Finite Group Databasebeta

← #36364047

Subgroups — #36364047

Order 22144.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic, complemented
4C2×D8×D173C211yesmaximal, characteristic, complemented
5#26012059C211yesmaximal, characteristic, complemented
6#26012054C211yesmaximal, characteristic, complemented
7#26011880C211yesmaximal, characteristic, complemented
8#26011879C211yesmaximal, characteristic, complemented
9#26011877C211yesmaximal, characteristic, complemented
10#26011873C211yesmaximal, characteristic, complemented
11#26011872C211yesmaximal, characteristic, complemented
12#26011867C211yesmaximal, characteristic, complemented
13#26011866C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic
20#19149706—24no—
21#19149703—24no—
22#19149700—24no—
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic, complemented
25C2×#14583815C2211yescharacteristic, complemented
26C2×#14583814C2211yescharacteristic, complemented
27C2×#14583813C2211yescharacteristic, complemented
28C2×D1384C2211yescharacteristic
29C2×D1384C2211yescharacteristic, complemented
30C2×#14583811C2211yescharacteristic
31C2×#14583811C2211yescharacteristic
32C2×#14583811C2211yescharacteristic, complemented
33C2×#14583811C2211yescharacteristic, complemented
34C2×#14583811C2211yescharacteristic, complemented
35D8×D173—24no—
36C2×#14583809C2211yescharacteristic
37C2×C8×D173C2211yescharacteristic
38#19149676C2211yescharacteristic, complemented
39#19149675C2211yescharacteristic, complemented
40#19149675C2211yescharacteristic, complemented
41#19149668C2211yescharacteristic, complemented
42#19149667C2211yescharacteristic, complemented
43#19149666C2211yescharacteristic, complemented
44#19149652—22no—
45#19149651—22no—
46#19149649—22no—
47#19149647—22no—
48#19149636C2211yescharacteristic
49#19149589C2211yescharacteristic, complemented
50C4×#13063436C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic, complemented
57#124×C173C2211yescharacteristic, complemented
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61#14583827—22no—
62#14583827—22no—
63#14583825—22no—
64#14583825—22no—
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692—22no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—22no—
75C2×C22⋊D173—22no—
76D4×D173—22no—
77D4×D173—22no—
78D4×D173—22no—
79D4×D173—22no—
80C22×Dic173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83#14583816—22no—
84#14583815—22no—
85#14583815—22no—
86#14583814—22no—
87#14583813—22no—
88D1384—22no—
89D1384—22no—
90#14583811—22no—
91#14583811—22no—
92#14583811—22no—
93#14583811—22no—
94#14583811—22no—
95#14583809D414yescomplemented
96C2×#13063436C2311yescharacteristic
97C2×#13063436C2311yescharacteristic
98C8×D173D414yescomplemented
99#14583805D412yes—
100C22⋊Dic173—22no—
101#14583803—22no—
102#14583803—22no—
103#14583803D412yes—
104C4×Dic173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×D4×C173—22no—
107C2×D4×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110SD16×C173—22no—
111SD16×C173—22no—
112D8×C173—22no—
113D8×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C22⋊C4×C173—22no—
117C42×C173C2311yescharacteristic
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—42no—
123C22×D173—82no—
124C22×D173—82no—
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346—21no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692—42no—
135D692—82no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—82no—
140C22⋊D173—82no—
141C2×Dic173—21no—
142C2×Dic173—42no—
143C2×Dic173C2×D412yes—
144C2×Dic173C2×D411yescharacteristic
145C4×D173—42no—
146C4×D173C2×D412yes—
147C4×D173C2×D412yes—
148#13063436C2×D412yes—
149#13063436C2×D412yes—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—82no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D412yes—
164C2×C4×C173C2411yescharacteristic
165C8×C173C2×D412yes—
166C8×C173C2×D412yes—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—42no—
171C22×C173—42no—
172C22×C173—42no—
173C22×C173C22×D411yescharacteristic
174C2×D173—22no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—42no—
179C2×D173—42no—
180C2×D173—42no—
181C2×D173—84no—
182C2×D173—84no—
183C2×D173#12712yes—
184C4×C173—22no—
185C4×C173—41no—
186C4×C173C22×D411yescharacteristic
187C4×C173C22×D411yescharacteristic
188Dic173—22no—
189Dic173—41no—
190Dic173#12712yes—
191C2×C173—81no—
192C2×C173—81no—
193C2×C173—81no—
194C2×C173C2×#12711yescharacteristic
195C2×C173#57612yes—
196D173—22no—
197D173—81no—
198D173—81no—
199D173—81no—
200C173#315511yescharacteristic, complemented, Sylow
201#3155—1731nomaximal, Sylow
202C2×AGL(1,ℤ/8)—1731no—
203C2×AGL(1,ℤ/8)—1731no—
204C2×#135—1731no—
205C22×D8—1731no—
206#535—1731no—
207#535—1731no—
208#496—1731no—
209#496—1731no—
210#496—1731no—
211#496D17311yescharacteristic, complemented
212#495—1731no—
213#495—1731no—
214#493—1731no—
215#493—1731no—
216#405—1731no—
217C2×#41—1731no—
218C2×#41—1731no—
219C22×D4—1731no—
220C22×D4—1731no—
221AGL(1,ℤ/8)—3464no—
222AGL(1,ℤ/8)—3464no—
223#135—3464no—
224C2×Q16—1731no—
225C2×SD16—1731no—
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×SD16C2×D17311yescharacteristic, complemented
230C2×SD16C2×D17311yescharacteristic, complemented
231C2×D8—3464no—
232C2×D8—1731no—
233C2×D8—1731no—
234C2×D8—1731no—
235C2×D8—1731no—
236C2×D8—1731no—
237C2×D8—1731no—
238C2×D8—1731no—
239C2×D8C2×D17311yescharacteristic
240C2×D8C2×D17311yescharacteristic, complemented
241C2×M16—1731no—
242C22×C8—1731no—
243#127—1731no—
244#127—1731no—
245#127—1731no—
246#127C2×D17311yescharacteristic, complemented
247#124—1731no—
248#124—1731no—
249#124—1731no—
250#124C2×D17311yescharacteristic, complemented
251#121—3462no—
252#121—3462no—
253#120—3462no—
254#120—3462no—
255#117—1731no—
256#97—1731no—
257#97C2×D17311yescharacteristic, complemented
258C4×C8—1731no—
259C4×C8—1731no—
260C24—3461no—
261C24—3461no—
262#41—3462no—
263#41—3462no—
264#41—3462no—
265#41—3462no—
266C2×Q8—1731no—
267C2×Q8C22×D17311yescharacteristic
268C2×D4—22no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3462no—
272C2×D4—3462no—
273C2×D4—3462no—
274C2×D4—3462no—
275C2×D4—3462no—
276C2×D4—3462no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3462no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—1731no—
283C2×D4C22×D17311yescharacteristic
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289Q16—3462no—
290SD16—22no—
291SD16—22no—
292SD16—3462no—
293SD16—3462no—
294SD16—3462no—
295SD16—3462no—
296D8—22no—
297D8—22no—
298D8—3462no—
299D8—3462no—
300D8—3462no—
301D8—3462no—
302D8—3462no—
303D8—3462no—
304D8—3462no—
305M16—1734no—
306C2×C8—1731no—
307C2×C8—1731no—
308C2×C8—1734no—
309C2×C8C22×D17311yescharacteristic
310C2×C8C22×D17311yescharacteristic
311C4⋊C4—1732no—
312C22⋊C4—22no—
313C22⋊C4—3462no—
314C22⋊C4—3462no—
315C22⋊C4—3462no—
316C22⋊C4—1732no—
317C42—1731no—
318C42C22×D17311yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—3461no—
323C23—3461no—
324C23—3461no—
325C23—13842no—
326C23—13842no—
327C23—1731no—
328C23—6922no—
329C23—6922no—
330Q8—21no—
331Q8—21no—
332Q8—3461no—
333Q8—3461no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—3461no—
342D4—3461no—
343D4—3461no—
344D4—3461no—
345D4—3461no—
346D4—3461no—
347D4—13842no—
348D4—13842no—
349D4—13842no—
350D4—6922no—
351D4—6922no—
352D4—6922no—
353D4—6922no—
354C2×C4—21no—
355C2×C4—3461no—
356C2×C4—1732no—
357C2×C4—1731no—
358C2×C4—1732no—
359C2×C4—1732no—
360C2×C4—6922no—
361C2×C4—6922no—
362C2×C4D4×D17312yes—
363C2×C4C23×D17311yescharacteristic
364C8—1732no—
365C8—1732no—
366C8D4×D17312yes—
367C8D4×D17312yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—3462no—
375C22—13844no—
376C22—13844no—
377C22—1732no—
378C22—6921no—
379C22—6921no—
380C22—6921no—
381C22—6922no—
382C22—6922no—
383C22—6922no—
384C22C2×D4×D17311yescharacteristic
385C4—22no—
386C4—41no—
387C4—3462no—
388C4—1732no—
389C4—6921no—
390C4C2×D4×D17311yescharacteristic
391C4C2×D4×D17311yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—3462no—
396C2—13841no—
397C2—13841no—
398C2—13841no—
399C2#127×D17311yescharacteristic
400C2#2601229512yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3155
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149703C2semidirect
3. C2×#19149700C2semidirect
4. C2×D8×D173C2semidirect
5. #26012059C2semidirect
6. #26012054C2semidirect
7. #26011880C2semidirect
8. #26011879C2semidirect
9. #26011877C2semidirect
10. #26011873C2semidirect
11. #26011872C2semidirect
12. #26011867C2semidirect
13. #26011866C2semidirect
14. #26011448C2semidirect
15. #496×C173C2semidirect
23. C2×#14583816C22semidirect
24. C2×#14583815C22semidirect
25. C2×#14583815C22semidirect
26. C2×#14583814C22semidirect
27. C2×#14583813C22semidirect
29. C2×D1384C22semidirect
32. C2×#14583811C22semidirect
33. C2×#14583811C22semidirect
34. C2×#14583811C22semidirect
38. #19149676C22semidirect
39. #19149675C22semidirect
40. #19149675C22semidirect
41. #19149668C22semidirect
42. #19149667C22semidirect
43. #19149666C22semidirect
49. #19149589C22semidirect
50. C4×#13063436C22semidirect
51. C8×Dic173C22semidirect
52. C2×SD16×C173C22semidirect
53. C2×SD16×C173C22semidirect
55. C2×D8×C173C22semidirect
56. #127×C173C22semidirect
57. #124×C173C22semidirect
58. #97×C173C22semidirect
95. #14583809D4semidirect
98. C8×D173D4semidirect
200. C173#3155semidirect
211. #496D173semidirect
229. C2×SD16C2×D173semidirect
230. C2×SD16C2×D173semidirect
240. C2×D8C2×D173semidirect
246. #127C2×D173semidirect
250. #124C2×D173semidirect
257. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497032
C2×#191497002
C2×D8×D1732
#260120592
#260120542
#260118802
#260118792
#260118772
#260118732
#260118722
#260118672
#260118662
#260114482
#496×C1732
#3155173