Finite Group Databasebeta

← #36362394

Subgroups — #36362394

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×#19149706C211yesmaximal, characteristic
2#26011973C211yesmaximal, characteristic, complemented
3#26011963C211yesmaximal, characteristic, complemented
4#26011604C211yesmaximal, characteristic, complemented
5#26011600C211yesmaximal, characteristic, complemented
6#26011594C211yesmaximal, characteristic, complemented
7SD16×Dic173C211yesmaximal, characteristic, complemented
8C4×#14583813C211yesmaximal, characteristic, complemented
9#26011573C211yesmaximal, characteristic, complemented
10C4×#14583811C211yesmaximal, characteristic, complemented
11#26011532C211yesmaximal, characteristic, complemented
12#26011486C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18#19149706C418yescomplemented
19C2×#14583813C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C2×#14583809C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24Q8×Dic173C2211yescharacteristic, complemented
25C4×D692—21no—
26C4×D692C2211yescharacteristic
27C4×D692C2211yescharacteristic, complemented
28D4×Dic173C2211yescharacteristic
29#19149641—21no—
30#19149641C2211yescharacteristic
31#19149641C2211yescharacteristic, complemented
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic
35#19149636—21no—
36#19149636C2211yescharacteristic
37C4⋊C4×D173—21no—
38C4⋊C4×D173C2211yescharacteristic
39C2×#14583803—21no—
40C22⋊C4×D173—21no—
41#19149612C2211yescharacteristic, complemented
42#19149611C2211yescharacteristic, complemented
43#19149610C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic, complemented
50#19149589C2211yescharacteristic, complemented
51C4×#13063436C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59#97×C173C2211yescharacteristic, complemented
60C23×D173—21no—
61#14583827—22no—
62#14583827C2×C414yescomplemented
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67D4×D173—22no—
68D4×D173C2×C414yescomplemented
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583813C2×C414yescomplemented
78D1384C2×C414yescomplemented
79#14583811C2×C414yescomplemented
80#14583809C2×C414yescomplemented
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—22no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104SD16×C173C2×C414yescomplemented
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C42×C173—21no—
112C42×C173C2311yescharacteristic
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—24no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119D692—24no—
120D692—21no—
121D692—21no—
122D692C22×C412yes—
123D692C22×C412yes—
124C22⋊D173—24no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—81no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—24no—
132C4×D173—81no—
133C4×D173—81no—
134C4×D173—81no—
135C4×D173—81no—
136C4×D173—81no—
137C4×D173C22×C412yes—
138C4×D173#4112yes—
139#13063436—21no—
140#13063436C22×C412yes—
141C23×C173—21no—
142Q8×C173—21no—
143Q8×C173C22×C412yes—
144D4×C173—21no—
145D4×C173C22×C412yes—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
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—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—22no—
162C2×D173—22no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C2×D173C4×D412yes—
169C4×C173—22no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173C23×C411yescharacteristic
175C4×C173C2×#4111yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—41no—
179Dic173—41no—
180Dic173C4×D412yes—
181C2×C173—42no—
182C2×C173C2×C4×D411yescharacteristic
183C2×C173#57611yescharacteristic
184C2×C173#57711yescharacteristic
185D173—22no—
186D173—42no—
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#442—1731no—
195#440—1731no—
196#440D17311yescharacteristic, complemented
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×SD16C2×D17311yescharacteristic
210C2×D8—1731no—
211C2×D8—1731no—
212C2×M16—1731no—
213C22×C8—1731no—
214C4×Q8—1731no—
215C4×Q8C2×D17311yescharacteristic, complemented
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×D4—1731no—
225C4×D4C2×D17311yescharacteristic
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#107C2×D17311yescharacteristic, complemented
235#106—1731no—
236#106—1731no—
237#103—1731no—
238#103C2×D17311yescharacteristic, complemented
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102C2×D17311yescharacteristic
245#97—1731no—
246#97C2×D17311yescharacteristic, complemented
247C4×C8—1731no—
248C4×C8—1731no—
249C24—3461no—
250#41—3462no—
251#41—1734no—
252C2×Q8C22×D17311yescharacteristic
253C2×D4—3461no—
254C2×D4—3462no—
255C2×D4—3461no—
256C2×D4—1734no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4C22×D17311yescharacteristic
260C22×C4—21no—
261C22×C4—3461no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—1731no—
269SD16—1734no—
270SD16C4×D17314yescomplemented
271D8—1734no—
272D8—1734no—
273M16—1734no—
274C2×C8—3462no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C4⋊C4—21no—
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—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—1731no—
296C22⋊C4—1731no—
297C42—21no—
298C42—3461no—
299C42—3461no—
300C42—1731no—
301C42C22×D17311yescharacteristic
302C23—21no—
303C23—3464no—
304C23—3461no—
305C23—3461no—
306C23—13841no—
307C23—13841no—
308C23—1731no—
309Q8—21no—
310Q8C2×C4×D17312yes—
311D4—21no—
312D4—3464no—
313D4—3464no—
314D4—3461no—
315D4—3461no—
316D4—1732no—
317D4—1732no—
318D4C2×C4×D17312yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—3464no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
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—41no—
346C22—41no—
347C22—3461no—
348C22—3461no—
349C22—3462no—
350C22—3462no—
351C22—13844no—
352C22—1732no—
353C22—6921no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22C2×D4×D17311yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—3461no—
364C4—3461no—
365C4—1732no—
366C4—6921no—
367C4—6921no—
368C4C22×C4×D17311yescharacteristic
369C4#41×D17311yescharacteristic
370C2—42no—
371C2—3462no—
372C2—6922no—
373C2—6922no—
374C2C4×D4×D17311yescharacteristic
375C2#2601229511yescharacteristic
376C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149706C2
cosocleC2×C4×C173C24
hypercenter#440D173
hypocenterC173#2946
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #26011973C2semidirect
3. #26011963C2semidirect
4. #26011604C2semidirect
5. #26011600C2semidirect
6. #26011594C2semidirect
7. SD16×Dic173C2semidirect
8. C4×#14583813C2semidirect
9. #26011573C2semidirect
10. C4×#14583811C2semidirect
11. #26011532C2semidirect
12. #26011486C2semidirect
13. #102×D173C2semidirect
14. #26011448C2semidirect
15. #440×C173C2semidirect
18. #19149706C4semidirect
24. Q8×Dic173C22semidirect
27. C4×D692C22semidirect
31. #19149641C22semidirect
41. #19149612C22semidirect
42. #19149611C22semidirect
43. #19149610C22semidirect
44. #19149603C22semidirect
46. #19149601C22semidirect
49. #19149599C22semidirect
50. #19149589C22semidirect
51. C4×#13063436C22semidirect
52. C8×Dic173C22semidirect
54. C4×Q8×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
59. #97×C173C22semidirect
62. #14583827C2×C4semidirect
68. D4×D173C2×C4semidirect
77. #14583813C2×C4semidirect
78. D1384C2×C4semidirect
79. #14583811C2×C4semidirect
80. #14583809C2×C4semidirect
104. SD16×C173C2×C4semidirect
188. C173#2946semidirect
196. #440D173semidirect
215. C4×Q8C2×D173semidirect
234. #107C2×D173semidirect
238. #103C2×D173semidirect
246. #97C2×D173semidirect
270. SD16C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260119732
#260119632
#260116042
#260116002
#260115942
SD16×Dic1732
C4×#145838132
#260115732
C4×#145838112
#260115322
#260114862
#102×D1732
#260114482
#440×C1732
#2946173