacteristic
264C2×C22⋊C4—5861no—
265#107—2931no—
266C4⋊C8C2×D29311yescharacteristic, complemented
267#103—2931no—
268#103—2931no—
269#102—2931no—
270#102—2931no—
271#102C2×D29311yescharacteristic, complemented
272#102C2×D29311yescharacteristic, complemented
273C22⋊C8—2931no—
274C22⋊C8—2931no—
275#97—2931no—
276C24—5861no—
277#41—5862no—
278#41—5862no—
279#41—5862no—
280#41—11721no—
281#41—11721no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8—2931no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2931no—
288C2×D4—2931no—
289C2×D4—5861no—
290C2×D4—5861no—
291C2×D4—5861no—
292C2×D4—5861no—
293C2×D4—5862no—
294C2×D4—5861no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—11721no—
298C2×D4C22×D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C2×D4C22×D29311yescharacteristic
301C22×C4—21no—
302C22×C4—2931no—
303C22×C4—2931no—
304C22×C4—5861no—
305C22×C4—5861no—
306C22×C4—5861no—
307Q16—5862no—
308SD16—2934no—
309SD16—5862no—
310SD16—5862no—
311SD16—11721no—
312D8—22no—
313D8—22no—
314D8—5862no—
315D8—11721no—
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—5861no—
337C42C22×D29311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2931no—
342C23—2931no—
343C23—23441no—
344C23—5861no—
345C23—11722no—
346C23—11722no—
347Q8—2932no—
348Q8—5861no—
349Q8—5861no—
350Q8—5861no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—23441no—
359D4—23441no—
360D4—23441no—
361D4—5861no—
362D4—5861no—
363D4—11721no—
364D4—11722no—
365D4—11722no—
366D4—11722no—
367D4—11722no—
368D4D4×D29312yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—2931no—
372C2×C4—2931no—
373C2×C4—5861no—
374C2×C4—5861no—
375C2×C4—5861no—
376C2×C4—5861no—
377C2×C4—5861no—
378C2×C4—5861no—
379C2×C4—5861no—
380C2×C4—11722no—
381C2×C4—11722no—
382C2×C4—11722no—
383C2×C4D4×D29311yescharacteristic
384C2×C4D4×D29311yescharacteristic
385C2×C4C23×D29311yescharacteristic
386C8—21no—
387C8—21no—
388C8—2932no—
389C8—5861no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—23444no—
400C22—5861no—
401C22—5861no—
402C22—5861no—
403C22—5861no—
404C22—5861no—
405C22—5861no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22C2×D4×D29311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5861no—
414C4—5861no—
415C4—5862no—
416C4—11721no—
417C4—11721no—
418C4C2×D4×D29311yescharacteristic
419C4#2262767011yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—23441no—
424C2—11721no—
425C2—11721no—
426C2#3403676311yescharacteristic
427C2#3403692511yescharacteristic
428C2#3403693011yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#459×C293C2
socleC22×C293C22×D4
hypercenter#459D293
hypocenterC293#3268
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C2×#22627643C2semidirect
3. #34036799C2semidirect
4. #34036774C2semidirect
5. #34036516C2semidirect
6. #34036374C2semidirect
7. #34036364C2semidirect
8. #34036352C2semidirect
9. #34036292C2semidirect
10. #34036275C2semidirect
11. #34036271C2semidirect
12. #34036261C2semidirect
13. #34036229C2semidirect
14. #34036199C2semidirect
15. #459×C293C2semidirect
18. C2×#17989801C22semidirect
22. C2×Dic1172C22semidirect
25. C2×#17989790C22semidirect
26. C2×#17989790C22semidirect
30. C2×C8×D293C22semidirect
40. #22627595C22semidirect
42. C4×Dic586C22semidirect
46. #22627557C22semidirect
47. #22627549C22semidirect
48. #22627547C22semidirect
52. #22627536C22semidirect
53. #22627535C22semidirect
56. #127×C293C22semidirect
58. C4⋊C8×C293C22semidirect
59. #102×C293C22semidirect
60. #102×C293C22semidirect
90. #17989790D4semidirect
214. C293#3268semidirect
224. #459D293semidirect
249. #127C2×D293semidirect
266. C4⋊C8C2×D293semidirect
271. #102C2×D293semidirect
272. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
C2×#226276432
#340367992
#340367742
#340365162
#340363742
#340363642
#340363522
#340362922
#340362752
#340362712
#340362612
#340362292
#340361992
#459×C2932
#3268293