Finite Group Databasebeta

← #37545421

Subgroups — #37545421

Order 23168.

504 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×#19153839C211yesmaximal, characteristic, complemented
2C22×#15668436C211yesmaximal, characteristic, complemented
3#26041940C211yesmaximal, characteristic, complemented
4#26041920C211yesmaximal, characteristic, complemented
5#26041664C211yesmaximal, characteristic, complemented
6#26041511C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041503C211yesmaximal, characteristic, complemented
9#26041497C211yesmaximal, characteristic, complemented
10#26041440C211yesmaximal, characteristic, complemented
11#26041427C211yesmaximal, characteristic, complemented
12#26041415C211yesmaximal, characteristic, complemented
13#26041364C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#451×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic, complemented
18C22×D724C2211yescharacteristic
19#19153839—24no—
20C2×#15668436—24no—
21C2×#15668436C2211yescharacteristic
22C2×#15668436C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25C2×#15668436C2211yescharacteristic, complemented
26C2×D1448C2211yescharacteristic
27C2×D1448C2211yescharacteristic
28C22×#13089678C2211yescharacteristic
29C2×C8×D181C2211yescharacteristic, complemented
30#19153805—21no—
31#19153805C2211yescharacteristic
32#19153805C2211yescharacteristic, complemented
33#19153801—21no—
34#19153792—21no—
35#19153792C2211yescharacteristic, complemented
36#19153791—21no—
37#19153791C2211yescharacteristic
38#19153790—21no—
39#19153790C2211yescharacteristic
40#19153786—21no—
41Q8×Dic181C2211yescharacteristic, complemented
42#19153778C2211yescharacteristic
43C4×C22⋊D181—21no—
44C4×C22⋊D181C2211yescharacteristic
45C22⋊C4×D181—21no—
46#19153747C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153740C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153736C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C22×Q8×C181C2211yescharacteristic
55C2×Q16×C181C2211yescharacteristic, complemented
56C2×Q16×C181C2211yescharacteristic, complemented
57C22⋊Q8×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60C22⋊C8×C181C2211yescharacteristic
61C23×D181—21no—
62#15668450—22no—
63#15668450—22no—
64#15668450—41no—
65#15668450—41no—
66C2×D724—22no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724—21no—
70C2×D724—21no—
71C2×D724—41no—
72C2×D724C2311yescharacteristic
73C2×D724C2311yescharacteristic
74C2×D724C2311yescharacteristic
75C2×C22⋊D181—21no—
76C2×C22⋊D181D411yescharacteristic
77C2×C4×D181—21no—
78C2×C4×D181—21no—
79C2×C4×D181—21no—
80C2×C4×D181C2311yescharacteristic
81#15668436—22no—
82#15668436—22no—
83#15668436—22no—
84#15668436—22no—
85#15668436D414yescomplemented
86D1448—22no—
87D1448—41no—
88C2×#13089678—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C8×D181—22no—
92C4⋊Dic181—21no—
93C4⋊Dic181C2311yescharacteristic
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181D411yescharacteristic
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108C2×Q8×C181—21no—
109C2×Q8×C181—22no—
110C2×Q8×C181—41no—
111C2×Q8×C181C2311yescharacteristic
112C2×Q8×C181C2311yescharacteristic
113C2×Q8×C181C2311yescharacteristic
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116Q16×C181—22no—
117Q16×C181—41no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C4⋊C4×C181—21no—
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—81no—
129C22×D181C2×D411yescharacteristic
130D724—21no—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—42no—
136D724—42no—
137D724—42no—
138D724—81no—
139D724C2×D412yes—
140C22⋊D181—22no—
141C22⋊D181—42no—
142C2×Dic181—21no—
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C4×D181—21no—
147C4×D181—21no—
148C4×D181—42no—
149C4×D181—42no—
150C4×D181—81no—
151#13089678—21no—
152#13089678C2×D412yes—
153C23×C181C2×D411yescharacteristic
154Q8×C181—21no—
155Q8×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159Q8×C181—42no—
160Q8×C181C2×D412yes—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—42no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181#13512yescomplemented
175C22×C181C22×D411yescharacteristic
176C2×D181—21no—
177C2×D181—22no—
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—84no—
189C4×C181—21no—
190C4×C181—22no—
191C4×C181—41no—
192C4×C181—41no—
193C4×C181—41no—
194C4×C181C22×D411yescharacteristic
195C4×C181H(5,𝔽2)11yescharacteristic
196Dic181—21no—
197Dic181—21no—
198Dic181—21no—
199Dic181—41no—
200C2×C181—22no—
201C2×C181D4211yescharacteristic
202C2×C181C2×#13511yescharacteristic
203C2×C181#57711yescharacteristic
204D181—41no—
205D181—42no—
206D181—81no—
207D181—81no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546—1811no—
214#495—1811no—
215#465—1811no—
216#465—1811no—
217#464—1811no—
218#463—1811no—
219#452—1811no—
220#451D18111yescharacteristic, complemented
221#449—1811no—
222#447—1811no—
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C22×Q8C2×D18111yescharacteristic
228C22×D4—1811no—
229#135—3624no—
230C2×Q16C2×D18111yescharacteristic, complemented
231C2×Q16C2×D18111yescharacteristic, complemented
232C2×SD16—3624no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16—1811no—
238C2×D8—1811no—
239C2×D8—1811no—
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124—1811no—
244#124—1811no—
245#123—3621no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8C2×D18111yescharacteristic
249#121—3621no—
250#121—3621no—
251#121—1811no—
252#121—1811no—
253#120—3621no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4—1811no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103C2×D18111yescharacteristic, complemented
263#103C2×D18111yescharacteristic, complemented
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8—1811no—
268C22⋊C8C2×D18111yescharacteristic
269C4×C8—1811no—
270C24—3621no—
271#41—3622no—
272#41—3622no—
273#41—7241no—
274#41—7241no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D18111yescharacteristic
279C2×Q8C22×D18111yescharacteristic
280C2×Q8C22×D18111yescharacteristic
281C2×D4—3622no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—1811no—
290C2×D4—1811no—
291C2×D4—7241no—
292C22×C4—21no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—22no—
299Q16—41no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—3622no—
306D8—7241no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C4⋊C4C22×D18111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—3621no—
331C42—1811no—
332C23—3621no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—7242no—
338C23—7242no—
339C23D4×D18111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8D4×D18112yes—
347D4—3622no—
348D4—3621no—
349D4—3621no—
350D4—3621no—
351D4—3621no—
352D4—3621no—
353D4—14481no—
354D4—1812no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358D4—7242no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—14481no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—1811no—
373C2×C4—7242no—
374C2×C4—7242no—
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—21no—
379C8—3621no—
380C8—1812no—
381C22—21no—
382C22—21no—
383C22—3621no—
384C22—3622no—
385C22—3621no—
386C22—3621no—
387C22—14484no—
388C22—7241no—
389C22—7241no—
390C22—7241no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22#1915383912yescomplemented
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3621no—
404C4—3621no—
405C4—3621no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—22no—
410C2—14481no—
411C2—14481no—
412C2—7241no—
413C2—7242no—
414C2#2604191111yescharacteristic
415C2C2×#1915383911yescharacteristic
416C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#451×C181C2
socleC22×C181C22×D4
hypercenter#451D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19153839C2semidirect
2. C22×#15668436C2semidirect
3. #26041940C2semidirect
4. #26041920C2semidirect
5. #26041664C2semidirect
6. #26041511C2semidirect
7. #26041510C2semidirect
8. #26041503C2semidirect
9. #26041497C2semidirect
10. #26041440C2semidirect
11. #26041427C2semidirect
12. #26041415C2semidirect
13. #26041364C2semidirect
14. #26041342C2semidirect
15. #451×C181C2semidirect
17. C2×#15668450C22semidirect
25. C2×#15668436C22semidirect
29. C2×C8×D181C22semidirect
32. #19153805C22semidirect
35. #19153792C22semidirect
41. Q8×Dic181C22semidirect
46. #19153747C22semidirect
47. #19153745C22semidirect
48. #19153740C22semidirect
51. #19153736C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
55. C2×Q16×C181C22semidirect
56. C2×Q16×C181C22semidirect
58. #103×C181C22semidirect
59. #103×C181C22semidirect
85. #15668436D4semidirect
174. C22×C181#135semidirect
208. C181#3282semidirect
220. #451D181semidirect
230. C2×Q16C2×D181semidirect
231. C2×Q16C2×D181semidirect
262. #103C2×D181semidirect
263. #103C2×D181semidirect
396. C22#19153839semidirect

Maximal subgroups

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

subgroupindex
C2×#191538392
C22×#156684362
#260419402
#260419202
#260416642
#260415112
#260415102
#260415032
#260414972
#260414402
#260414272
#260414152
#260413642
#260413422
#451×C1812
#3282181