Finite Group Databasebeta

← #36365171

Subgroups — #36365171

Order 22144.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149707C211yesmaximal, characteristic, complemented
2C2×#19149706C211yesmaximal, characteristic, complemented
3#26012138C211yesmaximal, characteristic, complemented
4#26012132C211yesmaximal, characteristic, complemented
5#26011872C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011573C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C22×D692C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic, complemented
20#19149707—24no—
21#19149706—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic, complemented
25C2×#14583813C2211yescharacteristic, complemented
26C2×D1384C2211yescharacteristic
27C2×D1384C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583811C2211yescharacteristic
30C2×#14583809C2211yescharacteristic, complemented
31#19149668C2211yescharacteristic, complemented
32#19149666C2211yescharacteristic, complemented
33#19149664—21no—
34#19149661—21no—
35#19149659C2211yescharacteristic, complemented
36#19149654—21no—
37#19149654C2211yescharacteristic
38#19149650—21no—
39#19149649—21no—
40#19149647—21no—
41#19149641C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic, complemented
43C4×C22⋊D173C2211yescharacteristic
44C22⋊C4×D173—21no—
45C22⋊C4×D173—21no—
46#19149608C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51C22⋊#13063436C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C8×Dic173C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×M16×C173C2211yescharacteristic, complemented
56C22⋊Q8×C173C2211yescharacteristic, complemented
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583827—22no—
64#14583827—41no—
65#14583826—22no—
66#14583826—41no—
67C2×Dic346C2311yescharacteristic
68C2×D692—22no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—41no—
73C2×D692C2311yescharacteristic
74C2×D692C2311yescharacteristic
75C2×D692C2311yescharacteristic
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173D411yescharacteristic
79D4×D173—22no—
80D4×D173—41no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173—21no—
85C2×C4×D173C2311yescharacteristic
86#14583816—41no—
87#14583815—22no—
88#14583815—22no—
89#14583813—22no—
90D1384—22no—
91D1384D414yescomplemented
92#14583811—22no—
93#14583811—41no—
94#14583809—22no—
95C2×#13063436C2311yescharacteristic
96C2×#13063436C2311yescharacteristic
97C4⋊Dic173—21no—
98#14583805D411yescharacteristic
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803D411yescharacteristic
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114SD16×C173—22no—
115M16×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—81no—
131C22×D173—81no—
132C22×D173C2×D411yescharacteristic
133Dic346—21no—
134Dic346—21no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—21no—
139D692—42no—
140D692—42no—
141D692—81no—
142D692C2×D412yes—
143D692C2×D412yes—
144C22⋊D173—41no—
145C22⋊D173—42no—
146C22⋊D173—42no—
147C2×Dic173—21no—
148C2×Dic173C2×D411yescharacteristic
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—42no—
154C4×D173—42no—
155C4×D173—81no—
156C4×D173—81no—
157#13063436—21no—
158#13063436—21no—
159C23×C173—21no—
160C23×C173C2×D411yescharacteristic
161Q8×C173—21no—
162Q8×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—81no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173C2×D412yes—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—21no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—22no—
184C2×D173—22no—
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—84no—
195C2×D173—84no—
196C4×C173—21no—
197C4×C173—41no—
198C4×C173—41no—
199C4×C173—41no—
200C4×C173C22×D411yescharacteristic
201C4×C173H(5,𝔽2)11yescharacteristic
202Dic173—21no—
203Dic173—21no—
204Dic173—21no—
205Dic173—41no—
206C2×C173—41no—
207C2×C173—81no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—42no—
213D173—42no—
214D173—81no—
215C173#327311yescharacteristic, complemented, Sylow
216#3273—1731nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1731no—
218C2×AGL(1,ℤ/8)—1731no—
219#546—1731no—
220#546—1731no—
221#495—1731no—
222#468—1731no—
223#468D17311yescharacteristic, complemented
224#463—1731no—
225#463—1731no—
226#452—1731no—
227#452—1731no—
228#447—1731no—
229#447—1731no—
230C4×D8—1731no—
231#436—1731no—
232C2×#41—1731no—
233C2×#41—1731no—
234C22×D4—1731no—
235C22×D4—1731no—
236AGL(1,ℤ/8)—3464no—
237AGL(1,ℤ/8)—3464no—
238C2×Q16—1731no—
239C2×SD16—1731no—
240C2×SD16—1731no—
241C2×SD16—1731no—
242C2×SD16C2×D17311yescharacteristic, complemented
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8—1731no—
246C2×D8—1731no—
247C2×M16—1731no—
248C2×M16C2×D17311yescharacteristic, complemented
249#124—1731no—
250#124—1731no—
251#123—3461no—
252#123—3461no—
253C22⋊Q8—1731no—
254C22⋊Q8C2×D17311yescharacteristic, complemented
255#121—3461no—
256#121—3461no—
257#121—1731no—
258#121C2×D17311yescharacteristic
259#120—3461no—
260#120—3461no—
261C4×D4—1731no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265C2×C22⋊C4—3461no—
266#107C2×D17311yescharacteristic, complemented
267C4⋊C8—1731no—
268#103—1731no—
269#103C2×D17311yescharacteristic, complemented
270#102—1731no—
271#102—1731no—
272#102—1731no—
273#102C2×D17311yescharacteristic, complemented
274C22⋊C8—1731no—
275C22⋊C8—1731no—
276C4×C8—1731no—
277C24—3461no—
278C24—3461no—
279#41—3462no—
280#41—3462no—
281#41—6921no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—3462no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—3462no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—1731no—
294C2×D4—1731no—
295C2×D4—1731no—
296C2×D4—1731no—
297C2×D4—6921no—
298C2×D4—6921no—
299C2×D4C22×D17311yescharacteristic
300C22×C4—3461no—
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—1731no—
305C22×C4C22×D17311yescharacteristic
306Q16—6921no—
307SD16—22no—
308SD16—3462no—
309SD16—3462no—
310SD16—3462no—
311D8—3462no—
312D8—3462no—
313D8—1734no—
314D8—6921no—
315M16—22no—
316M16—3462no—
317C2×C8—1731no—
318C2×C8—1731no—
319C2×C8C22×D17311yescharacteristic
320C2×C8C22×D17311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3461no—
323C4⋊C4—1731no—
324C4⋊C4C22×D17311yescharacteristic
325C4⋊C4C22×D17311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—1731no—
337C22⋊C4—1731no—
338C42—1731no—
339C23—21no—
340C23—3461no—
341C23—3461no—
342C23—3461no—
343C23—13841no—
344C23—13841no—
345C23—1731no—
346C23—6922no—
347C23—6922no—
348C23—6922no—
349C23—6922no—
350C23D4×D17311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3461no—
354Q8—3461no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—3461no—
359D4—3461no—
360D4—3461no—
361D4—3461no—
362D4—13841no—
363D4—1732no—
364D4—1732no—
365D4—6922no—
366D4—6922no—
367D4—6921no—
368D4—6922no—
369D4—6922no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—13841no—
379C2×C4—13841no—
380C2×C4—1731no—
381C2×C4—1731no—
382C2×C4—1731no—
383C2×C4—6922no—
384C2×C4—6922no—
385C2×C4D4×D17311yescharacteristic
386C2×C4C23×D17311yescharacteristic
387C8—21no—
388C8—3461no—
389C8—3461no—
390C8D4×D17312yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3461no—
398C22—3461no—
399C22—3461no—
400C22—3462no—
401C22—3462no—
402C22—13844no—
403C22—13844no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22C2×D4×D17311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3461no—
417C4—3461no—
418C4—3461no—
419C4—6921no—
420C4C2×D4×D17311yescharacteristic
421C4#1914972511yescharacteristic
422C2—41no—
423C2—81no—
424C2—13841no—
425C2—6921no—
426C2—6922no—
427C2—6922no—
428C2#2601212511yescharacteristic
429C2#2601229011yescharacteristic
430C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3273
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. C2×#19149706C2semidirect
3. #26012138C2semidirect
4. #26012132C2semidirect
5. #26011872C2semidirect
6. #26011738C2semidirect
7. #26011709C2semidirect
8. #26011705C2semidirect
9. #26011656C2semidirect
10. #26011651C2semidirect
11. #26011626C2semidirect
12. #26011624C2semidirect
13. #26011573C2semidirect
14. #26011567C2semidirect
15. #468×C173C2semidirect
18. C22×D692C22semidirect
19. C2×D4×D173C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583813C22semidirect
30. C2×#14583809C22semidirect
31. #19149668C22semidirect
32. #19149666C22semidirect
35. #19149659C22semidirect
41. #19149641C22semidirect
42. #19149641C22semidirect
46. #19149608C22semidirect
47. #19149602C22semidirect
48. #19149599C22semidirect
49. #19149599C22semidirect
50. #19149599C22semidirect
51. C22⋊#13063436C22semidirect
52. #19149591C22semidirect
53. C8×Dic173C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×M16×C173C22semidirect
56. C22⋊Q8×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
91. D1384D4semidirect
215. C173#3273semidirect
223. #468D173semidirect
242. C2×SD16C2×D173semidirect
248. C2×M16C2×D173semidirect
254. C22⋊Q8C2×D173semidirect
266. #107C2×D173semidirect
269. #103C2×D173semidirect
273. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C2×#191497062
#260121382
#260121322
#260118722
#260117382
#260117092
#260117052
#260116562
#260116512
#260116262
#260116242
#260115732
#260115672
#468×C1732
#3273173