Finite Group Databasebeta

← #36364378

Subgroups — #36364378

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#26012004C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic, complemented
3#26011820C211yesmaximal, characteristic, complemented
4#26011818C211yesmaximal, characteristic, complemented
5#26011814C211yesmaximal, characteristic, complemented
6#26011806C211yesmaximal, characteristic, complemented
7#26011744C211yesmaximal, characteristic, complemented
8#26011742C211yesmaximal, characteristic, complemented
9#26011739C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011639C211yesmaximal, characteristic, complemented
12#26011635C211yesmaximal, characteristic, complemented
13#26011626C211yesmaximal, characteristic, complemented
14#26011454C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C2×#14583813C2211yescharacteristic, complemented
21C2×D1384C2211yescharacteristic, complemented
22C2×#14583811C2211yescharacteristic, complemented
23C2×#14583810C2211yescharacteristic, complemented
24#19149664—21no—
25#19149655C2211yescharacteristic
26#19149654C2211yescharacteristic
27#19149654C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic, complemented
30#19149652—21no—
31#19149651—21no—
32#19149650C2211yescharacteristic
33#19149649—21no—
34#19149641—21no—
35#19149639—21no—
36#19149637C2211yescharacteristic, complemented
37C4⋊C4×D173C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×M16×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173D411yescharacteristic
61#14583827—41no—
62#14583827—41no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72D4×D173—41no—
73D4×D173—41no—
74D4×D173—44no—
75C22×Dic173D411yescharacteristic
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—41no—
79C2×C4×D173—41no—
80C2×C4×D173D411yescharacteristic
81C2×C4×D173D411yescharacteristic
82C2×C4×D173C2311yescharacteristic
83C2×C4×D173C2311yescharacteristic
84#14583813—41no—
85D1384—41no—
86#14583811—41no—
87#14583810—41no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94C22⋊Dic173—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101C4×Dic173—21no—
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106SD16×C173—41no—
107M16×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173—21no—
111C4⋊C4×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22⋊C4×C173—21no—
115C22×D173—22no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—41no—
119C22×D173—41no—
120C22×D173—42no—
121C22×D173—42no—
122C22×D173—42no—
123C22×D173C2×D411yescharacteristic
124C22×D173C2×D411yescharacteristic
125D692—21no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130D692—42no—
131D692—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—42no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—41no—
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—21no—
144C4×D173—21no—
145C4×D173—41no—
146C4×D173—41no—
147C4×D173—42no—
148C4×D173—42no—
149#13063436—21no—
150#13063436—21no—
151C23×C173—21no—
152C23×C173C2×D411yescharacteristic
153Q8×C173—21no—
154Q8×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—42no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
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×C173C22×D411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—44no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
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×C173—41no—
192C4×C173H(5,𝔽2)11yescharacteristic
193C4×C173H(5,𝔽2)11yescharacteristic
194Dic173—21no—
195Dic173—21no—
196Dic173—41no—
197Dic173—41no—
198C2×C173—41no—
199C2×C173—81no—
200C2×C173#53411yescharacteristic
201C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
202C2×C173#57511yescharacteristic
203D173—22no—
204D173—41no—
205D173—81no—
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#468D17311yescharacteristic, complemented
219#450—1731no—
220#449—1731no—
221#449—1731no—
222#447—1731no—
223#407—1731no—
224C2×#41—1731no—
225C22×D4—3461no—
226C22×D4—1731no—
227C23×C4—1731no—
228C2×SD16—1731no—
229C2×SD16C2×D17311yescharacteristic, complemented
230C2×D8—1731no—
231C2×D8—1731no—
232C2×M16—1731no—
233C2×M16C2×D17311yescharacteristic, complemented
234#123—3461no—
235C22⋊Q8—1731no—
236C22⋊Q8C2×D17311yescharacteristic, complemented
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#107C2×D17311yescharacteristic, complemented
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⋊C8—1731no—
267C24—3461no—
268C24—1731no—
269#41—6921no—
270#41—6921no—
271C2×Q8C22×D17311yescharacteristic
272C2×D4—21no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—6921no—
283C2×D4—6921no—
284C2×D4—6924no—
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
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—6921no—
299D8—6921no—
300M16—41no—
301M16—6921no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4C22×D17311yescharacteristic
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
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—3462no—
325C23—3461no—
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—42no—
340D4—3461no—
341D4—3461no—
342D4—3461no—
343D4—3461no—
344D4—6922no—
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—21no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—1731no—
363C2×C4—1731no—
364C2×C4—6921no—
365C2×C4—6921no—
366C2×C4—6922no—
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—3461no—
382C22—3461no—
383C22—3461no—
384C22—3461no—
385C22—3461no—
386C22—13844no—
387C22—1732no—
388C22—6924no—
389C22—6921no—
390C22—6921no—
391C22—6921no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—3461no—
401C4—3461no—
402C4—6921no—
403C4—6921no—
404C4#1914972411yescharacteristic
405C4#1914972511yescharacteristic
406C2—41no—
407C2—81no—
408C2—3462no—
409C2—13841no—
410C2—13841no—
411C2—6921no—
412C2#2601204911yescharacteristic
413C2AGL(1,ℤ/8)×D17311yescharacteristic
414C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3194
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. #26012004C2semidirect
2. #121×D173C2semidirect
3. #26011820C2semidirect
4. #26011818C2semidirect
5. #26011814C2semidirect
6. #26011806C2semidirect
7. #26011744C2semidirect
8. #26011742C2semidirect
9. #26011739C2semidirect
10. #26011640C2semidirect
11. #26011639C2semidirect
12. #26011635C2semidirect
13. #26011626C2semidirect
14. #26011454C2semidirect
15. #468×C173C2semidirect
16. C2×#14583827C22semidirect
20. C2×#14583813C22semidirect
21. C2×D1384C22semidirect
22. C2×#14583811C22semidirect
23. C2×#14583810C22semidirect
29. #19149653C22semidirect
36. #19149637C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149603C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149599C22semidirect
47. #19149599C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
52. C2×SD16×C173C22semidirect
53. C2×M16×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
207. C173#3194semidirect
218. #468D173semidirect
229. C2×SD16C2×D173semidirect
233. C2×M16C2×D173semidirect
236. C22⋊Q8C2×D173semidirect
252. #107C2×D173semidirect
256. #103C2×D173semidirect
262. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
#121×D1732
#260118202
#260118182
#260118142
#260118062
#260117442
#260117422
#260117392
#260116402
#260116392
#260116352
#260116262
#260114542
#468×C1732
#3194173