Finite Group Databasebeta

← #36362374

Subgroups — #36362374

Order 22144.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic
2C4×#14583825C211yesmaximal, characteristic, complemented
3C4×D4×D173C211yesmaximal, characteristic, complemented
4#26011601C211yesmaximal, characteristic, complemented
5#26011593C211yesmaximal, characteristic, complemented
6#26011591C211yesmaximal, characteristic, complemented
7#26011582C211yesmaximal, characteristic, complemented
8#26011578C211yesmaximal, characteristic, complemented
9D8×Dic173C211yesmaximal, characteristic, complemented
10#26011571C211yesmaximal, characteristic, complemented
11#26011530C211yesmaximal, characteristic, complemented
12#26011494C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#26011443C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C22×C4×D173—21no—
19#19149703C418yescomplemented
20C2×#14583815C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23C2×#14583809C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25C4×Dic346C2211yescharacteristic, complemented
26#19149644C2211yescharacteristic, complemented
27C4×D692C2211yescharacteristic
28D4×Dic173C2211yescharacteristic
29D4×Dic173C2211yescharacteristic, complemented
30#19149641C2211yescharacteristic
31C4×C22⋊D173—21no—
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149639—21no—
35#19149637C2211yescharacteristic
36#19149635—21no—
37C4⋊C4×D173C2211yescharacteristic
38C22⋊C4×D173—21no—
39C2×C4×Dic173—21no—
40C42×D173C2211yescharacteristic
41#19149613C2211yescharacteristic, complemented
42#19149612C2211yescharacteristic, complemented
43#19149610C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic
50#19149590C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×D8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic, complemented
56#106×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic, complemented
59#97×C173C2211yescharacteristic, complemented
60C23×D173—21no—
61#14583825—22no—
62#14583825C2×C414yescomplemented
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67D4×D173—22no—
68D4×D173C2×C414yescomplemented
69C22×Dic173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—44no—
76C2×C4×D173D411yescharacteristic
77C2×C4×D173D411yescharacteristic
78C2×C4×D173C2311yescharacteristic
79#14583815C2×C414yescomplemented
80#14583814C2×C414yescomplemented
81#14583811C2×C414yescomplemented
82#14583809—22no—
83#14583809C2×C414yescomplemented
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173D411yescharacteristic
100C4×Dic173D411yescharacteristic
101C4×Dic173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105C22×C4×C173—21no—
106D8×C173C2×C414yescomplemented
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—24no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173C2×D411yescharacteristic
119Dic346—21no—
120Dic346C22×C412yescomplemented
121D692—21no—
122D692C22×C412yes—
123C22⋊D173—24no—
124C22⋊D173—24no—
125C2×Dic173—21no—
126C2×Dic173—24no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—42no—
130C2×Dic173—42no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—22no—
135C4×D173—41no—
136C4×D173—42no—
137C4×D173—42no—
138C4×D173—42no—
139C4×D173C22×C412yes—
140C4×D173#4112yes—
141#13063436—21no—
142#13063436C22×C412yescomplemented
143C23×C173—21no—
144C23×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173C22×C412yes—
148D4×C173C22×C412yescomplemented
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—42no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173C22×C412yescomplemented
158C22×C173—22no—
159C22×C173—22no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—22no—
168C2×D173—48no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173C4×D412yes—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173C23×C411yescharacteristic
177C4×C173C2×#4111yescharacteristic
178Dic173—22no—
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182Dic173C4×D412yes—
183Dic173AGL(1,ℤ/8)12yescomplemented
184C2×C173—42no—
185C2×C173—42no—
186C2×C173C2×C4×D411yescharacteristic
187C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C173#57511yescharacteristic
189D173—22no—
190D173—42no—
191C173C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
192C4×AGL(1,ℤ/8)—1731nomaximal, Sylow
193C2×AGL(1,ℤ/8)—1731no—
194C4×#41—1731no—
195C2×C4×D4—1731no—
196#442—1731no—
197#442D17311yescharacteristic, complemented
198#440—1731no—
199#440—1731no—
200C4×SD16—1731no—
201C4×SD16—1731no—
202C4×D8—1731no—
203C4×D8—1731no—
204#428—1731no—
205#418—1731no—
206#417—1731no—
207C4×M16—1731no—
208C2×#41—1731no—
209C22×D4—1731no—
210C23×C4—3461no—
211AGL(1,ℤ/8)—1738no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×D8—1731no—
215C2×D8C2×D17311yescharacteristic
216C2×M16—1731no—
217C2×M16—1731no—
218C4×Q8—1731no—
219C4×Q8—1731no—
220C4×D4—3461no—
221C4×D4—3461no—
222C4×D4—3461no—
223C4×D4—3461no—
224C4×D4—1731no—
225C4×D4—1731no—
226C4×D4—1731no—
227C4×D4—1731no—
228C4×D4C2×D17311yescharacteristic
229C4×D4C2×D17311yescharacteristic, complemented
230#117—3461no—
231#117—1731no—
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
234C2×C42—3461no—
235C2×C42—1731no—
236#107—1731no—
237#107—1731no—
238#106—1731no—
239#106C2×D17311yescharacteristic, complemented
240#103—1731no—
241#103—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102C2×D17311yescharacteristic
247#102C2×D17311yescharacteristic, complemented
248#97—1731no—
249#97C2×D17311yescharacteristic, complemented
250C4×C8—1731no—
251C4×C8—1731no—
252C24—3461no—
253#41—3462no—
254#41—1734no—
255C2×Q8—1731no—
256C2×D4—3461no—
257C2×D4—3462no—
258C2×D4—3461no—
259C2×D4—1734no—
260C2×D4—1731no—
261C2×D4C22×D17311yescharacteristic
262C2×D4C22×D17311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—6924no—
275SD16—1734no—
276SD16—1734no—
277D8—1734no—
278D8C4×D17314yescomplemented
279M16—3462no—
280M16—1734no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8C22×D17311yescharacteristic
284C2×C8C22×D17311yescharacteristic
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—1731no—
289C4⋊C4—1731no—
290C4⋊C4C22×D17311yescharacteristic
291C4⋊C4C22×D17311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C42—3461no—
300C42—3461no—
301C42—3461no—
302C42—1731no—
303C42—1731no—
304C42—1731no—
305C42C22×D17311yescharacteristic
306C23—21no—
307C23—21no—
308C23—3464no—
309C23—3461no—
310C23—1731no—
311C23—6922no—
312C23—6922no—
313Q8—3461no—
314Q8—1732no—
315D4—21no—
316D4—21no—
317D4—3464no—
318D4—3464no—
319D4—3461no—
320D4—1732no—
321D4C2×C4×D17312yes—
322D4C2×C4×D17312yescomplemented
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—3462no—
328C2×C4—3461no—
329C2×C4—3464no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1732no—
336C2×C4—1732no—
337C2×C4—6921no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4—6922no—
341C2×C4—6922no—
342C2×C4—6922no—
343C2×C4D4×D17311yescharacteristic
344C2×C4D4×D17311yescharacteristic
345C2×C4C23×D17311yescharacteristic
346C8—21no—
347C8—3461no—
348C8—1732no—
349C8C2×C4×D17312yescomplemented
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3461no—
357C22—3461no—
358C22—3462no—
359C22—1732no—
360C22—6928no—
361C22—6921no—
362C22—6921no—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3462no—
369C4—3461no—
370C4—1732no—
371C4—1732no—
372C4—6921no—
373C4—6921no—
374C4C22×C4×D17311yescharacteristic
375C4#41×D17311yescharacteristic
376C2—42no—
377C2—42no—
378C2—3462no—
379C2—6922no—
380C2C4×D4×D17311yescharacteristic
381C2AGL(1,ℤ/8)×D17311yescharacteristic
382C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149703C2
cosocleC2×C4×C173C24
hypercenter#442D173
hypocenterC173C4×AGL(1,ℤ/8)
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. C4×#14583825C2semidirect
3. C4×D4×D173C2semidirect
4. #26011601C2semidirect
5. #26011593C2semidirect
6. #26011591C2semidirect
7. #26011582C2semidirect
8. #26011578C2semidirect
9. D8×Dic173C2semidirect
10. #26011571C2semidirect
11. #26011530C2semidirect
12. #26011494C2semidirect
13. #26011489C2semidirect
14. #26011443C2semidirect
15. #442×C173C2semidirect
19. #19149703C4semidirect
25. C4×Dic346C22semidirect
26. #19149644C22semidirect
29. D4×Dic173C22semidirect
41. #19149613C22semidirect
42. #19149612C22semidirect
43. #19149610C22semidirect
44. #19149604C22semidirect
45. #19149602C22semidirect
48. #19149600C22semidirect
50. #19149590C22semidirect
51. C8×Dic173C22semidirect
52. C8×Dic173C22semidirect
55. C4×D4×C173C22semidirect
56. #106×C173C22semidirect
58. #102×C173C22semidirect
59. #97×C173C22semidirect
62. #14583825C2×C4semidirect
68. D4×D173C2×C4semidirect
79. #14583815C2×C4semidirect
80. #14583814C2×C4semidirect
81. #14583811C2×C4semidirect
83. #14583809C2×C4semidirect
106. D8×C173C2×C4semidirect
120. Dic346C22×C4semidirect
142. #13063436C22×C4semidirect
148. D4×C173C22×C4semidirect
157. C8×C173C22×C4semidirect
183. Dic173AGL(1,ℤ/8)semidirect
191. C173C4×AGL(1,ℤ/8)semidirect
197. #442D173semidirect
229. C4×D4C2×D173semidirect
239. #106C2×D173semidirect
247. #102C2×D173semidirect
249. #97C2×D173semidirect
278. D8C4×D173semidirect
322. D4C2×C4×D173semidirect
349. C8C2×C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C4×#145838252
C4×D4×D1732
#260116012
#260115932
#260115912
#260115822
#260115782
D8×Dic1732
#260115712
#260115302
#260114942
#260114892
#260114432
#442×C1732
C4×AGL(1,ℤ/8)173