Finite Group Databasebeta

← #36364032

Subgroups — #36364032

Order 22144.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic, complemented
2C2×#19149706C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic, complemented
4C2×SD16×D173C211yesmaximal, characteristic, complemented
5#26012066C211yesmaximal, characteristic, complemented
6#26012054C211yesmaximal, characteristic, complemented
7#26011883C211yesmaximal, characteristic, complemented
8#26011882C211yesmaximal, characteristic, complemented
9#26011878C211yesmaximal, characteristic, complemented
10#26011877C211yesmaximal, characteristic, complemented
11#26011875C211yesmaximal, characteristic, complemented
12#26011872C211yesmaximal, characteristic, complemented
13#26011862C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15#492×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149708—24no—
21#19149706—24no—
22#19149700—24no—
23C2×Dic692C2211yescharacteristic, complemented
24C2×#14583816C2211yescharacteristic, complemented
25C2×#14583816C2211yescharacteristic, complemented
26C2×#14583815C2211yescharacteristic
27C2×#14583815C2211yescharacteristic, complemented
28C2×#14583814C2211yescharacteristic
29C2×#14583814C2211yescharacteristic, complemented
30C2×#14583813C2211yescharacteristic
31C2×#14583813C2211yescharacteristic
32SD16×D173—24no—
33C2×D1384C2211yescharacteristic
34C2×#14583811C2211yescharacteristic
35C2×#14583811C2211yescharacteristic, complemented
36C2×#14583809C2211yescharacteristic
37C2×C8×D173C2211yescharacteristic
38#19149677C2211yescharacteristic, complemented
39#19149675C2211yescharacteristic
40#19149668C2211yescharacteristic, complemented
41#19149667C2211yescharacteristic
42#19149667C2211yescharacteristic, complemented
43#19149666C2211yescharacteristic
44#19149664—22no—
45#19149657—22no—
46#19149651—22no—
47#19149647—22no—
48#19149636C2211yescharacteristic
49#19149590C2211yescharacteristic, complemented
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic, complemented
56C4⋊Q8×C173C2211yescharacteristic, complemented
57#127×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—22no—
62#14583825—22no—
63#14583825—22no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—22no—
75D4×D173—22no—
76D4×D173—22no—
77C22×Dic173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81Dic692—22no—
82#14583816—22no—
83#14583816—22no—
84#14583815—22no—
85#14583815—22no—
86#14583814—22no—
87#14583814—22no—
88#14583813—22no—
89#14583813—22no—
90D1384—22no—
91#14583811—22no—
92#14583811—22no—
93#14583809D414yescomplemented
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173D414yescomplemented
97#14583805—22no—
98#14583805D412yes—
99C22⋊Dic173—22no—
100#14583803—22no—
101#14583803—22no—
102#14583803—22no—
103#14583803D412yes—
104C4×Dic173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×Q8×C173C2311yescharacteristic
107C2×D4×C173—22no—
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110SD16×C173—22no—
111SD16×C173—22no—
112SD16×C173—22no—
113SD16×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173—22no—
117C42×C173C2311yescharacteristic
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—42no—
121C22×D173—82no—
122C22×D173C2×D411yescharacteristic
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—82no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—42no—
139C2×Dic173C2×D412yes—
140C2×Dic173C2×D411yescharacteristic
141C4×D173—42no—
142C4×D173—42no—
143C4×D173C2×D412yes—
144C4×D173C2×D412yes—
145#13063436C2×D412yes—
146#13063436C2×D412yes—
147C23×C173—21no—
148C23×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—82no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D412yes—
161C2×C4×C173C2411yescharacteristic
162C8×C173C2×D412yes—
163C8×C173C2×D412yes—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—42no—
167C22×C173—42no—
168C22×C173C22×D411yescharacteristic
169C2×D173—22no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—42no—
173C2×D173—42no—
174C2×D173—84no—
175C2×D173#12712yes—
176C4×C173—22no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C22×D411yescharacteristic
180C4×C173C22×D411yescharacteristic
181Dic173—22no—
182Dic173—41no—
183Dic173—41no—
184Dic173#12712yes—
185C2×C173—81no—
186C2×C173—81no—
187C2×C173C2×#12711yescharacteristic
188C2×C173#57712yes—
189D173—22no—
190D173—81no—
191D173—81no—
192C173#315411yescharacteristic, complemented, Sylow
193#3154—1731nomaximal, Sylow
194C2×#137—1731no—
195C2×AGL(1,ℤ/8)—1731no—
196C2×#135—1731no—
197C22×SD16—1731no—
198#536—1731no—
199#535—1731no—
200#497—1731no—
201#497—1731no—
202#496—1731no—
203#496—1731no—
204#495—1731no—
205#495—1731no—
206#492—1731no—
207#492D17311yescharacteristic, complemented
208#405—1731no—
209C2×#41—1731no—
210C2×#41—1731no—
211C22×Q8—1731no—
212C22×D4—1731no—
213#137—3464no—
214AGL(1,ℤ/8)—3464no—
215#135—3464no—
216C2×Q16—1731no—
217C2×Q16—1731no—
218C2×Q16—1731no—
219C2×SD16—3464no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×SD16—1731no—
226C2×SD16C2×D17311yescharacteristic
227C2×SD16C2×D17311yescharacteristic
228C2×SD16C2×D17311yescharacteristic, complemented
229C2×SD16C2×D17311yescharacteristic, complemented
230C2×D8—1731no—
231C2×D8—1731no—
232C2×D8—1731no—
233C2×M16—1731no—
234C22×C8—1731no—
235C4⋊Q8—1731no—
236C4⋊Q8C2×D17311yescharacteristic, complemented
237#127—1731no—
238#127C2×D17311yescharacteristic
239#124—1731no—
240#124—1731no—
241#124—1731no—
242#124—1731no—
243#123—3462no—
244C22⋊Q8—3462no—
245#121—3462no—
246#120—3462no—
247#117—1731no—
248#97—1731no—
249#97—1731no—
250C4×C8—1731no—
251C4×C8C2×D17311yescharacteristic, complemented
252C24—3461no—
253#41—3462no—
254#41—3462no—
255#41—3462no—
256#41—3462no—
257C2×Q8—3461no—
258C2×Q8—3462no—
259C2×Q8—3462no—
260C2×Q8—1731no—
261C2×Q8—1731no—
262C2×Q8C22×D17311yescharacteristic
263C2×Q8C22×D17311yescharacteristic
264C2×D4—22no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3462no—
268C2×D4—3462no—
269C2×D4—3462no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—1731no—
279Q16—3462no—
280Q16—3462no—
281Q16—3462no—
282SD16—22no—
283SD16—22no—
284SD16—22no—
285SD16—22no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290SD16—3462no—
291SD16—3462no—
292D8—3462no—
293D8—3462no—
294D8—3462no—
295M16—1734no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—1734no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—3462no—
303C4⋊C4—1732no—
304C22⋊C4—3462no—
305C22⋊C4—3462no—
306C22⋊C4—3462no—
307C22⋊C4—3462no—
308C22⋊C4—1732no—
309C42—1731no—
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—21no—
313C23—3461no—
314C23—3461no—
315C23—13842no—
316C23—1731no—
317C23—6922no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324Q8—3461no—
325Q8—3461no—
326Q8—6922no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—82no—
332D4—3461no—
333D4—3461no—
334D4—3461no—
335D4—3461no—
336D4—13842no—
337D4—6922no—
338D4—6922no—
339D4—6922no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—1732no—
345C2×C4—1731no—
346C2×C4—1732no—
347C2×C4—1732no—
348C2×C4—6922no—
349C2×C4—6922no—
350C2×C4—6922no—
351C2×C4D4×D17312yes—
352C2×C4C23×D17311yescharacteristic
353C8—1732no—
354C8—1732no—
355C8D4×D17312yes—
356C8D4×D17312yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—3462no—
362C22—13844no—
363C22—1732no—
364C22—6921no—
365C22—6921no—
366C22—6922no—
367C22—6922no—
368C22C2×D4×D17311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—3462no—
373C4—1732no—
374C4—6921no—
375C4—6921no—
376C4C2×D4×D17311yescharacteristic
377C4C2×D4×D17311yescharacteristic
378C2—81no—
379C2—81no—
380C2—3462no—
381C2—13841no—
382C2—13841no—
383C2#127×D17311yescharacteristic
384C2#2601229712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#492×C173C2
socleC22×C173C22×D4
hypercenter#492D173
hypocenterC173#3154
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C2×#19149706C2semidirect
3. C2×#19149700C2semidirect
4. C2×SD16×D173C2semidirect
5. #26012066C2semidirect
6. #26012054C2semidirect
7. #26011883C2semidirect
8. #26011882C2semidirect
9. #26011878C2semidirect
10. #26011877C2semidirect
11. #26011875C2semidirect
12. #26011872C2semidirect
13. #26011862C2semidirect
14. #26011445C2semidirect
15. #492×C173C2semidirect
23. C2×Dic692C22semidirect
24. C2×#14583816C22semidirect
25. C2×#14583816C22semidirect
27. C2×#14583815C22semidirect
29. C2×#14583814C22semidirect
35. C2×#14583811C22semidirect
38. #19149677C22semidirect
40. #19149668C22semidirect
42. #19149667C22semidirect
49. #19149590C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×SD16×C173C22semidirect
56. C4⋊Q8×C173C22semidirect
58. C4×C8×C173C22semidirect
93. #14583809D4semidirect
96. C8×D173D4semidirect
192. C173#3154semidirect
207. #492D173semidirect
228. C2×SD16C2×D173semidirect
229. C2×SD16C2×D173semidirect
236. C4⋊Q8C2×D173semidirect
251. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×#191497062
C2×#191497002
C2×SD16×D1732
#260120662
#260120542
#260118832
#260118822
#260118782
#260118772
#260118752
#260118722
#260118622
#260114452
#492×C1732
#3154173