Finite Group Databasebeta

← #13149833

Subgroups — #13149833

Order 1728.

1014 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 145 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#11722493C212yesmaximal, complemented
2C2×#88529C211yesmaximal, characteristic, complemented
3#64113⋊C3C211yesmaximal, characteristic, complemented
4#64110⋊C3C211yesmaximal, characteristic, complemented
5#11721068C212yesmaximal, complemented
6#11720850C212yesmaximal, complemented
7#11720848C214yesmaximal, complemented
8#11720847C212yesmaximal, complemented
9#10594687—31nomaximal
10#10594678—31nomaximal
11C2×C4⋊#272C2212yescomplemented
12#3828⋊C3C2211yescharacteristic
13#3828⋊C3C2212yescomplemented
14#88677—24no—
15#88654—22no—
16#88654C2214yescomplemented
17#88652—22no—
18#88652C2214yescomplemented
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20#88529C2212yescomplemented
21#88529C2212yescomplemented
22#3748⋊C3C2211yescharacteristic, complemented
23#3745⋊C3C2211yescharacteristic, complemented
24#3744⋊C3C2212yescomplemented
25#3743⋊C3C2212yescomplemented
26C8×(C3⋊D3)⋊C3C2212yescomplemented
27#3746⋊C3C2211yescharacteristic, complemented
28#88437C2214yescomplemented
29#88436C2214yescomplemented
30#64307—32no—
31#64307—32no—
32#64113D311yescharacteristic, complemented
33#64110D311yescharacteristic, complemented
34C3×#904—31no—
35C2×C3×D24—31no—
36C2×C3×D24—31no—
37C2×C3×D24—61no—
38C3×#901—31no—
39#63830—32no—
40#63823—32no—
41#63810—32no—
42#63807—32no—
43#63798—34no—
44#63797—32no—
45#63795—34no—
46D3×D24—32no—
47C22×#272—22no—
48C2×((C3⋊D3)⋊C3)⋊C2—24no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C4⋊#272—22no—
51C4⋊#272C2314yes—
52#638⋊C3—22no—
53#638⋊C3D412yes—
54C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
55C4⋊((C3⋊D3)⋊C3)C2312yes—
56C4×(C3⋊D3)⋊C3—22no—
57C4×(C3⋊D3)⋊C3C2312yes—
58#634⋊C3D411yescharacteristic
59#634⋊C3C2312yes—
60#6200—24no—
61#6200—24no—
62#6199—24no—
63C8×H(3,𝔽3)C2311yescharacteristic
64C8×H(3,𝔽3)C2311yescharacteristic
65(C3⋊(C3⋊C8))⋊C3C2312yes—
66#5769—91no—
67#5757—91no—
68#5752—91no—
69#3828C2×D311yescharacteristic
70#3828C2×D312yescomplemented
71C3×#235—31no—
72C3×#235—32no—
73C2×C3×D12—32no—
74C2×C3×D12—32no—
75C2×C3×D12—62no—
76D3×#67—64no—
77D3×#67—64no—
78#3802—34no—
79#3802—62no—
80D3×D12—34no—
81D3×D12—62no—
82#3796—34no—
83#3796—62no—
84#3795—34no—
85#3795—62no—
86C2×C32×C8—21no—
87C2×C32×C8C2×D311yescharacteristic, complemented
88C2×C32×C8C2×D311yescharacteristic, complemented
89#3748C2×D311yescharacteristic, complemented
90#3746C2×D311yescharacteristic, complemented
91#3745C2×D311yescharacteristic, complemented
92#3744C2×D312yescomplemented
93#3743C2×D312yescomplemented
94C8×C3⋊D3C2×D312yescomplemented
95C3×#208—31no—
96C3×Dic12—31no—
97C3×D24—32no—
98C3×D24—32no—
99C3×D24—32no—
100C3×D24—32no—
101C3×D24—31no—
102C3×D24—62no—
103C3×D24—62no—
104C3×#204—32no—
105C3×ΓL(1,𝔽25)—32no—
106C3×D3×C8—32no—
107#3716—34no—
108#3715—34no—
109#3714—34no—
110#3713—34no—
111C22×H(3,𝔽3)C2×D411yescharacteristic
112C2×#272—24no—
113C2×#272—82no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3C2×D412yes—
116((C3⋊D3)⋊C3)⋊C2—84no—
117C4×H(3,𝔽3)C2×D411yescharacteristic
118C4×H(3,𝔽3)C2411yescharacteristic
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3C2×D412yes—
121#1009—92no—
122#1009—92no—
123#1004—92no—
124C3×#135—91no—
125C2×C3×D8—91no—
126C3×#131—91no—
127#949—92no—
128#932—92no—
129#919—92no—
130#914—94no—
131#912—92no—
132#911—94no—
133D3×D8—92no—
134D3×D8—92no—
135#908—94no—
136#908—92no—
137#904—31no—
138#904—91no—
139#904—92no—
140C2×D24—31no—
141C2×D24—31no—
142C2×D24—61no—
143C2×D24—92no—
144#901—31no—
145#901—91no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—122no—
149C2×D32—64no—
150C2×D32—64no—
151C2×C32×C4—21no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154#638—22no—
155#638D1212yes—
156C4⋊(C3⋊D3)D1211yescharacteristic
157C4⋊(C3⋊D3)C22×D312yes—
158C4×C3⋊D3—22no—
159C4×C3⋊D3C22×D312yes—
160#634D1211yescharacteristic
161#634C22×D312yes—
162C3×#67—32no—
163C3×#67—62no—
164C3×D12—31no—
165C3×D12—32no—
166C3×D12—34no—
167C3×D12—34no—
168C3×D12—64no—
169C3×D12—62no—
170C3×D12—62no—
171C3×D12—122no—
172C3×C4×D3—32no—
173C3×C4×D3—62no—
174C3×Dic6—31no—
175C3×Dic6—32no—
176#626—64no—
177#626—64no—
178#625—64no—
179#625—64no—
180D3×Dic3—64no—
181D3×Dic3—64no—
182C32×C8—21no—
183C32×C8—21no—
184C32×C8C22×D311yescharacteristic
185C32×C8C22×D311yescharacteristic
186C32×C8C22×D311yescharacteristic
187C32×C8C22×D311yescharacteristic
188C3⋊(C3⋊C8)C22×D312yes—
189C3×C3⋊C8—32no—
190#576—271noSylow
191C2×H(3,𝔽3)—21no—
192C2×H(3,𝔽3)C22×D411yescharacteristic
193#272—44no—
194(C3⋊D3)⋊C3—22no—
195(C3⋊D3)⋊C3—42no—
196C3×#41—91no—
197C3×#41—92no—
198C2×C3×D4—92no—
199C2×#67—184no—
200#239—94no—
201#239—182no—
202#237—94no—
203#237—182no—
204D3×D4—94no—
205D3×D4—94no—
206D3×D4—182no—
207D3×D4—182no—
208D3×D4—184no—
209D3×D4—184no—
210#235—31no—
211#235—32no—
212#235—91no—
213#235—92no—
214#235—94no—
215#235—182no—
216C2×D12—32no—
217C2×D12—32no—
218C2×D12—62no—
219C2×D12—94no—
220C2×D12—182no—
221C3×Q16—91no—
222C3×SD16—92no—
223C3×D8—91no—
224C3×D8—92no—
225C3×D8—92no—
226C3×M16—92no—
227C3×M16—91no—
228C2×C3×C8—31no—
229C2×C3×C8—31no—
230C2×C3×C8—61no—
231C2×C3×C8—92no—
232C2×C3×C8D3211yescharacteristic
233#215—94no—
234#214—94no—
235#213—94no—
236#213—94no—
237#208—31no—
238#208—91no—
239Dic12—31no—
240Dic12—91no—
241D24—32no—
242D24—32no—
243D24—32no—
244D24—32no—
245D24—31no—
246D24—62no—
247D24—62no—
248D24—91no—
249D24—94no—
250#204—32no—
251#204—92no—
252#204—94no—
253ΓL(1,𝔽25)—32no—
254ΓL(1,𝔽25)—92no—
255D3×C8—32no—
256D3×C8—92no—
257C22×C32—21no—
258C22×C32C2×D1211yescharacteristic
259C22×C32D3×D411yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3C2×D1212yes—
262C2×C3×D3—32no—
263C2×C3×D3—62no—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C2×C3×D3—124no—
267C2×C3×D3—242no—
268C2×C3×D3—242no—
269C2×C3×D3—244no—
270D32—244no—
271D32—244no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4C2×D1211yescharacteristic
275C32×C4D3×D411yescharacteristic
276C32×C4C23×D311yescharacteristic
277C32×C4C23×D311yescharacteristic
278C3⋊Dic3—22no—
279C3⋊Dic3C2×D1212yes—
280C3×Dic3—32no—
281C3×Dic3—62no—
282H(5,𝔽2)—272no—
283AGL(1,ℤ/8)—274no—
284AGL(1,ℤ/8)—272no—
285#135—91no—
286#135—272no—
287C2×D8—91no—
288C2×D8—272no—
289#131—91no—
290H(3,𝔽3)#57611yescharacteristic, complemented, Sylow
291C23×C3—182no—
292C22×D3—62no—
293C22×D3—62no—
294C22×D3—122no—
295C22×D3—184no—
296C22×D3—184no—
297C22×D3—184no—
298C3×Q8—91no—
299C3×Q8—92no—
300C3×D4—91no—
301C3×D4—92no—
302C3×D4—92no—
303C3×D4—94no—
304C3×D4—182no—
305C3×D4—182no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—61no—
309C2×C3×C4—92no—
310C2×C3×C4—182no—
311C2×C3×C4C2×D3211yescharacteristic
312#67—32no—
313#67—62no—
314#67—92no—
315#67—182no—
316#67—184no—
317#67—184no—
318#67—184no—
319#67—184no—
320C2×Dic3—184no—
321D12—34no—
322D12—34no—
323D12—32no—
324D12—31no—
325D12—64no—
326D12—62no—
327D12—62no—
328D12—91no—
329D12—92no—
330D12—122no—
331D12—184no—
332D12—184no—
333C4×D3—32no—
334C4×D3—62no—
335C4×D3—92no—
336C4×D3—182no—
337C4×D3—184no—
338C4×D3—184no—
339Dic6—31no—
340Dic6—32no—
341Dic6—91no—
342Dic6—92no—
343C3×C8—31no—
344C3×C8—31no—
345C3×C8—31no—
346C3×C8—31no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—92no—
350C3×C8C2×D3211yescharacteristic
351C3×C8C2×D3211yescharacteristic
352C3⋊C8—32no—
353C3⋊C8—92no—
354C2×C32—22no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32C22×D1211yescharacteristic
359C2×C32C2×D3×D411yescharacteristic
360C3⋊D3—22no—
361C3⋊D3—42no—
362C3×D3—62no—
363C3×D3—122no—
364C3×D3—124no—
365C3×D3—124no—
366C3×D3—244no—
367#41—91no—
368#41—92no—
369#41—274no—
370#41—542no—
371C2×D4—92no—
372C2×D4—274no—
373C2×D4—542no—
374C2×D4—544no—
375Q16—91no—
376SD16—92no—
377SD16—274no—
378D8—92no—
379D8—92no—
380D8—91no—
381D8—274no—
382M16—91no—
383M16—92no—
384C2×C8—92no—
385C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—61no—
389C22×C3—92no—
390C22×C3—182no—
391C22×C3—184no—
392C22×C3—722no—
393C22×C3D3×D1211yescharacteristic
394C2×D3—32no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—62no—
398C2×D3—92no—
399C2×D3—124no—
400C2×D3—182no—
401C2×D3—242no—
402C2×D3—242no—
403C2×D3—244no—
404C2×D3—724no—
405C2×D3—724no—
406C2×D3—724no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—92no—
414C3×C4—182no—
415C3×C4D3×D1211yescharacteristic
416C3×C4C22×D3211yescharacteristic
417Dic3—32no—
418Dic3—62no—
419Dic3—92no—
420Dic3—182no—
421C32—21no—
422C32#575211yescharacteristic, complemented
423C32#576911yescharacteristic, complemented
424C23—182no—
425C23—544no—
426Q8—91no—
427Q8—92no—
428D4—92no—
429D4—94no—
430D4—92no—
431D4—91no—
432D4—182no—
433D4—182no—
434D4—544no—
435D4—544no—
436C2×C4—92no—
437C2×C4—182no—
438C2×C4—544no—
439C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C8—92no—
441C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
442C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C2×C3—21no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—62no—
449C2×C3—61no—
450C2×C3—182no—
451C2×C3—362no—
452C2×C3—364no—
453C2×C3C2×D3×D1211yescharacteristic
454D3—62no—
455D3—124no—
456D3—124no—
457D3—122no—
458D3—182no—
459D3—244no—
460D3—362no—
461C22—92no—
462C22—182no—
463C22—184no—
464C22—722no—
465C22—2164no—
466C22#8865411yescharacteristic
467C4—92no—
468C4—182no—
469C4#8865411yescharacteristic
470C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
471C3—31no—
472C3—31no—
473C3—61no—
474C3#1059468011yescharacteristic
475C2—21no—
476C2—182no—
477C2—364no—
478C2—362no—
479C2C2×#8865411yescharacteristic
subgroupG/H
centerC2C2×#88654
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#576
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11722493C2semidirect
2. C2×#88529C2semidirect
3. #64113⋊C3C2semidirect
4. #64110⋊C3C2semidirect
5. #11721068C2semidirect
6. #11720850C2semidirect
7. #11720848C2semidirect
8. #11720847C2semidirect
11. C2×C4⋊#272C22semidirect
13. #3828⋊C3C22semidirect
16. #88654C22semidirect
18. #88652C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. #88529C22semidirect
21. #88529C22semidirect
22. #3748⋊C3C22semidirect
23. #3745⋊C3C22semidirect
24. #3744⋊C3C22semidirect
25. #3743⋊C3C22semidirect
26. C8×(C3⋊D3)⋊C3C22semidirect
27. #3746⋊C3C22semidirect
28. #88437C22semidirect
29. #88436C22semidirect
32. #64113D3semidirect
33. #64110D3semidirect
70. #3828C2×D3semidirect
87. C2×C32×C8C2×D3semidirect
88. C2×C32×C8C2×D3semidirect
89. #3748C2×D3semidirect
90. #3746C2×D3semidirect
91. #3745C2×D3semidirect
92. #3744C2×D3semidirect
93. #3743C2×D3semidirect
94. C8×C3⋊D3C2×D3semidirect
290. H(3,𝔽3)#576semidirect
385. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
422. C32#5752semidirect
423. C32#5769semidirect

Maximal subgroups

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

subgroupindex
#117224932
C2×#885292
#64113⋊C32
#64110⋊C32
#117210682
#117208502
#117208482
#117208472
#105946873
#105946783