Finite Group Databasebeta

← #72220102

Subgroups — #72220102

Order 5184.

5597 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 5597 conjugacy classes of subgroups — too many to draw. Only its 269 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72220064C216yesmaximal, complemented
2C9×#63515C212yesmaximal, complemented
3#72220046C213yesmaximal, complemented
4C24×C9⋊D9C211yesmaximal, characteristic, complemented
5C2×#13047160C213yesmaximal, complemented
6#13152041—32nomaximal
7C24×C92C2211yescharacteristic, complemented
8C23×C9⋊D9—212no—
9C23×C9⋊D9C2213yes—
10C23×C9⋊D9C2213yescomplemented
11C22×D92C2211yescharacteristic, complemented
12C2×C9×#611C2216yescomplemented
13#13047160—212no—
14C2×#10619530C2216yescomplemented
15C2×#10619528C2213yescomplemented
16C9×#3676C2216yescomplemented
17#13046463C2216yescomplemented
18C24×C3⋊D9—32no—
19C24×C3⋊D9—61no—
20C2×#88668—36no—
21C9×#953—32no—
22C3×#63515D312yes—
23#11721291—36no—
24#11721278—36no—
25#11721274—36no—
26C23×C92—26no—
27C23×C92C2313yes—
28C22×C9⋊D9—232no—
29C22×C9⋊D9—26no—
30C22×C9⋊D9—212no—
31C22×C9⋊D9—212no—
32C22×C9⋊D9D4112yescomplemented
33C22×C9⋊D9C2313yes—
34C22×C9⋊D9C2311yescharacteristic
35C22×C9×D9C2312yescomplemented
36C2×D92—26no—
37C9×#611—212no—
38C2×C9×Dic9C2316yescomplemented
39#10619530—212no—
40#10619528—26no—
41#10595527—91no—
42D9×#120—92no—
43C24×C3×C9—21no—
44C24×C3×C9C2×D312yes—
45C23×C3⋊D9—36no—
46C23×C3⋊D9—36no—
47C23×C3⋊D9—63no—
48C23×C3⋊D9—63no—
49C23×C3⋊D9—624no—
50C23×C3⋊D9—624no—
51C22×D3×D9—32no—
52C2×C9×#67—36no—
53C2×C3×#611C2×D316yes—
54#88668—624no—
55C2×#6195—36no—
56C2×#6192—36no—
57C2×(C3⋊D9)⋊C4—36no—
58C9×#217—36no—
59C3×#3676C2×D316yes—
60#88445—312no—
61C22×C92—28no—
62C22×C92—26no—
63C22×C92C2×D416yescomplemented
64C22×C92C2411yescharacteristic
65C2×C9⋊D9—224no—
66C2×C9⋊D9—26no—
67C2×C9⋊D9—224no—
68C2×C9⋊D9C2×D416yes—
69C2×C9⋊D9C2×D416yescomplemented
70C2×C9×D9—26no—
71D92—44no—
72C9×Dic9—26no—
73C24×C3⋊D3—91no—
74C2×#3811—93no—
75C24×D9—92no—
76C24×D9—182no—
77C24×D9—183no—
78C3×#953—32no—
79#63981—96no—
80#63974—93no—
81C2×D4×D9—96no—
82C9×#120—92no—
83#63515D912yescomplemented
84#63499—96no—
85#63444—96no—
86C22⋊C4×D9—96no—
87C23×C3×C9—212no—
88C23×C3×C9—23no—
89C23×C3×C9—212no—
90C23×C3×C9C22×D316yes—
91C22×C3⋊D9—32no—
92C22×C3⋊D9—324no—
93C22×C3⋊D9—36no—
94C22×C3⋊D9—624no—
95C22×C3⋊D9—624no—
96C22×C3⋊D9—612no—
97C22×C3⋊D9—61no—
98C22×C3⋊D9—63no—
99C22×C3⋊D9—664no—
100C22×C3⋊D9—624no—
101C22×C3⋊D9—624no—
102C22×C3⋊D9—664no—
103C22×C3⋊D9—612no—
104C22×C3⋊D9—612no—
105C22×D3×C9—32no—
106C22×C3×D9C22×D312yes—
107C2×D3×D9—612no—
108C9×#67—612no—
109C3×#611—212no—
110C2×C9×Dic3—36no—
111C2×C3×Dic9C22×D316yes—
112#6195—612no—
113#6192—612no—
114(C3⋊D9)⋊C4—612no—
115D3×#120—272no—
116C2×C92—26no—
117C2×C92C22×D413yes—
118C9⋊D9—26no—
119C9⋊D9#12014yescomplemented
120C9×D9—42no—
121C24×C32D3211yescharacteristic
122C23×C3⋊D3—93no—
123C23×C3⋊D3—93no—
124C23×C3⋊D3—1812no—
125C22×D32—91no—
126C2×C3×#67—36no—
127#3811—1812no—
128C2×#626—96no—
129C2×#624—93no—
130C24×C9—23no—
131C24×C9—22no—
132C24×C9C2×D912yescomplemented
133C23×D9—96no—
134C23×D9—96no—
135C23×D9—92no—
136C23×D9—1824no—
137C23×D9—186no—
138C23×D9—186no—
139C23×D9—1848no—
140C23×D9—189no—
141C23×D9—189no—
142C23×D9—1872no—
143C3×#217—36no—
144#3722—96no—
145C2×D4×C9—96no—
146C2×#611—96no—
147C2×#611C2×D916yescomplemented
148D4×D9—1824no—
149C2×D36—96no—
150C2×C4×D9—96no—
151C9×C22⋊C4—96no—
152#3676C2×D916yescomplemented
153(C2×D9)⋊C4—912no—
154C22×C3×C9—216no—
155C22×C3×C9—212no—
156C22×C3×C9—216no—
157C22×C3×C9—21no—
158C22×C3×C9—26no—
159C22×C3×C9—212no—
160C22×C3×C9D3×D4112yes—
161C22×C3×C9C23×D312yes—
162C2×C3⋊D9—312no—
163C2×C3⋊D9—312no—
164C2×C3⋊D9—648no—
165C2×C3⋊D9—66no—
166C2×C3⋊D9—66no—
167C2×C3⋊D9—648no—
168C2×C3⋊D9—648no—
169C2×C3⋊D9—612no—
170C2×C3⋊D9—648no—
171C2×C3⋊D9—612no—
172C2×D3×C9—66no—
173C2×C3×D9—26no—
174D3×D9—128no—
175C9×Dic3—66no—
176C3×Dic9—26no—
177C24×D3—272no—
178C24×D3—541no—
179C2×D3×D4—276no—
180C3×#120—92no—
181#953—32no—
182#937—276no—
183#882—276no—
184D3×C22⋊C4—276no—
185C92C2×#12011yescharacteristic, complemented, Sylow
186C23×C32—26no—
187C23×C32C2×D3213yes—
188C22×C3⋊D3—91no—
189C22×C3⋊D3—93no—
190C22×C3⋊D3—912no—
191C22×C3⋊D3—1812no—
192C22×C3⋊D3—1812no—
193C22×C3⋊D3—1832no—
194C22×C3⋊D3—186no—
195C22×C3×D3—32no—
196C2×D32—186no—
197C3×#67—612no—
198C2×C3×Dic3—36no—
199#626—1812no—
200#624—186no—
201C23×C9—236no—
202C23×C9—224no—
203C23×C9—29no—
204C23×C9—212no—
205C23×C9—26no—
206C23×C9—92no—
207C23×C9C22×D916yes—
208C22×D9—92no—
209C22×D9—924no—
210C22×D9—96no—
211C22×D9—182no—
212C22×D9—186no—
213C22×D9—1824no—
214C22×D9—1812no—
215C22×D9—1824no—
216C22×D9—1824no—
217C22×D9—1836no—
218C22×D9—183no—
219C22×D9—189no—
220C22×D9—1848no—
221C22×D9—1848no—
222C22×D9—1864no—
223C22×D9—1872no—
224C22×D9—1872no—
225C22×D9—18128no—
226C22×D9—18192no—
227C22×D9—1836no—
228C22×D9—1824no—
229C22×D9—1812no—
230C22×D9C22×D912yescomplemented
231D4×C9—1812no—
232C2×C4×C9—96no—
233#611—212no—
234#611—1812no—
235C2×Dic9C22×D916yescomplemented
236D36—1812no—
237C4×D9—1812no—
238C2×#120—811noSylow
239C2×C3×C9—212no—
240C2×C3×C9—212no—
241C2×C3×C9—23no—
242C2×C3×C9C2×D3×D416yes—
243C3⋊D9—38no—
244C3⋊D9—64no—
245C3⋊D9—612no—
246C3⋊D9—612no—
247D3×C9—122no—
248C3×D9—42no—
249C24×C3—21no—
250C24×C3D3×D912yes—
251C23×D3—272no—
252C23×D3—276no—
253C23×D3—276no—
254C23×D3—543no—
255C23×D3—543no—
256C23×D3—5424no—
257C23×D3—5424no—
258C2×C3×D4—96no—
259C2×#67—36no—
260C2×#67—276no—
261D3×D4—5424no—
262C2×D12—276no—
263C2×C4×D3—276no—
264C3×C22⋊C4—96no—
265#217—36no—
266(C2×D3)⋊C4—2712no—
267C22×C32—28no—
268C22×C32—26no—
269C22×C32#381116yes—
270C22×C32C22×D3211yescharacteristic
271C2×C3⋊D3—96no—
272C2×C3⋊D3—96no—
273C2×C3⋊D3—1824no—
274C2×C3⋊D3—1824no—
275C2×C3⋊D3—186no—
276C2×C3×D3—66no—
277D32—364no—
278C3×Dic3—66no—
279C22×C9—248no—
280C22×C9—232no—
281C22×C9—236no—
282C22×C9—216no—
283C22×C9—224no—
284C22×C9—23no—
285C22×C9—218no—
286C22×C9—212no—
287C22×C9—22no—
288C22×C9—212no—
289C22×C9—186no—
290C22×C9D4×D9112yescomplemented
291C22×C9C23×D912yes—
292C2×D9—26no—
293C2×D9—912no—
294C2×D9—912no—
295C2×D9—1812no—
296C2×D9—1812no—
297C2×D9—1848no—
298C2×D9—1818no—
299C2×D9—1818no—
300C2×D9—1896no—
301C2×D9—18144no—
302C2×D9—18144no—
303C2×D9—1896no—
304C2×D9—1836no—
305C2×D9—1848no—
306C2×D9—1824no—
307C2×D9—1812no—
308C2×D9—368no—
309C4×C9—186no—
310Dic9—26no—
311C25—811no—
312C22×D4—813no—
313#120—92no—
314#120—816no—
315C2×C22⋊C4—813no—
316C3×C9—21no—
317C3×C9D3×#12012yes—
318C23×C3—212no—
319C23×C3—212no—
320C23×C3—23no—
321C23×C3—92no—
322C23×C3C2×D3×D916yes—
323C22×D3—32no—
324C22×D3—272no—
325C22×D3—276no—
326C22×D3—2724no—
327C22×D3—541no—
328C22×D3—543no—
329C22×D3—5412no—
330C22×D3—5412no—
331C22×D3—5424no—
332C22×D3—5424no—
333C22×D3—5424no—
334C22×D3—5424no—
335C22×D3—5464no—
336C22×D3—5464no—
337C22×D3—5412no—
338C22×D3—5412no—
339C3×D4—1812no—
340C2×C3×C4—96no—
341#67—612no—
342#67—5412no—
343C2×Dic3—36no—
344D12—5412no—
345C4×D3—5412no—
346C2×C32—26no—
347C2×C32C2×#381113yes—
348C3⋊D3—94no—
349C3⋊D3—186no—
350C3×D3—122no—
351C2×C9—236no—
352C2×C9—224no—
353C2×C9—29no—
354C2×C9—212no—
355C2×C9—26no—
356C2×C9—362no—
357C2×C9C2×D4×D916yes—
358D9—42no—
359D9—98no—
360D9—188no—
361D9—1812no—
362D9—1836no—
363D9—1824no—
364D9—1812no—
365C24—811no—
366C24—813no—
367C24—813no—
368C24—16212no—
369C24D9211yescharacteristic, complemented
370C2×D4—96no—
371C2×D4—816no—
372C2×D4—16212no—
373C22×C4—813no—
374C22⋊C4—96no—
375C22⋊C4—816no—
376C22×C3—216no—
377C22×C3—216no—
378C22×C3—212no—
379C22×C3—212no—
380C22×C3—21no—
381C22×C3—26no—
382C22×C3—186no—
383C22×C3#88668112yes—
384C22×C3C22×D3×D912yes—
385C2×D3—66no—
386C2×D3—2712no—
387C2×D3—2712no—
388C2×D3—546no—
389C2×D3—546no—
390C2×D3—5448no—
391C2×D3—5448no—
392C2×D3—5448no—
393C2×D3—5448no—
394C2×D3—5412no—
395C2×D3—5412no—
396C2×D3—1088no—
397C3×C4—186no—
398Dic3—66no—
399C32#1059552711yescharacteristic
400C9—23no—
401C9—22no—
402C9D9×#12012yescomplemented
403C23—26no—
404C23—92no—
405C23—811no—
406C23—813no—
407C23—8112no—
408C23—16212no—
409C23—16212no—
410C23—16232no—
411C23—1626no—
412C23—1626no—
413C23C2×D9213yes—
414D4—1812no—
415D4—16212no—
416C2×C4—96no—
417C2×C4—1626no—
418C2×C3—212no—
419C2×C3—212no—
420C2×C3—23no—
421C2×C3—362no—
422C2×C3C2×#8866816yes—
423D3—122no—
424D3—278no—
425D3—544no—
426D3—5412no—
427D3—5412no—
428C22—28no—
429C22—26no—
430C22—186no—
431C22—3244no—
432C22—816no—
433C22—816no—
434C22—16224no—
435C22—16224no—
436C22—1626no—
437C22#1304716016yescomplemented
438C22C22×D9211yescharacteristic
439C4—186no—
440C3—21no—
441C3#1315204112yes—
442C2—26no—
443C2—362no—
444C2—814no—
445C2—1626no—
446C2C2×#1304716013yes—
subgroupG/H
centerC22C22×D92
commutator, cosocleC22×C92C24
Frattini, socleC22×C32C22×D32
FittingC24×C92C22
hypercenterC24D92
hypocenterC92C2×#120
LCSG ▷ C22×C92 ▷ C92
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. #72220064C2semidirect
2. C9×#63515C2semidirect
3. #72220046C2semidirect
4. C24×C9⋊D9C2semidirect
5. C2×#13047160C2semidirect
7. C24×C92C22semidirect
10. C23×C9⋊D9C22semidirect
11. C22×D92C22semidirect
12. C2×C9×#611C22semidirect
14. C2×#10619530C22semidirect
15. C2×#10619528C22semidirect
16. C9×#3676C22semidirect
17. #13046463C22semidirect
32. C22×C9⋊D9D4semidirect
35. C22×C9×D9C23semidirect
38. C2×C9×Dic9C23semidirect
63. C22×C92C2×D4semidirect
69. C2×C9⋊D9C2×D4semidirect
83. #63515D9semidirect
119. C9⋊D9#120semidirect
132. C24×C9C2×D9semidirect
147. C2×#611C2×D9semidirect
152. #3676C2×D9semidirect
185. C92C2×#120semidirect
230. C22×D9C22×D9semidirect
235. C2×Dic9C22×D9semidirect
290. C22×C9D4×D9semidirect
369. C24D92semidirect
402. C9D9×#120semidirect
437. C22#13047160semidirect

Maximal subgroups

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

subgroupindex
#722200642
C9×#635152
#722200462
C24×C9⋊D92
C2×#130471602
#131520413