Finite Group Databasebeta

← #49170310

Subgroups — #49170310

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×#22627648C211yesmaximal, characteristic, complemented
2#34036778C211yesmaximal, characteristic, complemented
3#34036767C211yesmaximal, characteristic, complemented
4#34036522C211yesmaximal, characteristic, complemented
5#34036384C211yesmaximal, characteristic, complemented
6#34036358C211yesmaximal, characteristic, complemented
7#34036346C211yesmaximal, characteristic, complemented
8#34036332C211yesmaximal, characteristic, complemented
9#34036291C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11#34036278C211yesmaximal, characteristic, complemented
12#34036274C211yesmaximal, characteristic, complemented
13#34036241C211yesmaximal, characteristic, complemented
14#34036200C211yesmaximal, characteristic, complemented
15C2×#135×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×D4×D293C2211yescharacteristic
18#22627648—24no—
19C2×#17989792C2211yescharacteristic, complemented
20C2×#17989791C2211yescharacteristic
21C2×#17989790C2211yescharacteristic
22C2×#17989789C2211yescharacteristic
23C2×#17989787C2211yescharacteristic
24C2×#17989785C2211yescharacteristic
25#22627622C2211yescharacteristic, complemented
26#22627613C2211yescharacteristic
27#22627607—21no—
28#22627607—21no—
29#22627605—21no—
30#22627603C2211yescharacteristic, complemented
31#22627600C2211yescharacteristic
32#22627600C2211yescharacteristic
33#22627598C2211yescharacteristic
34#22627596—21no—
35#22627596—21no—
36#22627592—21no—
37D4×Dic293C2211yescharacteristic
38D4×Dic293C2211yescharacteristic, complemented
39C4×C22⋊D293C2211yescharacteristic
40C2×#17989781—21no—
41C2×#17989779—21no—
42#22627556C2211yescharacteristic, complemented
43#22627553C2211yescharacteristic, complemented
44#22627549C2211yescharacteristic, complemented
45#22627548C2211yescharacteristic, complemented
46#22627548C2211yescharacteristic, complemented
47#22627546C2211yescharacteristic
48#22627545C2211yescharacteristic
49#22627545C2211yescharacteristic, complemented
50C22⋊#14556873C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic
52#22627534C2211yescharacteristic, complemented
53C2×#41×C293C2211yescharacteristic
54C2×#41×C293C2211yescharacteristic, complemented
55#135×C293—24no—
56C2×Q16×C293C2211yescharacteristic, complemented
57C2×SD16×C293C2211yescharacteristic
58C2×SD16×C293C2211yescharacteristic, complemented
59C2×D8×C293C2211yescharacteristic
60C22×C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989801—22no—
63#17989801—41no—
64C2×Dic586C2311yescharacteristic
65C2×D1172C2311yescharacteristic
66C2×C22⋊D293—21no—
67C2×C22⋊D293—21no—
68C2×C22⋊D293—21no—
69C2×C22⋊D293D411yescharacteristic
70D4×D293—22no—
71D4×D293—41no—
72C22×Dic293—21no—
73C22×Dic293—21no—
74C22×Dic293—21no—
75C2×C4×D293C2311yescharacteristic
76#17989792—41no—
77#17989791—22no—
78#17989790—22no—
79#17989789—41no—
80#17989787—22no—
81#17989785—22no—
82C2×#14556873C2311yescharacteristic
83C2×#14556873C2311yescharacteristic
84C4⋊Dic293C2311yescharacteristic
85C4⋊Dic293C2311yescharacteristic
86#17989781—21no—
87#17989781—21no—
88#17989781—21no—
89#17989781—21no—
90#17989781—21no—
91#17989781D411yescharacteristic
92C22⋊Dic293—21no—
93C22⋊Dic293—21no—
94C22⋊Dic293—21no—
95C22⋊Dic293D411yescharacteristic
96#17989779—21no—
97#17989779—21no—
98#17989779—21no—
99#17989779—21no—
100#17989779—21no—
101#17989779D411yescharacteristic
102C4×Dic293C2311yescharacteristic
103#41×C293—22no—
104#41×C293—22no—
105#41×C293—41no—
106#41×C293—41no—
107C2×Q8×C293C2311yescharacteristic
108C2×Q8×C293C2311yescharacteristic
109C2×D4×C293—21no—
110C2×D4×C293—21no—
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114C22×C4×C293—21no—
115C22×C4×C293C2311yescharacteristic
116Q16×C293—22no—
117SD16×C293—22no—
118SD16×C293—22no—
119D8×C293D414yescomplemented
120C2×C8×C293—22no—
121C2×C8×C293C2311yescharacteristic
122C2×C8×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128Dic586—21no—
129Dic586—21no—
130D1172—21no—
131D1172—21no—
132C22⋊D293—41no—
133C22⋊D293—42no—
134C22⋊D293—42no—
135C22⋊D293—81no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—42no—
140C2×Dic293—42no—
141C2×Dic293—81no—
142C2×Dic293C2×D411yescharacteristic
143C2×Dic293C2×D411yescharacteristic
144C2×Dic293C2×D411yescharacteristic
145C4×D293—21no—
146C4×D293—21no—
147#14556873—21no—
148#14556873—21no—
149C23×C293—21no—
150C23×C293—21no—
151C23×C293C2×D411yescharacteristic
152Q8×C293—21no—
153Q8×C293—21no—
154Q8×C293—21no—
155Q8×C293—21no—
156D4×C293—21no—
157D4×C293—21no—
158D4×C293—42no—
159D4×C293—42no—
160D4×C293C2×D412yes—
161D4×C293C2×D412yes—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—42no—
167C2×C4×C293—42no—
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2411yescharacteristic
170C8×C293—21no—
171C8×C293C2×D412yes—
172C22×C293—21no—
173C22×C293—22no—
174C22×C293—22no—
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—41no—
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293C22×D411yescharacteristic
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—21no—
185C2×D293—41no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—84no—
189C4×C293—21no—
190C4×C293—41no—
191C4×C293—41no—
192C4×C293C22×D411yescharacteristic
193C4×C293H(5,𝔽2)11yescharacteristic
194Dic293—21no—
195Dic293—21no—
196Dic293—21no—
197Dic293—41no—
198Dic293—41no—
199Dic293—41no—
200C2×C293—42no—
201C2×C293—41no—
202C2×C293—42no—
203C2×C293D4211yescharacteristic
204C2×C293#57511yescharacteristic
205C2×C293#57711yescharacteristic
206D293—41no—
207D293—81no—
208C293#326711yescharacteristic, complemented, Sylow
209#3267—2931nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2931no—
211C2×#135D29311yescharacteristic, complemented
212#547—2931no—
213#546—2931no—
214#497—2931no—
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×#41C2×D29311yescharacteristic
227C2×#41C2×D29311yescharacteristic, complemented
228C22×D4—2931no—
229AGL(1,ℤ/8)—5864no—
230#135—24no—
231C2×Q16—2931no—
232C2×Q16C2×D29311yescharacteristic, complemented
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—2931no—
236C2×SD16C2×D29311yescharacteristic
237C2×SD16C2×D29311yescharacteristic, complemented
238C2×D8—2931no—
239C2×D8C2×D29311yescharacteristic
240C2×M16—2931no—
241C22×C8C2×D29311yescharacteristic, complemented
242C4⋊Q8—2931no—
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×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#97—2931no—
270C24—5861no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—5862no—
276#41—11721no—
277C2×Q8—2931no—
278C2×Q8C22×D29311yescharacteristic
279C2×Q8C22×D29311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2931no—
283C2×D4—2931no—
284C2×D4—5861no—
285C2×D4—5861no—
286C2×D4—5862no—
287C2×D4—5861no—
288C2×D4—11721no—
289C2×D4C22×D29311yescharacteristic
290C2×D4C22×D29311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2931no—
294C22×C4—5861no—
295C22×C4—5861no—
296C22×C4—5861no—
297C22×C4C22×D29311yescharacteristic
298Q16—22no—
299Q16—11721no—
300SD16—22no—
301SD16—22no—
302SD16—5862no—
303SD16—5862no—
304SD16—11721no—
305D8—5862no—
306D8C22⋊D29314yescomplemented
307M16—5862no—
308C2×C8—22no—
309C2×C8—2931no—
310C2×C8—2931no—
311C2×C8C22×D29311yescharacteristic
312C2×C8C22×D29311yescharacteristic
313C4⋊C4—2931no—
314C4⋊C4—2931no—
315C4⋊C4—2931no—
316C4⋊C4—5861no—
317C4⋊C4—5861no—
318C4⋊C4—5861no—
319C4⋊C4—5861no—
320C4⋊C4—5861no—
321C22⋊C4—2931no—
322C22⋊C4—2931no—
323C22⋊C4—5861no—
324C22⋊C4—5861no—
325C22⋊C4—5861no—
326C22⋊C4—5861no—
327C22⋊C4—5861no—
328C22⋊C4—5861no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C42—2931no—
332C23—21no—
333C23—21no—
334C23—2931no—
335C23—23441no—
336C23—5861no—
337C23—11722no—
338C23—11722no—
339C23D4×D29311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5861no—
345Q8—5861no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—23441no—
351D4—5861no—
352D4—5861no—
353D4—11721no—
354D4—11722no—
355D4—11722no—
356D4C2×C22⋊D29312yes—
357D4C2×C22⋊D29312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—2931no—
365C2×C4—2931no—
366C2×C4—2931no—
367C2×C4—23441no—
368C2×C4—5861no—
369C2×C4—5861no—
370C2×C4—5861no—
371C2×C4—5861no—
372C2×C4—5861no—
373C2×C4—11722no—
374C2×C4—11722no—
375C2×C4D4×D29311yescharacteristic
376C2×C4C23×D29311yescharacteristic
377C8—21no—
378C8—5861no—
379C8—5861no—
380C8C2×C22⋊D29312yes—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—23444no—
391C22—5861no—
392C22—5861no—
393C22—5861no—
394C22—11721no—
395C22—11721no—
396C22—11721no—
397C22C2×D4×D29311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—5861no—
402C4—5861no—
403C4—5861no—
404C4—11721no—
405C4—11721no—
406C4—11721no—
407C4C22×C22⋊D29311yescharacteristic
408C4#2262766911yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—23441no—
413C2—11721no—
414C2D4×C22⋊D29311yescharacteristic
415C2#3403692611yescharacteristic
416C2#3403693611yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#135×C293C2
socleC22×C293C22×D4
hypercenterC2×#135D293
hypocenterC293#3267
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. #34036778C2semidirect
3. #34036767C2semidirect
4. #34036522C2semidirect
5. #34036384C2semidirect
6. #34036358C2semidirect
7. #34036346C2semidirect
8. #34036332C2semidirect
9. #34036291C2semidirect
10. #34036288C2semidirect
11. #34036278C2semidirect
12. #34036274C2semidirect
13. #34036241C2semidirect
14. #34036200C2semidirect
15. C2×#135×C293C2semidirect
19. C2×#17989792C22semidirect
25. #22627622C22semidirect
30. #22627603C22semidirect
38. D4×Dic293C22semidirect
42. #22627556C22semidirect
43. #22627553C22semidirect
44. #22627549C22semidirect
45. #22627548C22semidirect
46. #22627548C22semidirect
49. #22627545C22semidirect
50. C22⋊#14556873C22semidirect
52. #22627534C22semidirect
54. C2×#41×C293C22semidirect
56. C2×Q16×C293C22semidirect
58. C2×SD16×C293C22semidirect
60. C22×C8×C293C22semidirect
119. D8×C293D4semidirect
208. C293#3267semidirect
211. C2×#135D293semidirect
227. C2×#41C2×D293semidirect
232. C2×Q16C2×D293semidirect
237. C2×SD16C2×D293semidirect
241. C22×C8C2×D293semidirect
306. D8C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
#340367782
#340367672
#340365222
#340363842
#340363582
#340363462
#340363322
#340362912
#340362882
#340362782
#340362742
#340362412
#340362002
C2×#135×C2932
#3267293