Finite Group Databasebeta

← #468×D173 #36363147

Subgroups — #468×D173 #36363147

Order 22144.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D173C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3C22⋊Q8×D173C211yesmaximal, characteristic, complemented
4#121×D173C211yesmaximal, characteristic
5#26011740C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011738C211yesmaximal, characteristic, complemented
8#26011737C211yesmaximal, characteristic, complemented
9#26011736C211yesmaximal, characteristic, complemented
10#26011735C211yesmaximal, characteristic, complemented
11#26011734C211yesmaximal, characteristic, complemented
12#107×D173C211yesmaximal, characteristic, complemented
13#103×D173C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583813C2211yescharacteristic, complemented
23SD16×D173—24no—
24C2×#14583810C2211yescharacteristic, complemented
25C2×#14583809C2211yescharacteristic, complemented
26C2×#14583809C2211yescharacteristic, complemented
27M16×D173—24no—
28C2×C8×D173C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic
30#19149659C2211yescharacteristic, complemented
31#19149658C2211yescharacteristic, complemented
32#19149657—21no—
33#19149656—21no—
34#19149655C2211yescharacteristic, complemented
35#19149654C2211yescharacteristic
36#19149653C2211yescharacteristic
37#19149652—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40C4⋊C4×D173—21no—
41C4⋊C4×D173C2211yescharacteristic
42C4⋊C4×D173C2211yescharacteristic
43C22⋊C4×D173—21no—
44C22⋊C4×D173—21no—
45#19149613C2211yescharacteristic, complemented
46#19149612C2211yescharacteristic, complemented
47#19149612C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic, 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×D173D411yescharacteristic
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65Q8×D173—22no—
66Q8×D173—22no—
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72D4×D173—22no—
73D4×D173—22no—
74D4×D173—44no—
75C22×Dic173D411yescharacteristic
76C2×C4×D173—22no—
77C2×C4×D173—22no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173D411yescharacteristic
82C2×C4×D173D411yescharacteristic
83C2×C4×D173C2311yescharacteristic
84C2×C4×D173C2311yescharacteristic
85#14583815—22no—
86#14583814—22no—
87#14583813—22no—
88#14583810—22no—
89#14583809—22no—
90#14583809—22no—
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C8×D173—22no—
94C8×D173D414yescomplemented
95C4⋊Dic173—21no—
96C4⋊Dic173C2311yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805—21no—
100#14583805—21no—
101#14583805—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112SD16×C173—22no—
113M16×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173—21no—
117C4⋊C4×C173C2311yescharacteristic
118C4⋊C4×C173C2311yescharacteristic
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22×D173—22no—
122C22×D173—22no—
123C22×D173—22no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173C2×D411yescharacteristic
129C22×D173C2×D411yescharacteristic
130Dic346—21no—
131Dic346—21no—
132Dic346—42no—
133D692—21no—
134D692—21no—
135D692—42no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C2×Dic173—22no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—22no—
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—22no—
149C4×D173—42no—
150C4×D173—42no—
151C4×D173—42no—
152C4×D173C2×D412yes—
153C4×D173#4112yes—
154#13063436—21no—
155#13063436C2×D412yescomplemented
156C23×C173—21no—
157C23×C173C2×D411yescharacteristic
158Q8×C173—21no—
159Q8×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yescomplemented
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
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×D173#12112yes—
188C2×D173AGL(1,ℤ/8)12yes—
189C2×D173#13712yes—
190C4×C173—21no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173—41no—
194C4×C173C22×D411yescharacteristic
195C4×C173C2×#4111yescharacteristic
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200Dic173#12112yes—
201C2×C173—41no—
202C2×C173—81no—
203C2×C173C2×#12111yescharacteristic
204C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C173C2×#13711yescharacteristic
206D173—41no—
207D173—81no—
208D173#46814yesdirect product factor
209C173C2×#46811yescharacteristic, complemented, Sylow
210C2×#468—1731nomaximal, Sylow
211C22×SD16—1731no—
212C22×M16—1731no—
213C2×C22⋊Q8—1731no—
214C2×#121—1731no—
215#468—1731no—
216#468—1731no—
217#468—1731no—
218#468—1731no—
219#468—1731no—
220#468—1731no—
221#468—1731no—
222#468D17311yescharacteristic, direct product factor
223C2×#107—1731no—
224C2×#103—1731no—
225C2×#102—1731no—
226C22×Q8—1731no—
227C22×D4—3461no—
228C22×D4—1731no—
229C23×C4—1731no—
230C2×SD16—3464no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16C2×D17311yescharacteristic, complemented
235C2×M16—3464no—
236C2×M16—1731no—
237C2×M16—1731no—
238C2×M16—1731no—
239C2×M16C2×D17311yescharacteristic, complemented
240C22×C8—1731no—
241C22×C8—1731no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8—1731no—
247C22⋊Q8C2×D17311yescharacteristic, complemented
248#121—3461no—
249#121—3461no—
250#121—1731no—
251#121—1731no—
252#121—1731no—
253#121C2×D17311yescharacteristic
254C2×C4⋊C4—3461no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
257C2×C22⋊C4—3461no—
258C2×C22⋊C4—3461no—
259#107—1731no—
260#107—1731no—
261#107—1731no—
262#107C2×D17311yescharacteristic, complemented
263#103—1731no—
264#103—1731no—
265#103—1731no—
266#103C2×D17311yescharacteristic, complemented
267#102—1731no—
268#102—1731no—
269#102—1731no—
270#102C2×D17311yescharacteristic, complemented
271C24—3461no—
272C24—1731no—
273C2×Q8—3461no—
274C2×Q8—3462no—
275C2×Q8—3462no—
276C2×Q8—1731no—
277C2×Q8C22×D17311yescharacteristic
278C2×D4—21no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3462no—
283C2×D4—3462no—
284C2×D4—3461no—
285C2×D4—1731no—
286C2×D4—6924no—
287C2×D4C22×D17311yescharacteristic
288C22×C4—3462no—
289C22×C4—3462no—
290C22×C4—3461no—
291C22×C4—3461no—
292C22×C4—3461no—
293C22×C4—1731no—
294C22×C4—1731no—
295C22×C4—1731no—
296C22×C4—1731no—
297C22×C4—1731no—
298C22×C4C22×D17311yescharacteristic
299SD16—22no—
300SD16—3462no—
301SD16—3462no—
302SD16—3462no—
303M16—22no—
304M16—3462no—
305M16—3462no—
306M16—3462no—
307C2×C8—3462no—
308C2×C8—1731no—
309C2×C8—1731no—
310C2×C8—1734no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—3461no—
318C4⋊C4—3461no—
319C4⋊C4—1731no—
320C4⋊C4—1731no—
321C4⋊C4C22×D17311yescharacteristic
322C4⋊C4C22×D17311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C23—21no—
332C23—3462no—
333C23—3462no—
334C23—3462no—
335C23—3461no—
336C23—1731no—
337C23—1731no—
338C23—6922no—
339C23—6922no—
340C23—6922no—
341C23D4×D17311yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—3461no—
345Q8—3461no—
346Q8—6922no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—3461no—
351D4—3461no—
352D4—6922no—
353D4—6922no—
354D4—6922no—
355D4—6922no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—3462no—
362C2×C4—3462no—
363C2×C4—3462no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—1732no—
372C2×C4—1732no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4—6922no—
376C2×C4D4×D17311yescharacteristic
377C2×C4C23×D17311yescharacteristic
378C8—21no—
379C8—3461no—
380C8—1732no—
381C8D4×D17312yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—3461no—
389C22—3461no—
390C22—3461no—
391C22—13844no—
392C22—1732no—
393C22—1732no—
394C22—1732no—
395C22—6924no—
396C22—6921no—
397C22—6921no—
398C22—6921no—
399C22C2×D4×D17311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—3461no—
405C4—1732no—
406C4—6921no—
407C4—6921no—
408C4—6921no—
409C4C2×D4×D17311yescharacteristic
410C4#41×D17311yescharacteristic
411C2—41no—
412C2—81no—
413C2—13841no—
414C2—1734no—
415C2—6921no—
416C2#121×D17311yescharacteristic
417C2AGL(1,ℤ/8)×D17311yescharacteristic
418C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#468D173
hypocenterC173C2×#468
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×SD16×D173C2semidirect
2. C2×M16×D173C2semidirect
3. C22⋊Q8×D173C2semidirect
5. #26011740C2semidirect
6. #26011739C2semidirect
7. #26011738C2semidirect
8. #26011737C2semidirect
9. #26011736C2semidirect
10. #26011735C2semidirect
11. #26011734C2semidirect
12. #107×D173C2semidirect
13. #103×D173C2semidirect
14. #102×D173C2semidirect
15. #468×C173C2semidirect
20. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583813C22semidirect
24. C2×#14583810C22semidirect
25. C2×#14583809C22semidirect
26. C2×#14583809C22semidirect
30. #19149659C22semidirect
31. #19149658C22semidirect
34. #19149655C22semidirect
45. #19149613C22semidirect
46. #19149612C22semidirect
47. #19149612C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
51. #19149601C22semidirect
52. #19149600C22semidirect
53. #19149599C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×M16×C173C22semidirect
56. C22⋊Q8×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
94. C8×D173D4semidirect
155. #13063436C2×D4semidirect
171. C8×C173C2×D4semidirect
208. D173#468direct (N × H)
209. C173C2×#468semidirect
222. #468D173direct (N × H)
234. C2×SD16C2×D173semidirect
239. C2×M16C2×D173semidirect
247. C22⋊Q8C2×D173semidirect
262. #107C2×D173semidirect
266. #103C2×D173semidirect
270. #102C2×D173semidirect
381. C8D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1732
C2×M16×D1732
C22⋊Q8×D1732
#121×D1732
#260117402
#260117392
#260117382
#260117372
#260117362
#260117352
#260117342
#107×D1732
#103×D1732
#102×D1732
#468×C1732
C2×#468173