Finite Group Databasebeta

← #36365169

Subgroups — #36365169

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×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3#26012136C211yesmaximal, characteristic, complemented
4#26012134C211yesmaximal, characteristic, complemented
5#26011873C211yesmaximal, characteristic, complemented
6#26011740C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011568C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic, complemented
20#19149706—24no—
21#19149703—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic, complemented
25C2×#14583813C2211yescharacteristic, complemented
26C2×D1384C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583809C2211yescharacteristic, complemented
30C2×#14583809C2211yescharacteristic, complemented
31#19149667C2211yescharacteristic, complemented
32#19149663—21no—
33#19149661—21no—
34#19149659C2211yescharacteristic, complemented
35#19149655C2211yescharacteristic, complemented
36#19149654C2211yescharacteristic
37#19149653—21no—
38#19149652—21no—
39#19149650C2211yescharacteristic
40#19149649—21no—
41#19149649—21no—
42C4×D692C2211yescharacteristic
43C4×D692C2211yescharacteristic, complemented
44C2×#14583803—21no—
45C22⋊C4×D173—21no—
46#19149612C2211yescharacteristic, complemented
47#19149604C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C4×#13063436C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583827—22no—
64#14583827—41no—
65#14583825—22no—
66#14583825—41no—
67C2×Dic346C2311yescharacteristic
68C2×D692—21no—
69C2×D692—21no—
70C2×D692D411yescharacteristic
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77D4×D173—22no—
78D4×D173—22no—
79D4×D173—41no—
80D4×D173—41no—
81C22×Dic173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85C2×C4×D173C2311yescharacteristic
86#14583816—41no—
87#14583815—22no—
88#14583814—22no—
89#14583813—22no—
90D1384—22no—
91#14583811—41no—
92#14583811D414yescomplemented
93#14583809—22no—
94#14583809—22no—
95C2×#13063436C2311yescharacteristic
96C2×#13063436C2311yescharacteristic
97C4⋊Dic173—21no—
98C4⋊Dic173D411yescharacteristic
99C4⋊Dic173C2311yescharacteristic
100#14583805—21no—
101C22⋊Dic173—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803D411yescharacteristic
109#14583803D411yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114SD16×C173—22no—
115D8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173C2311yescharacteristic
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—81no—
130C22×D173—81no—
131C22×D173C2×D411yescharacteristic
132C22×D173C2×D411yescharacteristic
133Dic346—21no—
134Dic346—21no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—41no—
139D692—42no—
140D692—81no—
141D692C2×D412yes—
142C22⋊D173—42no—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C22⋊D173—81no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—42no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—21no—
154C4×D173—21no—
155C4×D173—42no—
156C4×D173—81no—
157#13063436—21no—
158#13063436C2×D412yes—
159C23×C173—21no—
160C23×C173—21no—
161Q8×C173—21no—
162Q8×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—21no—
166D4×C173C2×D412yes—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—81no—
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173—21no—
175C22×C173—22no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—22no—
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—21no—
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—21no—
198C4×C173—41no—
199C4×C173—41no—
200C4×C173C22×D411yescharacteristic
201C4×C173H(5,𝔽2)11yescharacteristic
202Dic173—21no—
203Dic173—21no—
204Dic173—41no—
205Dic173—41no—
206C2×C173—42no—
207C2×C173—81no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—41no—
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#468—1731no—
224#463—1731no—
225#463D17311yescharacteristic, complemented
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×D8C2×D17311yescharacteristic
247C2×M16—1731no—
248C2×M16—1731no—
249#124—1731no—
250#124C2×D17311yescharacteristic, complemented
251#123—3461no—
252#123—3461no—
253C22⋊Q8—1731no—
254C22⋊Q8—1731no—
255#121—3461no—
256#121—3461no—
257#121—1731no—
258#121—1731no—
259#120—3461no—
260#120—3461no—
261C4×D4—1731no—
262C4×D4—1731no—
263C4×D4C2×D17311yescharacteristic, complemented
264C2×C22⋊C4—3461no—
265C2×C22⋊C4—3461no—
266#107—1731no—
267C4⋊C8C2×D17311yescharacteristic, complemented
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—3461no—
286C2×D4—3461no—
287C2×D4—3462no—
288C2×D4—3462no—
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—1731no—
294C2×D4—1731no—
295C2×D4—1731no—
296C2×D4—6921no—
297C2×D4—6921no—
298C2×D4C22×D17311yescharacteristic
299C2×D4C22×D17311yescharacteristic
300C22×C4—21no—
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—1731no—
305C22×C4—1731no—
306Q16—6921no—
307SD16—22no—
308SD16—3462no—
309SD16—3462no—
310SD16—3462no—
311D8—22no—
312D8—3462no—
313D8—1734no—
314D8—6921no—
315M16—3462no—
316M16—3462no—
317C2×C8—1731no—
318C2×C8—1731no—
319C2×C8C22×D17311yescharacteristic
320C2×C8C22×D17311yescharacteristic
321C4⋊C4—3461no—
322C4⋊C4—3461no—
323C4⋊C4—1731no—
324C4⋊C4—1731no—
325C4⋊C4C22×D17311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
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—
338C42C22×D17311yescharacteristic
339C23—21no—
340C23—21no—
341C23—3461no—
342C23—3461no—
343C23—13841no—
344C23—13841no—
345C23—1731no—
346C23—1731no—
347C23—6922no—
348C23—6922no—
349C23—6922no—
350C23—6922no—
351Q8—21no—
352Q8—21no—
353Q8—3461no—
354Q8—3461no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—3461no—
359D4—3461no—
360D4—3461no—
361D4—13841no—
362D4—13841no—
363D4—1732no—
364D4—6921no—
365D4—6922no—
366D4—6922no—
367D4—6922no—
368D4—6922no—
369D4D4×D17312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—13841no—
380C2×C4—1731no—
381C2×C4—1731no—
382C2×C4—6922no—
383C2×C4—6922no—
384C2×C4D4×D17311yescharacteristic
385C2×C4D4×D17311yescharacteristic
386C2×C4C23×D17311yescharacteristic
387C8—21no—
388C8—21no—
389C8—3461no—
390C8—1732no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3461no—
398C22—3461no—
399C22—3461no—
400C22—3461no—
401C22—3461no—
402C22—3461no—
403C22—3462no—
404C22—13844no—
405C22—13844no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22C2×D4×D17311yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3461no—
417C4—3461no—
418C4—6921no—
419C4—6921no—
420C4C2×D4×D17311yescharacteristic
421C4#1914972511yescharacteristic
422C2—42no—
423C2—81no—
424C2—13841no—
425C2—6921no—
426C2—6921no—
427C2—6922no—
428C2#2601212511yescharacteristic
429C2#2601228811yescharacteristic
430C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3273
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149703C2semidirect
3. #26012136C2semidirect
4. #26012134C2semidirect
5. #26011873C2semidirect
6. #26011740C2semidirect
7. #26011736C2semidirect
8. #26011707C2semidirect
9. #26011656C2semidirect
10. #26011650C2semidirect
11. #26011626C2semidirect
12. #26011623C2semidirect
13. C4×#14583811C2semidirect
14. #26011568C2semidirect
15. #463×C173C2semidirect
18. C2×D4×D173C22semidirect
19. C2×D4×D173C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583814C22semidirect
25. C2×#14583813C22semidirect
29. C2×#14583809C22semidirect
30. C2×#14583809C22semidirect
31. #19149667C22semidirect
34. #19149659C22semidirect
35. #19149655C22semidirect
43. C4×D692C22semidirect
46. #19149612C22semidirect
47. #19149604C22semidirect
48. #19149601C22semidirect
49. #19149600C22semidirect
50. #19149599C22semidirect
51. #19149591C22semidirect
52. #19149591C22semidirect
53. C4×#13063436C22semidirect
54. C2×SD16×C173C22semidirect
56. #124×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
92. #14583811D4semidirect
215. C173#3273semidirect
225. #463D173semidirect
242. C2×SD16C2×D173semidirect
250. #124C2×D173semidirect
263. C4×D4C2×D173semidirect
267. C4⋊C8C2×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×#191497062
C2×#191497032
#260121362
#260121342
#260118732
#260117402
#260117362
#260117072
#260116562
#260116502
#260116262
#260116232
C4×#145838112
#260115682
#463×C1732
#3273173