Finite Group Databasebeta

← #36364375

Subgroups — #36364375

Order 22144.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011998C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic, complemented
3#26011820C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011741C211yesmaximal, characteristic, complemented
9#26011740C211yesmaximal, characteristic, complemented
10#26011734C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011453C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173—21no—
17C2×D4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×#14583813C2211yescharacteristic, complemented
20C2×#14583811C2211yescharacteristic, complemented
21C2×#14583809C2211yescharacteristic, complemented
22C2×#14583809C2211yescharacteristic, complemented
23#19149660—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic, complemented
26#19149654C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149652—21no—
29#19149651—21no—
30#19149650C2211yescharacteristic
31#19149650C2211yescharacteristic
32#19149650C2211yescharacteristic, complemented
33#19149647—21no—
34C4×C22⋊D173—21no—
35C4×C22⋊D173—21no—
36#19149638C2211yescharacteristic, complemented
37C4⋊C4×D173C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173D411yescharacteristic
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68D4×D173—41no—
69D4×D173—41no—
70D4×D173—44no—
71C22×Dic173D411yescharacteristic
72C2×C4×D173—21no—
73C2×C4×D173—41no—
74C2×C4×D173—41no—
75C2×C4×D173D411yescharacteristic
76C2×C4×D173D411yescharacteristic
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2311yescharacteristic
79#14583813—41no—
80#14583811—41no—
81#14583809—41no—
82#14583809—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99C4×Dic173—21no—
100#41×C173—41no—
101#41×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173C2311yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173—21no—
108C22×C4×C173C2311yescharacteristic
109SD16×C173—41no—
110D8×C173—41no—
111C2×C8×C173C2311yescharacteristic
112C2×C8×C173C2311yescharacteristic
113C4⋊C4×C173C2311yescharacteristic
114C22⋊C4×C173—21no—
115C22×D173—22no—
116C22×D173—21no—
117C22×D173—41no—
118C22×D173—41no—
119C22×D173—42no—
120C22×D173—42no—
121C22×D173—42no—
122C22×D173C2×D411yescharacteristic
123C22×D173C2×D411yescharacteristic
124D692—21no—
125D692—21no—
126D692—42no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—41no—
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—41no—
143C4×D173—41no—
144C4×D173—42no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173C2×D411yescharacteristic
150Q8×C173—21no—
151Q8×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—42no—
157D4×C173—42no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—42no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2411yescharacteristic
165C8×C173—21no—
166C8×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173C22×D411yescharacteristic
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—44no—
183C2×D173—41no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—84no—
187C2×D173AGL(1,ℤ/8)12yes—
188C4×C173—21no—
189C4×C173—41no—
190C4×C173—41no—
191C4×C173H(5,𝔽2)11yescharacteristic
192C4×C173H(5,𝔽2)11yescharacteristic
193Dic173—21no—
194Dic173—21no—
195Dic173—41no—
196Dic173—41no—
197Dic173—41no—
198C2×C173—41no—
199C2×C173—81no—
200C2×C173—81no—
201C2×C173#53411yescharacteristic
202C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C173#57511yescharacteristic
204D173—22no—
205D173—41no—
206D173—81no—
207C173#319411yescharacteristic, complemented, Sylow
208#3194—1731nomaximal, Sylow
209#525—1731no—
210C2×#121—1731no—
211#483—1731no—
212#482—1731no—
213#482—1731no—
214#480—1731no—
215#469—1731no—
216#469—1731no—
217#468—1731no—
218#468—1731no—
219#450—1731no—
220#449—1731no—
221#449D17311yescharacteristic, complemented
222#447—1731no—
223#407—1731no—
224C2×#41C2×D17311yescharacteristic, complemented
225C22×D4—3461no—
226C22×D4—1731no—
227C23×C4—1731no—
228C2×SD16—1731no—
229C2×SD16C2×D17311yescharacteristic, complemented
230C2×D8—1731no—
231C2×D8C2×D17311yescharacteristic, complemented
232C2×M16—1731no—
233C2×M16—1731no—
234#123—3461no—
235C22⋊Q8—1731no—
236C22⋊Q8—1731no—
237#121—3461no—
238#121—3461no—
239#121—1731no—
240#121—1731no—
241#121—1731no—
242#121—1731no—
243#121—1731no—
244#121C2×D17311yescharacteristic
245#120—3461no—
246C4×D4—3461no—
247C4×D4—3461no—
248#117—1731no—
249C2×C4⋊C4—1731no—
250C2×C22⋊C4—3461no—
251#107—1731no—
252#107—1731no—
253#106—1731no—
254#106—1731no—
255#103—1731no—
256#103C2×D17311yescharacteristic, complemented
257#102—1731no—
258#102—1731no—
259#102—1731no—
260#102—1731no—
261#102—1731no—
262#102C2×D17311yescharacteristic, complemented
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265C22⋊C8—1731no—
266C22⋊C8C2×D17311yescharacteristic, complemented
267C24—3461no—
268C24—1731no—
269#41—41no—
270#41—41no—
271C2×Q8C22×D17311yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4—6921no—
282C2×D4—6921no—
283C2×D4—6924no—
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—21no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4—1731no—
290C22×C4—1731no—
291C22×C4—1731no—
292C22×C4—1731no—
293C22×C4—6921no—
294C22×C4—6921no—
295C22×C4C22×D17311yescharacteristic
296SD16—41no—
297SD16—6921no—
298D8—41no—
299D8—6921no—
300M16—6921no—
301M16—6921no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C42—3461no—
323C23—21no—
324C23—21no—
325C23—3462no—
326C23—3461no—
327C23—1731no—
328C23—1731no—
329C23—6921no—
330C23—6921no—
331C23—6922no—
332C23—6922no—
333C23—6922no—
334C23D4×D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—3461no—
344D4—3461no—
345D4—6922no—
346D4—6922no—
347D4—6922no—
348D4—6922no—
349D4—6922no—
350D4—6922no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—1731no—
364C2×C4—1731no—
365C2×C4—6921no—
366C2×C4—6921no—
367C2×C4—6922no—
368C2×C4—6921no—
369C2×C4D4×D17311yescharacteristic
370C2×C4C23×D17311yescharacteristic
371C8—21no—
372C8—21no—
373C8—3461no—
374C8—3461no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—3461no—
385C22—3461no—
386C22—3461no—
387C22—3461no—
388C22—3461no—
389C22—13844no—
390C22—1732no—
391C22—6924no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—3461no—
400C4—3461no—
401C4—6921no—
402C4—6921no—
403C4—6921no—
404C4#1914972411yescharacteristic
405C4#1914972411yescharacteristic
406C2—41no—
407C2—81no—
408C2—81no—
409C2—3462no—
410C2—13841no—
411C2—6921no—
412C2#2601205111yescharacteristic
413C2AGL(1,ℤ/8)×D17311yescharacteristic
414C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3194
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26011998C2semidirect
2. #121×D173C2semidirect
3. #26011820C2semidirect
4. #26011816C2semidirect
5. #26011812C2semidirect
6. #26011803C2semidirect
7. #26011743C2semidirect
8. #26011741C2semidirect
9. #26011740C2semidirect
10. #26011734C2semidirect
11. #26011640C2semidirect
12. #26011636C2semidirect
13. #26011623C2semidirect
14. #26011453C2semidirect
15. #449×C173C2semidirect
19. C2×#14583813C22semidirect
20. C2×#14583811C22semidirect
21. C2×#14583809C22semidirect
22. C2×#14583809C22semidirect
25. #19149655C22semidirect
32. #19149650C22semidirect
36. #19149638C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149603C22semidirect
44. #19149601C22semidirect
45. #19149601C22semidirect
46. #19149600C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
49. C22⋊#13063436C22semidirect
52. C2×#41×C173C22semidirect
53. C2×SD16×C173C22semidirect
54. C2×D8×C173C22semidirect
56. #103×C173C22semidirect
57. #102×C173C22semidirect
58. C22⋊C8×C173C22semidirect
207. C173#3194semidirect
221. #449D173semidirect
224. C2×#41C2×D173semidirect
229. C2×SD16C2×D173semidirect
231. C2×D8C2×D173semidirect
256. #103C2×D173semidirect
262. #102C2×D173semidirect
266. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
#121×D1732
#260118202
#260118162
#260118122
#260118032
#260117432
#260117412
#260117402
#260117342
#260116402
#260116362
#260116232
#260114532
#449×C1732
#3194173