Finite Group Databasebeta

← #13155237

Subgroups — #13155237

Order 1728.

512 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D3×#3745C211yesmaximal, characteristic, complemented
2C3×#63807C211yesmaximal, characteristic, complemented
3#11722068C211yesmaximal, characteristic, complemented
4#11722012C211yesmaximal, characteristic, complemented
5#11722005C211yesmaximal, characteristic, complemented
6C3×#63546C211yesmaximal, characteristic, complemented
7C3×#63543C211yesmaximal, characteristic, complemented
8#10591024—31nomaximal
9#10590634—31nomaximal
10#10590232—31nomaximal
11D3×C4⋊(C3⋊D3)—21no—
12C3×#3795—21no—
13#88852C2211yescharacteristic
14C3×#3746C2211yescharacteristic, complemented
15C3×#3745C2211yescharacteristic
16C32×D24C2211yescharacteristic, complemented
17C32×D3×C8C2211yescharacteristic
18#88791—21no—
19C3×#3716—21no—
20C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
21C32×C3⋊C16C2211yescharacteristic, complemented
22C2×#3745—31no—
23C3×#912—31no—
24C3×#904—31no—
25#63807D311yescharacteristic, complemented
26D3×D24—31no—
27D3×D24—31no—
28D3×D24—61no—
29#63653—31no—
30#63627—31no—
31C3×#826—31no—
32C3×#812—31no—
33C3×D48—31no—
34C3×#798—31no—
35#63549—31no—
36#63549—31no—
37#63549—31no—
38#63549—61no—
39#63548—31no—
40#63547—31no—
41#63546—31no—
42#63546—31no—
43#63546—61no—
44#63546D311yescharacteristic, complemented
45#63543D311yescharacteristic, complemented
46C2×D3×C3⋊D3—41no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×#634—21no—
50C32×D12—21no—
51C32×C4×D3D411yescharacteristic
52#6292—41no—
53C3×#625—41no—
54C3×D3×Dic3—41no—
55C33×C8C2311yescharacteristic
56C32×C3⋊C8D411yescharacteristic
57#5193—91no—
58#4911—91no—
59#4908—91no—
60C2×C4⋊(C3⋊D3)—61no—
61C3×#237—61no—
62C3×#235—61no—
63D3×D12—62no—
64D3×D12—61no—
65D3×D12—61no—
66#3795—21no—
67C32×D8—31no—
68C2×C32×C8—31no—
69#3746C2×D311yescharacteristic, complemented
70#3745—31no—
71#3745—31no—
72#3745—31no—
73#3745—61no—
74#3745—61no—
75#3745—61no—
76#3745—61no—
77#3745—121no—
78#3745C2×D311yescharacteristic
79C3×#214—61no—
80C3×Dic12—31no—
81C3×Dic12—31no—
82C3×Dic12—61no—
83C3×D24—21no—
84C3×D24—31no—
85C3×D24—31no—
86C3×D24—61no—
87C3×D24C2×D311yescharacteristic, complemented
88C3×D24C2×D311yescharacteristic, complemented
89C3×#204—61no—
90C3×D3×C8—21no—
91C3×D3×C8C2×D311yescharacteristic
92C3×D3×C8C2×D311yescharacteristic
93#3716—21no—
94#3714—62no—
95#3714—61no—
96#3714—61no—
97C32×C16—31no—
98C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
99C3×C3⋊C16—21no—
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C3×C3⋊C16C2×D311yescharacteristic, complemented
104C3×C3⋊C16C2×D311yescharacteristic, complemented
105C2×C3⋊(C3⋊D3)—41no—
106C2×C3×C3⋊D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3D811yescharacteristic
109D3×C3⋊D3—82no—
110C33×C4C2×D411yescharacteristic
111C3×C3⋊Dic3—41no—
112C32×Dic3D811yescharacteristic
113C3×#135—91no—
114#912—31no—
115D3×D8—91no—
116#904—31no—
117C2×D24—91no—
118C2×D24—91no—
119C2×D24—181no—
120C3×D16—91no—
121C3×M32—91no—
122#828—91no—
123#828—91no—
124#827—91no—
125#826—31no—
126#826—91no—
127#826—91no—
128#826—181no—
129#812—31no—
130#800—91no—
131#800—91no—
132#800—91no—
133#800—181no—
134D48—31no—
135D48—91no—
136#798—31no—
137C22×C3⋊D3—121no—
138C2×D32—122no—
139C2×D32—121no—
140C2×D32—121no—
141C32×D4—61no—
142C2×C32×C4—31no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—62no—
146C4⋊(C3⋊D3)—61no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—121no—
149C4⋊(C3⋊D3)—121no—
150C4⋊(C3⋊D3)—122no—
151C4⋊(C3⋊D3)—121no—
152#634—21no—
153C3×#67—121no—
154C3×#67—121no—
155C2×C3×Dic3—121no—
156C3×D12—21no—
157C3×D12—21no—
158C3×D12—41no—
159C3×D12—62no—
160C3×D12—61no—
161C3×D12—61no—
162C3×C4×D3—21no—
163C3×C4×D3—121no—
164C3×C4×D3D1211yescharacteristic
165C3×C4×D3#6711yescharacteristic
166C3×Dic6—61no—
167C3×Dic6—61no—
168C3×Dic6—121no—
169#626—122no—
170#626—121no—
171#626—121no—
172#625—41no—
173D3×Dic3—41no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—21no—
177C32×C8—31no—
178C32×C8—41no—
179C32×C8C22×D311yescharacteristic
180C32×C8C22×D311yescharacteristic
181C32×C8C22×D311yescharacteristic
182C3×C3⋊C8—21no—
183C3×C3⋊C8D1211yescharacteristic
184C3×C3⋊C8#6711yescharacteristic
185#509—271noSylow
186C2×C33C2×D811yescharacteristic
187C3⋊(C3⋊D3)—81no—
188C3×C3⋊D3—81no—
189C32×D3—21no—
190C32×D3—81no—
191C3×#41—181no—
192#237—61no—
193D3×D4—181no—
194#235—61no—
195C2×D12—181no—
196C2×D12—181no—
197C2×D12—182no—
198C3×Q16—91no—
199C3×SD16—181no—
200C3×D8—31no—
201C3×D8—31no—
202C3×D8—61no—
203C3×D8—91no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207#214—61no—
208#213—181no—
209Dic12—31no—
210Dic12—31no—
211Dic12—61no—
212D24—31no—
213D24—31no—
214D24—61no—
215D24—91no—
216D24—91no—
217D24—91no—
218D24—181no—
219D24—181no—
220D24—181no—
221D24—181no—
222D24—181no—
223D24—182no—
224D24—361no—
225D24D3211yescharacteristic, complemented
226#204—61no—
227D3×C8D3211yescharacteristic
228C3×C16—31no—
229C3×C16—31no—
230C3×C16—61no—
231C3×C16—91no—
232C3⋊C16—31no—
233C3⋊C16—31no—
234C3⋊C16—61no—
235C3⋊C16D3211yescharacteristic, complemented
236C22×C32—31no—
237C22×C32—121no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3—122no—
241C2×C3⋊D3—121no—
242C2×C3⋊D3—121no—
243C2×C3⋊D3—241no—
244C2×C3⋊D3—241no—
245C2×C3⋊D3—242no—
246C2×C3⋊D3—242no—
247C2×C3×D3—21no—
248C2×C3×D3—41no—
249C2×C3×D3—41no—
250C2×C3×D3—81no—
251C2×C3×D3—122no—
252C2×C3×D3—121no—
253C2×C3×D3—121no—
254C2×C3×D3D2411yescharacteristic
255C2×C3×D3#21311yescharacteristic
256D32—244no—
257D32—242no—
258D32—242no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—31no—
263C32×C4—41no—
264C32×C4C2×D1211yescharacteristic
265C32×C4D3×D411yescharacteristic
266C32×C4C2×#6711yescharacteristic
267C3⋊Dic3—41no—
268C3×Dic3—21no—
269C3×Dic3—121no—
270C3×Dic3—121no—
271C3×Dic3—241no—
272C3×Dic3D2411yescharacteristic
273C3×Dic3#21311yescharacteristic
274#135—91no—
275C2×D8—271no—
276SD32—271no—
277SD32—271no—
278D16—91no—
279D16—271no—
280M32—91no—
281C33#50911yescharacteristic, complemented, Sylow
282C22×D3—361no—
283C22×D3—361no—
284C22×D3—362no—
285C22×D3—361no—
286C3×Q8—181no—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4—121no—
290C3×D4—181no—
291C3×D4—361no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—361no—
296#67—121no—
297#67—121no—
298#67—361no—
299C2×Dic3—121no—
300D12—21no—
301D12—62no—
302D12—61no—
303D12—61no—
304D12—182no—
305D12—181no—
306D12—181no—
307D12—181no—
308D12—361no—
309D12—361no—
310D12—362no—
311D12—361no—
312D12—361no—
313D12—362no—
314C4×D3—121no—
315C4×D3#62611yescharacteristic
316Dic6—61no—
317Dic6—61no—
318Dic6—121no—
319C3×C8—21no—
320C3×C8—21no—
321C3×C8—21no—
322C3×C8—31no—
323C3×C8—31no—
324C3×C8—41no—
325C3×C8—61no—
326C3×C8C2×D3211yescharacteristic
327C3×C8C2×D3211yescharacteristic
328C3×C8C2×D3211yescharacteristic
329C3⋊C8#62611yescharacteristic
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—41no—
334C2×C32—61no—
335C2×C32—241no—
336C2×C32C2×D2411yescharacteristic
337C2×C32D3×D811yescharacteristic
338C2×C32C2×#21311yescharacteristic
339C3⋊D3—81no—
340C3⋊D3—241no—
341C3⋊D3—242no—
342C3⋊D3—241no—
343C3⋊D3—241no—
344C3⋊D3—481no—
345C3⋊D3—481no—
346C3⋊D3—482no—
347C3×D3—22no—
348C3×D3—21no—
349C3×D3—21no—
350C3×D3—81no—
351C3×D3—81no—
352C3×D3—161no—
353C3×D3—242no—
354C3×D3—241no—
355C3×D3—241no—
356#41—181no—
357C2×D4—541no—
358Q16—91no—
359SD16—181no—
360D8—31no—
361D8—91no—
362D8—271no—
363D8—541no—
364C2×C8—31no—
365C16—31no—
366C16—91no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—61no—
370C22×C3—121no—
371C22×C3—121no—
372C22×C3—241no—
373C22×C3—361no—
374C2×D3—41no—
375C2×D3—122no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—362no—
379C2×D3—361no—
380C2×D3—361no—
381C2×D3—361no—
382C2×D3—721no—
383C2×D3—721no—
384C2×D3—722no—
385C2×D3—722no—
386C2×D3—724no—
387C2×D3—722no—
388C2×D3—722no—
389C2×D3#371411yescharacteristic
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—41no—
396C3×C4—61no—
397C3×C4—361no—
398C3×C4D3×D1211yescharacteristic
399C3×C4C2×#62611yescharacteristic
400C3×C4D3×#6711yescharacteristic
401Dic3—121no—
402Dic3—121no—
403Dic3—241no—
404Dic3#371411yescharacteristic
405C32—21no—
406C32—21no—
407C32—21no—
408C32—41no—
409C32#490811yescharacteristic, complemented
410C32#491111yescharacteristic, complemented
411C32#519311yescharacteristic, complemented
412C23—1081no—
413Q8—181no—
414D4—61no—
415D4—181no—
416D4—361no—
417D4—541no—
418D4—1081no—
419C2×C4—31no—
420C2×C4—361no—
421C8—31no—
422C8D3311yescharacteristic
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—62no—
428C2×C3—61no—
429C2×C3—61no—
430C2×C3—241no—
431C2×C3—241no—
432C2×C3—481no—
433C2×C3—721no—
434C2×C3D3×D2411yescharacteristic
435C2×C3C2×#371411yescharacteristic
436C2×C3D3×#21311yescharacteristic
437D3—21no—
438D3—81no—
439D3—242no—
440D3—241no—
441D3—241no—
442D3—722no—
443D3—721no—
444D3—721no—
445D3—721no—
446D3—1441no—
447D3—1441no—
448D3—1442no—
449C22—31no—
450C22—121no—
451C22—361no—
452C22—1081no—
453C22—2162no—
454C4—31no—
455C4—361no—
456C4D3×#62611yescharacteristic
457C3—21no—
458C3—21no—
459C3—21no—
460C3—41no—
461C3#1059023211yescharacteristic, complemented
462C3#1059063411yescharacteristic, complemented
463C3#1059102411yescharacteristic, complemented
464C2—61no—
465C2—241no—
466C2—721no—
467C2—2161no—
468C2D3×#371411yescharacteristic
subgroupG/H
centerC2D3×#3714
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. D3×#3745C2semidirect
2. C3×#63807C2semidirect
3. #11722068C2semidirect
4. #11722012C2semidirect
5. #11722005C2semidirect
6. C3×#63546C2semidirect
7. C3×#63543C2semidirect
14. C3×#3746C22semidirect
16. C32×D24C22semidirect
20. C3×C3⋊(C3⋊C16)C22semidirect
21. C32×C3⋊C16C22semidirect
25. #63807D3semidirect
44. #63546D3semidirect
45. #63543D3semidirect
69. #3746C2×D3semidirect
87. C3×D24C2×D3semidirect
88. C3×D24C2×D3semidirect
98. C3⋊(C3⋊C16)C2×D3semidirect
103. C3×C3⋊C16C2×D3semidirect
104. C3×C3⋊C16C2×D3semidirect
225. D24D32semidirect
235. C3⋊C16D32semidirect
281. C33#509semidirect
409. C32#4908semidirect
410. C32#4911semidirect
411. C32#5193semidirect
461. C3#10590232semidirect
462. C3#10590634semidirect
463. C3#10591024semidirect

Maximal subgroups

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

subgroupindex
D3×#37452
C3×#638072
#117220682
#117220122
#117220052
C3×#635462
C3×#635432
#105910243
#105906343
#105902323