Finite Group Databasebeta

← #36365193

Subgroups — #36365193

Order 22144.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3#26012149C211yesmaximal, characteristic, complemented
4#26012134C211yesmaximal, characteristic, complemented
5#26011746C211yesmaximal, characteristic, complemented
6#26011741C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011635C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13C4×#14583815C211yesmaximal, characteristic, complemented
14#26011570C211yesmaximal, characteristic, complemented
15#495×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149711—24no—
21#19149703—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×#14583809C2211yescharacteristic, complemented
29#19149665—21no—
30#19149661—21no—
31#19149656—21no—
32#19149655C2211yescharacteristic
33#19149655C2211yescharacteristic
34#19149652—21no—
35#19149652—21no—
36#19149650C2211yescharacteristic
37#19149650C2211yescharacteristic, complemented
38#19149649—21no—
39C4×Dic346C2211yescharacteristic, complemented
40C4×D692C2211yescharacteristic
41C4×D692C2211yescharacteristic
42#19149639—21no—
43C22⋊C4×D173—21no—
44#19149611C2211yescharacteristic, complemented
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149604C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149600C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic, complemented
54C2×Q16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic, complemented
57C2×D8×C173C2211yescharacteristic
58#124×C173C2211yescharacteristic
59#124×C173C2211yescharacteristic, complemented
60C4×C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64#14583825—22no—
65#14583825—41no—
66C2×Dic346—21no—
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173—41no—
70C2×D692—21no—
71C2×D692D411yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77D4×D173—22no—
78D4×D173—41no—
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84C2×C4×D173C2311yescharacteristic
85#14583816—22no—
86#14583815D414yescomplemented
87#14583814—22no—
88#14583813—22no—
89#14583811—22no—
90#14583809—22no—
91#14583809—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C4⋊Dic173D411yescharacteristic
95C4⋊Dic173C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803D411yescharacteristic
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112Q16×C173—22no—
113SD16×C173—41no—
114SD16×C173—41no—
115D8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—42no—
131Dic346C2×D412yes—
132D692—21no—
133D692—41no—
134D692—42no—
135D692C2×D412yes—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—81no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—42no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—42no—
151C4×D173—42no—
152C4×D173—81no—
153#13063436—21no—
154#13063436—21no—
155C23×C173—21no—
156C23×C173—21no—
157Q8×C173—21no—
158Q8×C173—21no—
159Q8×C173—21no—
160Q8×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—22no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—21no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—84no—
189C4×C173—21no—
190C4×C173—21no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—21no—
196Dic173—21no—
197Dic173—22no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—81no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—41no—
207D173—42no—
208C173#327511yescharacteristic, complemented, Sylow
209#3275—1731nomaximal, Sylow
210C2×#137—1731no—
211C2×AGL(1,ℤ/8)—1731no—
212#548—1731no—
213#546—1731no—
214#495D17311yescharacteristic, complemented
215#470—1731no—
216#469—1731no—
217#464—1731no—
218#463—1731no—
219#452—1731no—
220#450—1731no—
221#449—1731no—
222#448—1731no—
223C4×SD16—1731no—
224#436—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C22×Q8—1731no—
228C22×D4—1731no—
229#137—3464no—
230AGL(1,ℤ/8)—3464no—
231C2×Q16—1731no—
232C2×Q16C2×D17311yescharacteristic, complemented
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16C2×D17311yescharacteristic
237C2×SD16C2×D17311yescharacteristic, complemented
238C2×D8—1731no—
239C2×D8C2×D17311yescharacteristic
240C2×M16—1731no—
241C2×M16—1731no—
242#124—3461no—
243#124C2×D17311yescharacteristic
244#124C2×D17311yescharacteristic, complemented
245#123—3461no—
246C22⋊Q8—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249#121—3461no—
250#121—3461no—
251#121—1731no—
252#121—1731no—
253#120—3461no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—1731no—
257C4×D4—1731no—
258C2×C22⋊C4—3461no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102—1731no—
266#102—1731no—
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269C4×C8C2×D17311yescharacteristic, complemented
270C24—3461no—
271#41—3462no—
272#41—3462no—
273#41—6921no—
274#41—6921no—
275C2×Q8—3461no—
276C2×Q8—3462no—
277C2×Q8—1731no—
278C2×Q8—6921no—
279C2×Q8C22×D17311yescharacteristic
280C2×Q8C22×D17311yescharacteristic
281C2×D4—3461no—
282C2×D4—3462no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4—1731no—
289C2×D4—6921no—
290C2×D4C22×D17311yescharacteristic
291C2×D4C22×D17311yescharacteristic
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4—1731no—
298Q16—22no—
299Q16—3462no—
300SD16—41no—
301SD16—41no—
302SD16—3462no—
303SD16—3462no—
304SD16—1734no—
305D8—22no—
306D8—3462no—
307M16—3462no—
308M16—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—3461no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—1731no—
317C4⋊C4—1731no—
318C4⋊C4—1731no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—1731no—
329C22⋊C4—1731no—
330C42—3461no—
331C42C22×D17311yescharacteristic
332C23—21no—
333C23—21no—
334C23—3461no—
335C23—13841no—
336C23—1731no—
337C23—1731no—
338C23—6922no—
339C23—6922no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3461no—
345Q8—1732no—
346Q8—6922no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—3461no—
352D4—13841no—
353D4—1732no—
354D4—6921no—
355D4—6922no—
356D4—6922no—
357D4—6922no—
358D4—6922no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—13841no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4D4×D17311yescharacteristic
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—3461no—
379C8—3461no—
380C8D4×D17312yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3461no—
388C22—3461no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—3461no—
393C22—3462no—
394C22—13844no—
395C22—6921no—
396C22—6921no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3461no—
403C4—3461no—
404C4—3462no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—81no—
410C2—81no—
411C2—6921no—
412C2—6921no—
413C2—6922no—
414C2#2601212611yescharacteristic
415C2#2601228811yescharacteristic
416C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3275
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×#19149703C2semidirect
3. #26012149C2semidirect
4. #26012134C2semidirect
5. #26011746C2semidirect
6. #26011741C2semidirect
7. #26011710C2semidirect
8. #26011703C2semidirect
9. #26011650C2semidirect
10. #26011640C2semidirect
11. #26011635C2semidirect
12. #26011628C2semidirect
13. C4×#14583815C2semidirect
14. #26011570C2semidirect
15. #495×C173C2semidirect
18. C2×Q8×D173C22semidirect
22. C2×#14583816C22semidirect
28. C2×#14583809C22semidirect
37. #19149650C22semidirect
39. C4×Dic346C22semidirect
44. #19149611C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149604C22semidirect
47. #19149603C22semidirect
48. #19149603C22semidirect
51. #19149600C22semidirect
53. #19149591C22semidirect
54. C2×Q16×C173C22semidirect
56. C2×SD16×C173C22semidirect
59. #124×C173C22semidirect
60. C4×C8×C173C22semidirect
86. #14583815D4semidirect
208. C173#3275semidirect
214. #495D173semidirect
232. C2×Q16C2×D173semidirect
237. C2×SD16C2×D173semidirect
244. #124C2×D173semidirect
269. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191497032
#260121492
#260121342
#260117462
#260117412
#260117102
#260117032
#260116502
#260116402
#260116352
#260116282
C4×#145838152
#260115702
#495×C1732
#3275173