Finite Group Databasebeta

← #36363082

Subgroups — #36363082

Order 22144.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012059C211yesmaximal, characteristic
4#26011967C211yesmaximal, characteristic, complemented
5#26011716C211yesmaximal, characteristic, complemented
6#26011715C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011692C211yesmaximal, characteristic, complemented
9#26011691C211yesmaximal, characteristic, complemented
10#26011690C211yesmaximal, characteristic, complemented
11#26011689C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#26011486C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149711—24no—
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583815C2211yescharacteristic
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583814C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×#14583813C2211yescharacteristic, complemented
26SD16×D173—24no—
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149676C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149668C2211yescharacteristic
32#19149652—21no—
33#19149652—21no—
34#19149649—21no—
35#19149649—21no—
36Q8×Dic173C2211yescharacteristic, complemented
37C4×Dic346—21no—
38C4×Dic346C2211yescharacteristic, complemented
39#19149644—21no—
40#19149644C2211yescharacteristic, complemented
41#19149637—21no—
42#19149637C2211yescharacteristic
43#19149636—21no—
44#19149636C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic
52#19149599C2211yescharacteristic
53#19149599C2211yescharacteristic, complemented
54C2×Q16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×Q8×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic
61C23×D173—21no—
62#14583827—22no—
63#14583827—22no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173D414yescomplemented
68C2×D692—21no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—22no—
77D4×D173—22no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81#14583816—22no—
82#14583815—22no—
83#14583815—22no—
84#14583814—22no—
85#14583813—22no—
86#14583813—22no—
87#14583809—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96#14583805D411yescharacteristic
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803D411yescharacteristic
101#14583803D411yescharacteristic
102C4×Dic173—21no—
103C4×Dic173—21no—
104C4×Dic173—21no—
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110Q16×C173—22no—
111SD16×C173—22no—
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C4⋊C4×C173—21no—
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C42×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—42no—
123C22×D173—81no—
124C22×D173—81no—
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346—24no—
128Dic346C2×D412yes—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134D692—81no—
135D692—81no—
136C22⋊D173—42no—
137C22⋊D173—81no—
138C22⋊D173—81no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—24no—
145C4×D173—42no—
146C4×D173C2×D412yes—
147C4×D173#4112yes—
148#13063436—21no—
149#13063436—21no—
150C23×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173C2×D412yes—
155D4×C173—21no—
156D4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—84no—
178C2×D173#12112yes—
179C4×C173—22no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173C22×D411yescharacteristic
185C4×C173C2×#4111yescharacteristic
186Dic173—22no—
187Dic173—21no—
188Dic173—21no—
189Dic173—41no—
190Dic173#12112yes—
191C2×C173—81no—
192C2×C173C2×#12111yescharacteristic
193C2×C173#57711yescharacteristic
194C2×C173#57711yescharacteristic
195D173—22no—
196D173—81no—
197D173—81no—
198C173#304311yescharacteristic, complemented, Sylow
199#3043—1731nomaximal, Sylow
200C2×#137—1731no—
201C22×SD16—1731no—
202#535—1731no—
203#519—1731no—
204#464—1731no—
205#464—1731no—
206#464—1731no—
207#464D17311yescharacteristic, complemented
208#460—1731no—
209#460—1731no—
210#460—1731no—
211#460—1731no—
212#424—1731no—
213#418—1731no—
214#416—1731no—
215C2×#41—1731no—
216C22×Q8—1731no—
217C22×D4—1731no—
218#137—3464no—
219C2×Q16—1731no—
220C2×Q16C2×D17311yescharacteristic, complemented
221C2×SD16—3464no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×SD16—1731no—
226C2×SD16—1731no—
227C2×SD16C2×D17311yescharacteristic
228C2×M16—1731no—
229C22×C8—1731no—
230#127—1731no—
231#127—1731no—
232#124—1731no—
233#124C2×D17311yescharacteristic
234#121—3461no—
235#121—3461no—
236#120—3461no—
237#120—3461no—
238C4×Q8—3461no—
239C4×Q8—3461no—
240C4×Q8—1731no—
241C4×Q8—1731no—
242C4×Q8—1731no—
243C4×Q8C2×D17311yescharacteristic, complemented
244#117—3461no—
245#117—3461no—
246#117—1731no—
247#117—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250C4⋊C8—1731no—
251C4⋊C8C2×D17311yescharacteristic, complemented
252#103—1731no—
253#103C2×D17311yescharacteristic, complemented
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102—1731no—
259#102C2×D17311yescharacteristic
260C24—3461no—
261#41—3462no—
262#41—3462no—
263C2×Q8—3461no—
264C2×Q8—3462no—
265C2×Q8—1734no—
266C2×Q8—1731no—
267C2×Q8C22×D17311yescharacteristic
268C2×Q8C22×D17311yescharacteristic
269C2×D4—3461no—
270C2×D4—3462no—
271C2×D4—3462no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4—1731no—
279C2×D4C22×D17311yescharacteristic
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—1731no—
283Q16—22no—
284Q16—3462no—
285SD16—22no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290SD16—3462no—
291M16—3462no—
292C2×C8—3462no—
293C2×C8—1731no—
294C2×C8—1731no—
295C2×C8C22×D17311yescharacteristic
296C2×C8C22×D17311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—1731no—
302C4⋊C4—1731no—
303C4⋊C4—1731no—
304C4⋊C4C22×D17311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—1731no—
311C22⋊C4—1731no—
312C42—21no—
313C42—3461no—
314C42—3461no—
315C42—3461no—
316C42—3461no—
317C42—1731no—
318C42C22×D17311yescharacteristic
319C23—21no—
320C23—3461no—
321C23—3461no—
322C23—13841no—
323C23—13841no—
324C23—1731no—
325C23—6922no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—3464no—
330Q8—3461no—
331Q8—1732no—
332Q8D4×D17312yes—
333D4—21no—
334D4—21no—
335D4—3461no—
336D4—3461no—
337D4—3461no—
338D4—3461no—
339D4—13841no—
340D4—13841no—
341D4—13841no—
342D4—13841no—
343D4—6922no—
344D4—6922no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—3464no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—1731no—
352C2×C4—1731no—
353C2×C4—1731no—
354C2×C4—1732no—
355C2×C4—1732no—
356C2×C4—6922no—
357C2×C4D4×D17311yescharacteristic
358C2×C4D4×D17311yescharacteristic
359C2×C4C23×D17311yescharacteristic
360C8—21no—
361C8—21no—
362C8—3461no—
363C8—3461no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—3461no—
368C22—3461no—
369C22—13844no—
370C22—1732no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22—6921no—
376C22—6921no—
377C22C2×D4×D17311yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—3462no—
384C4—3461no—
385C4—3461no—
386C4—1732no—
387C4—6921no—
388C4C2×D4×D17311yescharacteristic
389C4#41×D17311yescharacteristic
390C2—81no—
391C2—3462no—
392C2—13841no—
393C2—13841no—
394C2#121×D17311yescharacteristic
395C2#2601229811yescharacteristic
396C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radical#26012059C2
hypercenter#464D173
hypocenterC173#3043
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×SD16×D173C2semidirect
4. #26011967C2semidirect
5. #26011716C2semidirect
6. #26011715C2semidirect
7. #26011710C2semidirect
8. #26011692C2semidirect
9. #26011691C2semidirect
10. #26011690C2semidirect
11. #26011689C2semidirect
12. #26011516C2semidirect
13. #26011494C2semidirect
14. #26011486C2semidirect
15. #464×C173C2semidirect
20. C2×#14583816C22semidirect
22. C2×#14583815C22semidirect
25. C2×#14583813C22semidirect
36. Q8×Dic173C22semidirect
38. C4×Dic346C22semidirect
40. #19149644C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149603C22semidirect
50. #19149601C22semidirect
53. #19149599C22semidirect
54. C2×Q16×C173C22semidirect
57. C4×Q8×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
67. Q8×D173D4semidirect
198. C173#3043semidirect
207. #464D173semidirect
220. C2×Q16C2×D173semidirect
243. C4×Q8C2×D173semidirect
251. C4⋊C8C2×D173semidirect
253. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×SD16×D1732
#260120592
#260119672
#260117162
#260117152
#260117102
#260116922
#260116912
#260116902
#260116892
#260115162
#260114942
#260114862
#464×C1732
#3043173