Finite Group Databasebeta

← #36362392

Subgroups — #36362392

Order 22144.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic
2#26011970C211yesmaximal, characteristic, complemented
3#26011963C211yesmaximal, characteristic, complemented
4#26011600C211yesmaximal, characteristic, complemented
5#26011593C211yesmaximal, characteristic, complemented
6#26011588C211yesmaximal, characteristic, complemented
7#26011578C211yesmaximal, characteristic, complemented
8C4×#14583814C211yesmaximal, characteristic, complemented
9D8×Dic173C211yesmaximal, characteristic, complemented
10C4×#14583811C211yesmaximal, characteristic, complemented
11#26011530C211yesmaximal, characteristic, complemented
12#26011484C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18#19149703C418yescomplemented
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C2×#14583809C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24C4×Dic346C2211yescharacteristic, complemented
25#19149644C2211yescharacteristic, complemented
26C4×D692C2211yescharacteristic
27D4×Dic173C2211yescharacteristic
28D4×Dic173C2211yescharacteristic, complemented
29#19149641C2211yescharacteristic
30C4×C22⋊D173—21no—
31C4×C22⋊D173—21no—
32#19149639—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic
35#19149636C2211yescharacteristic
36#19149635—21no—
37C2×#14583805—21no—
38C4⋊C4×D173C2211yescharacteristic
39C2×#14583803—21no—
40C22⋊C4×D173—21no—
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#19149589C2211yescharacteristic, complemented
51C4×#13063436C2211yescharacteristic, 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×D173C2311yescharacteristic
76#14583815C2×C414yescomplemented
77#14583814C2×C414yescomplemented
78#14583811C2×C414yescomplemented
79#14583809C2×C414yescomplemented
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C8×D173—22no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173—21no—
105D8×C173C2×C414yescomplemented
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C42×C173C2311yescharacteristic
113C22×D173—21no—
114C22×D173—24no—
115C22×D173—81no—
116C22×D173—81no—
117C22×D173C2×D411yescharacteristic
118Dic346—21no—
119Dic346C22×C412yes—
120D692—21no—
121D692C22×C412yes—
122C22⋊D173—24no—
123C22⋊D173—24no—
124C2×Dic173—21no—
125C2×Dic173—24no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—81no—
129C2×Dic173—81no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—81no—
134C4×D173—81no—
135C4×D173—81no—
136C4×D173C22×C412yes—
137C4×D173#4112yes—
138#13063436—21no—
139#13063436C22×C412yes—
140C23×C173—21no—
141C23×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173C22×C412yes—
145D4×C173C22×C412yes—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—81no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173C22×C412yes—
155C22×C173—22no—
156C22×C173—22no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—22no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C2×D173C4×D412yes—
169C4×C173—21no—
170C4×C173—21no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173C23×C411yescharacteristic
174C4×C173C2×#4111yescharacteristic
175Dic173—22no—
176Dic173—21no—
177Dic173—21no—
178Dic173—41no—
179Dic173—41no—
180Dic173C4×D412yes—
181C2×C173—42no—
182C2×C173—42no—
183C2×C173C2×C4×D411yescharacteristic
184C2×C173#57611yescharacteristic
185C2×C173#57711yescharacteristic
186D173—22no—
187D173—42no—
188C173#294611yescharacteristic, complemented, Sylow
189#2946—1731nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1731no—
191#520—1731no—
192#518—1731no—
193#442—1731no—
194#442D17311yescharacteristic, complemented
195#440—1731no—
196#440—1731no—
197C4×SD16—1731no—
198C4×SD16—1731no—
199C4×D8—1731no—
200C4×D8—1731no—
201#428—1731no—
202#416—1731no—
203C2×#102—1731no—
204#405—1731no—
205C2×#41—1731no—
206C22×D4—1731no—
207AGL(1,ℤ/8)—1738no—
208C2×SD16—1731no—
209C2×SD16—1731no—
210C2×D8—1731no—
211C2×D8C2×D17311yescharacteristic
212C2×M16—1731no—
213C22×C8—1731no—
214C4×Q8—1731no—
215C4×Q8—1731no—
216C4×D4—3461no—
217C4×D4—3461no—
218C4×D4—3461no—
219C4×D4—3461no—
220C4×D4—1731no—
221C4×D4—1731no—
222C4×D4—1731no—
223C4×D4—1731no—
224C4×D4C2×D17311yescharacteristic
225C4×D4C2×D17311yescharacteristic, complemented
226#117—3461no—
227#117—1731no—
228#117—1731no—
229C2×C4⋊C4—3461no—
230C2×C4⋊C4—1731no—
231C2×C22⋊C4—3461no—
232C2×C22⋊C4—3461no—
233#107—1731no—
234#107—1731no—
235#106—1731no—
236#106C2×D17311yescharacteristic, complemented
237#103—1731no—
238#103—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102C2×D17311yescharacteristic
244#102C2×D17311yescharacteristic, complemented
245#97—1731no—
246#97C2×D17311yescharacteristic, complemented
247C4×C8—1731no—
248C4×C8—1731no—
249C24—3461no—
250#41—3462no—
251#41—1734no—
252C2×Q8—1731no—
253C2×D4—3461no—
254C2×D4—3462no—
255C2×D4—3461no—
256C2×D4—1734no—
257C2×D4—1731no—
258C2×D4C22×D17311yescharacteristic
259C2×D4C22×D17311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—1731no—
269SD16—1734no—
270SD16—1734no—
271D8—1734no—
272D8C4×D17314yescomplemented
273M16—1734no—
274C2×C8—3462no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—1731no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4C22×D17311yescharacteristic
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—1731no—
296C22⋊C4—1731no—
297C42—3461no—
298C42—3461no—
299C42—3461no—
300C42—1731no—
301C42C22×D17311yescharacteristic
302C23—21no—
303C23—21no—
304C23—3464no—
305C23—3461no—
306C23—13841no—
307C23—13841no—
308C23—1731no—
309Q8—3461no—
310Q8—1732no—
311D4—21no—
312D4—21no—
313D4—3464no—
314D4—3464no—
315D4—3461no—
316D4—1732no—
317D4C2×C4×D17312yes—
318D4C2×C4×D17312yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—3461no—
324C2×C4—3464no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—13841no—
328C2×C4—13841no—
329C2×C4—13841no—
330C2×C4—13841no—
331C2×C4—13841no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1732no—
336C2×C4—1732no—
337C2×C4D4×D17311yescharacteristic
338C2×C4D4×D17311yescharacteristic
339C2×C4C23×D17311yescharacteristic
340C8—21no—
341C8—3461no—
342C8—1732no—
343C8C2×C4×D17312yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—3461no—
351C22—3461no—
352C22—3462no—
353C22—13844no—
354C22—1732no—
355C22—6921no—
356C22—6921no—
357C22C2×D4×D17311yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—3462no—
363C4—3461no—
364C4—3461no—
365C4—1732no—
366C4—6921no—
367C4—6921no—
368C4C22×C4×D17311yescharacteristic
369C4#41×D17311yescharacteristic
370C2—42no—
371C2—42no—
372C2—3462no—
373C2—6922no—
374C2C4×D4×D17311yescharacteristic
375C2#2601229511yescharacteristic
376C2#2601230111yescharacteristic
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
hypocenterC173#2946
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. #26011970C2semidirect
3. #26011963C2semidirect
4. #26011600C2semidirect
5. #26011593C2semidirect
6. #26011588C2semidirect
7. #26011578C2semidirect
8. C4×#14583814C2semidirect
9. D8×Dic173C2semidirect
10. C4×#14583811C2semidirect
11. #26011530C2semidirect
12. #26011484C2semidirect
13. #102×D173C2semidirect
14. #26011448C2semidirect
15. #442×C173C2semidirect
18. #19149703C4semidirect
24. C4×Dic346C22semidirect
25. #19149644C22semidirect
28. D4×Dic173C22semidirect
41. #19149613C22semidirect
42. #19149612C22semidirect
43. #19149610C22semidirect
44. #19149604C22semidirect
45. #19149602C22semidirect
48. #19149600C22semidirect
50. #19149589C22semidirect
51. C4×#13063436C22semidirect
52. C8×Dic173C22semidirect
55. C4×D4×C173C22semidirect
56. #106×C173C22semidirect
58. #102×C173C22semidirect
59. #97×C173C22semidirect
62. #14583825C2×C4semidirect
68. D4×D173C2×C4semidirect
76. #14583815C2×C4semidirect
77. #14583814C2×C4semidirect
78. #14583811C2×C4semidirect
79. #14583809C2×C4semidirect
105. D8×C173C2×C4semidirect
188. C173#2946semidirect
194. #442D173semidirect
225. C4×D4C2×D173semidirect
236. #106C2×D173semidirect
244. #102C2×D173semidirect
246. #97C2×D173semidirect
272. D8C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
#260119702
#260119632
#260116002
#260115932
#260115882
#260115782
C4×#145838142
D8×Dic1732
C4×#145838112
#260115302
#260114842
#102×D1732
#260114482
#442×C1732
#2946173